The yoga of schemic Grothendieck rings, a topos-theoretical approach

Kontsevich [9] has formulated a general integration technique on (smooth) schemes over an algebraically closed field κ, modeled on p-adic integration and called motivic integration. This was then extended by Denef and Loeser [1,2,3] to achieve motivic rationality, by which they mean the fact that the rationality of a certain generating series from geometry or number-theory, like the Igusa-zeta series, is "motivated" by the rationality of its motivic counterpart which specializes to the given classical series via some multiplicative function (counting function, Euler characteristic, . . . ). The two main ingredients of this construction are the Grothendieck ring of varieties over κ (see below), in which the integration takes its values, and the arc space LÔXÕ of a variety X, that is to say, the reduced Hilbert scheme classifying all arcs Spec κÖÖξ×× X. Our aim is to extend this by replacing varieties by schemes, in such a way that killing the nilpotent structure reverts to the old theory. The classical Grothendieck ring GrÔÎ Ö κ Õ of an algebraically closed field κ is designed to encode both combinatorial and geometric properties of varieties. It is defined as the quotient of the free Abelian group on varieties over κ, modulo the relations ÖX× ¡ ÖX ½ ×, if X X ½ , and (1) if Y is a closed subvariety, for Y, X, X ½ varieties (=reduced, separated schemes of finite type over κ). We will refer to the former relations as isomorphism relations and to the latter as scissor relations, in the sense that we "cut out Y from X." Multiplication on GrÔÎ Ö κ Õ is then induced by the fiber product. In sum, the three main ingredients for building the Grothendieck ring are: scissor relations, isomorphism relations , and products. Only the former causes problems if one wants to extend the construction of the Grothendieck ring from varieties too arbitrary finitely generated schemes. Put bluntly, we cannot cut a scheme in two, as there is no notion of a scheme-theoretic complement, and so we ask what new objects we should add to make this work. Let us call these new objects tentatively motives, in the sense that their existence is motivated by a combinatorial necessity. There are now two approaches to construct these motives. The first one, discussed at length in [14], is based on definability, and was the original approach. The point of departure is the representation of a scheme by an equational (first-order) formula modulo the theory of local Artin algebras. The logical operations of conjunction, disjunction, and negation are then used to form the required new objects: motives arise as Boolean combinations of schemes. Scissor relations are now easily expressed in this formalism, whereas products are given by conjunction with respect to distinct variables, and isomorphism relations are phrased in terms of definable isomorphisms, cumulating in the construction of the schemic Grothendieck ring over κ. To obtain geometrically more significant motives, one is forced not only to introduce quantifiers, but also to resort to some infinitary logic via formularies, leading to larger Grothendieck rings, all of which still admit a natural homomorphism onto the classical Grothendieck ring. To define the analogue of arc spaces, one obtains arc formulae by interpreting the theory of a local Artin algebra in that of its residue field. The resulting arc operator is compatible with Boolean combinations and hence induces an endomorphism on the Grothendieck rings. This was the first striking success of the new theory: no such operation holds in the classical Grothendieck ring. Other main advantages of this approach are (i) the presence of negation, allowing one to "cut up" a scheme into motives, and (ii) the uniformity inherent in model-theory, allowing one to use parameters, and hence to work over an arbitrary base ring rather than just an algebraically closed field. The main disadvantage stems from non-functoriality, in particularly when dealing with morphisms. Nonetheless, the theory has been applied in [14] with success to establish motivic rationality in certain cases, even the, thus far illusive, positive characteristic case. However, soon after writing down this version, I came to realize that it might beneficial to sacrifice definability for functoriality. In this approach, which forms the content of this paper and is essentially topos-theoretical, schemes are viewed as (contravariant) functors. Traditionally, one views them as functors, called representable functors, from the category of κ-algebras to the category of sets, but the power of the present approach comes from narrowing down the former category to that of fat points, consisting only of one-point schemes over κ. Thus, given a scheme X of finite type over κ and a fat point z, we let XÔzÕ be the set of all z-rational points, that is to say, morphisms z X. The functor z XÔzÕ now determines the scheme X uniquely. Motives are then certain subfunctors of these representable functors, with morphisms between them given as natural transformations. Since these functors take values in the category of sets, all set-theoretic operations are available to us, such as union, intersection, and complement. However, complementation does not behave functorially, and so motives now only form a lattice, leading to the notion of a motivic site: apart from a Grothendieck topology inherited from the Zariski topology on the schemes, we also require a categorical lattice structure in order to formulate scissor relations. Defining multiplication by means of fiber products, we thus get the Grothendieck ring of a motivic site. Among the many tools from category theory and topos theory we can now resort to, adjunction takes a primary place: it allows us, for instance, to define, without much effort, arc schemes, which act again nicely on the corresponding Grothendieck rings. Let me now briefly discuss in more detail the content of the present paper. In §2, we discuss the functors that will play the role of motives. Borrowing terminology from topos theory, on the category of fat points, a subfunctor X of a representable functor given by a scheme X is called a sieve on X, and X is called an ambient space of X. We may do this over an arbitrary Noetherian base scheme V , provided it is also Jacobson. For the sake of this introduction, I will only treat the case of greatest interest to us, namely, when V is the spectrum of an algebraically closed field κ. A morphism of sieves is in principle any natural transformation, but often such a morphism extends to a morphism of the ambient spaces, in which case we call it algebraic. We turn this into a true topos in §3, by defining an admissible open of a sieve X to be its restriction to an open in its ambient space. We define a global section on a sieve X to be any morphism into the affine line. We establish an acyclicity result for global sections, allowing us to define the structure sheaf O X of X. In the next four sections, § §4-7, we introduce the Grothendieck ring of a motivic site, and discuss the three main cases. As already mentioned, a motivic site is for each scheme, a choice of lattice of sieves on that scheme, called the motives of the site. The associated Grothendieck ring is then defined as the free Abelian group on motives in the site modulo the isomorphism relations and the scissor relations, where the latter take the lattice form (2) ÖX× ÖY× ÖX Y× ÖX Y×, for any two motives X and Y on the same ambient space. The first motivic site of interest consists of the schemic motives, given on each scheme as the lattice of its closed subschemes (viewed as representable subfunctors). The resulting Grothendieck ring is too coarse, as it is freely generated as an additive group by the classes of irreducible schemic motives (Theorem 5.7). A larger, more interesting site is given by the sub-schemic motives, where we call a sieve X on X sub-schemic, if there is a morphism ϕ : Y X such that at each fat point z, the set XÔzÕ consists of all z-rational points z X that factor through ϕ, that is to say, XÔzÕ is the image of the induced map ϕÔzÕ : Y ÔzÕ XÔzÕ. Any locally closed subscheme is sub-schemic, so that in the corresponding Grothendieck ring, we may express the class of any separated scheme in terms of classes of affine schemes. Moreover, any morphism of sieves with domain a sub-schemic motif is algebraic (Theorem 3.7), from which it follows that the sub-schemic Grothendieck ring admits a natural homomorphism into the classical Grothendieck ring. Whereas in general the complement of a sieve is no longer a sieve (as functoriality fails), this does hold for any open subscheme. However, such a complement is in general no longer sub-schemic, but only what we will call a formal motif, that is to say, a sieve X that can be approximated by sub-schemic submotives in the sense that for each fat point z, one of its sub-schemic approximations has the same z-rational points as X. In case of an open U X ¡ Y , the complement is represented by the formal completion Ô X Y , whose approximations are the jet spaces J n Y X : SpecÔO X ßI n Y Õ. A morphism in the site of formal motives ÓÖÑ κ is approximated by algebraic morphisms, and therefore, the ensuing Grothendieck ring GrÔ ÓÖÑ κ Õ still admits a canonical homomorphism onto the classical Grothendieck ring GrÔÎ Ö κ Õ. In §8, we discuss adjunction of motivic sites over different base schemes. Formally, this consists of a pair of functors η : XÔηÔzÕÕ Ô∇ η XÕÔzÕ for any V -sieve X and any W -fat point z. We call such an adjunction (sub-)schemic or formal, if ∇ η X is respectively (sub-)schemic or formal, for any scheme X. Under this assumption, the adjunction induces a homomorphism from the corresponding Grothendieck ring over V to the Grothendieck ring over W . Whenever we have a morphism f : V W of finite type, we obtain an adjunction given by the pair Ôf ¦ , f ¦ ∇ f¦ Õ, called augmentation, where f ¦ means restriction of scalars and f ¦ means extension of scalars via f . If f is moreover finite and flat, then we can go the other way, called diminution, via the pair Ôf ¦ , ∇ f ¦Õ. Both adjunctions are schemic, and they are related by the projection formula where h : Ṽ V is of finite type, and f and h are the corresponding base changes of f and h with respect to the other (Theorem 8.11). Applying diminution to a rational point a È XÔκÕ, we may define for each morphism Y X and each motif Y on Y , the specialization Y a . In this way, Y becomes a family of motives. Another interesting application of adjunction is via the action of Frobenius F in characteristic p 0. If we let FÔzÕ be the fat point with coordinate ring the p-th powers of elements in the coordinate ring of z, then this yields an adjunction ÔF, ∇ F Õ which is only sub-schemic: ∇ F Y is given as the image of the relative Frobenius on Y (locally via some embedding in affine space; see Theorem 8.14 for a precise formulation). In §9, we apply the adjunction theory to the special case of the structure morphism j : z Spec κ of a fat point, and we define the arc functor ∇ z as the composition of augmentation and diminution along j, that is to say, This corresponds in the special case that z l n : Spec κÖξ×ßÔξ n Õ to the classical truncated arc space L n through the formula L n ÔXÕ Ô∇ ln XÕ red for any variety X. However, it should be noted that ∇ z does not commute with taking reductions, so that even if X and X ½ have the same underlying variety, they will in general have different arc schemes ∇ z X and ∇ z X ½ , even possibly of different dimension (see §10). In fact, the dimension of the arc scheme of a fat point over itself is an intriguing invariant. By the general theory of adjunction, ∇ z is an endomorphism on each Grothendieck ring. Arcs behave well over smooth varieties, as in the classical case (Theorem 9.10): the canonical morphism ∇ z X X is a locally trivial fibration over the non-singular locus of X, with general fiber some affine space. Hence, in the smooth case, Ö∇ z X× ÖX×L lÔd¡1Õ , where d and l are respectively the dimension of X and the length of (the coordinate ring of) z, and where L : ÖA 1 κ × is the Lefschetz class. Using the formalism of adjunction, we discuss some variants of arcs: deformed arcs in §11, and extendable arcs in §13. For the definition of the latter, we discuss in §12 a compactification of the category of fat points, the category of limit points, given as direct limits of fat points (e.g., the formal completion Ô Y P ). Although we can extend the notion of arcs to any limit point, the corresponding arc scheme is no longer of finite type. In §14, we discuss some of the motivic series that can now be defined using this formalism. As already mentioned, since they or their classical variants specialize to generating series that are known to be rational, we expect them to be already rational over the formal Grothendieck ring, or rather, over its localization GrÔ ÓÖÑ κ Õ L ; this is called motivic rationality. Let us briefly describe each of these series here. Igusa: with j n : J n P Y the n-th jet of a closed point P on a scheme Y , and X any scheme of dimension d, we set Igu ÔY,PÕ X mot ÔtÕ : L ¡dℓÔjnÕ Ö∇ jn X× t n Hilbert: with ÔY, P Õ and j n as above, we set Hilb mot ÔY,PÕ : L ¡dℓÔynÕ Ö∇ j ¦ yn X yn × t n ; Poincare: with ∇ Ô YP ßjn X the submotif of ∇ jn X consisting of all arcs that factor through the formal completion Ô Y P , we set X mot ÔtÕ : Hasse-Weil: with X ÔnÕ the n-fold symmetric product of X, that is to say, the Hilbert scheme of effective zero cycles of degree n on X, we set HW mot X : Note that the specializations of the motivic Hilbert and Hilbert-Kunz series to the classical Grothendieck ring are trivial since the underlying varieties are just points. Put differently, this kind of motivic rationality cannot even be phrased in the classical setup. However, specializing to the length of the motives, which is well-defined by Proposition 14.4, yields their classical counterparts. The final section, §15 is devoted to motivic integration. We only develop the finitistic theory, that is to say, over a fixed fat point, leaving the case of a limit point to a future paper. One of the great disadvantages of the categorical approach is that fibers are in general not functorial (after all, a fiber is the complement of the remaining fibers). We can overcome this by restricting to the category of split fat points, in which the morphisms are now assumed to be split. Our motivic integration will take values in the localization GrÔ ÓÖÑ κ Õ L . A functor s, viewed on the category of split fat points, from a formal motif X on X to the constant sheaf with values in this localization GrÔ ÓÖÑ κ Õ L is called a formal invariant if all its fibers are formal motives, with only finitely many non-empty. We then define where d is the dimension of X and l the length of z. This motivic integral can be calculated locally (Theorem 15.8). Varieties are assumed to be reduced, but not necessarily irreducible. Given a scheme X, we let X red denote its underlying variety or reduction. We often denote a morphism of affine schemes Spec B Spec A by the same letter as the corresponding ring homomorphism A B, whenever this causes no confusion. By a germ ÔX, Y Õ we mean a scheme X together with a closed subscheme Y ⊆ X. Most of the time Y is an irreducible subvariety, that is to say, the closure of a point y È X, and we simply write ÔX, yÕ for this germ. If Y is a closed point, we call the germ closed. The n-th , where I Y is the ideal of definition of Y . 1 The formal completion Ô X Y of the germ ÔX, Y Õ is the locally ringed space obtained as the direct limit of the J n Y X (see [7, II. §9]). For instance, if Y P is a closed point with maximal ideal m P , then the ring of global sections of Ô X P is the m P -adic completion Ô O X,P of O X,P . We often denote the affine line A L and L ¦ . Whereas in the classical Gro- thendieck ring, L L ¦ 1 (and Ô L is undefined), in the formal Grothendieck ring, we have L L ¦ Ô L (see Proposition 7.1), which, after taking global sections, takes the suggestive form "κÖx×" "κÖx, 1ßx×" "κÖÖx××". The n-th jet of ÔL, OÕ will be denoted l n : SpecÔκÖx×ßÔx n ÕÕ. Fix a Noetherian scheme V , to be used as our base space, and which, for reasons that will become apparent soon, will also be assumed to be Jacobson. Most often, V is just the spectrum of an algebraically closed field κ. By a V -scheme X, we mean a separated scheme X together with a morphism of finite type X V . We call X a fat V -point, if X V is finite and X has a unique point. In other words, X Spec R, for some Artinian local ring R which is finite as a O V -module. We call the length of R the length of the fat point and denote it ℓÔzÕ. We denote the subcategory of fat V -points by Ø V , and we will use letters z, v, . . . to denote fat points. An important example of fat points are the jets of a closed germ ÔX, P Õ, that is to say, given a V -scheme X and a closed point P on X with corresponding maximal ideal m P , we let J n P X, called the n-th jet of X along P , be the fat point with coordinate ring O X,P ßm n P . Let X be a V -scheme and z a fat point. A morphism of V -schemes a : z X will be called a z-rational point on X (over V ). The set of all z-rational points on X will be denoted by XÔzÕ. The image of the unique point of z under a is a closed point on X, called the center (or origin) of a. Indeed, since the composition z X V is the structure map whence finite, so is its first component a : z X. As finite morphisms are proper, closed points are mapped to closed points. Let x be the center of a z-rational point a on a V -scheme X. We will denote the residue field of x and z respectively by κÔxÕ and κÔzÕ. The z-rational point a induces a homomorphism of residue fields κÔxÕ κÔzÕ. By the Nullstellensatz this a is a finite extension, and its degree will be called the degree of a. In particular, if V is the spectrum of an algebraically closed field κ, then κ κÔxÕ κÔzÕ, so that any z-rational point has degree zero. The category of contravariant functors (with natural transformations as morphisms) from Ø V to the category of sets will be called the category of presheaves over V , fol- lowing standard practice in topos theory, notwithstanding the confusion this causes with the usual notion from algebraic geometry. The product X ¢ Y (respectively, the disjoint union X Y) of two presheaves X and Y is defined point-wise by the rule that a fat point z is mapped to the Cartesian product XÔzÕ ¢ YÔzÕ (respectively, to the disjoint union XÔzÕ YÔzÕ). Similarly, we say that X is a sub-presheaf of Y, symbolically X ⊆ Y, if for every fat point z, we have an inclusion XÔzÕ ⊆ YÔzÕ, and this inclusion is a natural transformation, meaning that for any morphism j : z z, we have a commutative diagram where the downward arrows are the maps induced functorially by j. We call a presheaf X on Ø V representable (respectively, pro-representable), if there exists a V -scheme X (respectively, a scheme X which is not necessarily of finite type over V ) such that XÔzÕ XÔzÕ, for all fat points z. To emphasize that we view the (V -)scheme X as a contravariant functor on Ø V , we will denote it by X ¥ : Mor V Ô¤, XÕ. Any morphism ϕ : Y X of (V -)schemes induces, by composition, a natural transformation ϕ ¥ : Y ¥ X ¥ , that is to say, a morphism in the category of presheaves on Ø V . More precisely, given a fat point Instead of ϕ ¥ ÔzÕ, we will simply write ϕÔzÕ for the induced map Y ÔzÕ XÔzÕ if there is no danger for confusion. 2.1. Lemma. Two closed closed subschemes X and Y of a V -scheme Z are distinct if and only if there is a fat point z such that XÔzÕ and Y ÔzÕ are distinct subsets of ZÔzÕ. Proof. One direction is immediate, so assume X and Y are distinct. Then their restriction to some affine open of Z remains distinct, and hence we may assume Z Spec A is affine. Let I and J be the ideals in A defining X and Y respectively. Since I J, there is a maximal ideal m ⊆ A such that IA m JA m . Hence, by Krull's Intersection Theorem in the Noetherian local ring A m , there is some n such that I and J remain distinct ideals in Aßm n . In particular, upon replacing I by J if necessary, z : SpecÔAßÔI m n ÕÕ is a closed subscheme of X, but not of Y , showing that the closed immersion z ⊆ Z lies in XÔzÕ ¡ Y ÔzÕ. The resulting map from the category of V -schemes to the category of presheaves on Ø V is a full embedding by Yoneda's Lemma, Lemma 2.1, and Proposition 2.7 below. Note that this is no longer true for schemes not of finite type, an obvious reason for this failure being that there might be no rational points at all: for instance, SpecÔCÕ has no rational points over any fat point defined over the algebraic closure Q. The product of two (pro-)representable presheaves is again (pro-)representable. More explicitly, the product is a morphism of presheaves, then we define its image ImÔsÕ and its graph ΓÔsÕ as the sub-presheaf of respectively X and X ¢Y, given at each fat point z as respectively the image and the graph of the map sÔzÕ : XÔzÕ YÔzÕ. Sieves. By a sieve, we mean a sub-presheaf X of some representable X ¥ . If we want to emphasize the underlying V -scheme X, we say that X is a sieve on X, or that X is an ambient space of X. Some examples of sieves: if ϕ : Y X is a morphism of V -schemes, then we let ImÔϕÕ be the image pre-sieve of the corresponding natural transformation ϕ ¥ : Y ¥ X ¥ , that is to say, ImÔϕÕÔzÕ, for a fat point z, consists of all z-rational points on X that lift to a z-rational point on Y , meaning that z X factors through Y . Any sieve of the form ImÔϕÕ for some morphism ϕ : Y X of V -schemes is called sub-schemic. If ϕ is a (locally) closed or open immersion, then ImÔϕÕ is equal to Y ¥ , whence is itself representable, and we call ImÔϕÕ Y ¥ respectively a (locally) closed or open subsieve on X. Let X be a sieve on X and z Spec R a fat point. A z-rational point a : z X belongs to XÔzÕ if and only if ImÔaÕ ⊆ X. If X Y ¥ is a closed subsieve, given by a closed subscheme Y ⊆ X, then this is also equivalent with z z ¢ X Y and also with a ¦ I Y 0, where I Y is the ideal sheaf of Y and a ¦ I Y its image in R. Proof. Suppose a È XÔzÕ. Let w be a fat point and b È ImÔaÕÔwÕ. Hence b : w X factors through a, that is to say, we can find a morphism i : w z such that the diagram (4) By functoriality, i induces a map XÔzÕ XÔwÕ, sending a to b, proving that b È XÔwÕ. Since this holds for all w and b, we showed ImÔaÕ ⊆ X. Conversely, assume the latter inclusion of sieves holds. In particular, the identity 1 z is a z-rational point whose image under aÔzÕ is just a, proving that a È ImÔaÕÔzÕ ⊆ XÔzÕ. To see the equivalence with the last two conditions if X Y ¥ , we may work locally and assume X Spec A. Let I be the ideal defining Y , and let A R be the homomorphism corresponding to a. Then a È Y ÔzÕ if and only if IR 0 if and only if AßI A R R, proving the desired equivalences. We may generalize the notion of an image sieve as follows. Given a sieve Y on an V -scheme Y and a morphism ϕ : Y X of V -schemes, we define the push-forward ϕ ¦ Y as the sieve on X given at a fat point z as the image of YÔzÕ ⊆ Y ÔzÕ under the map ϕÔzÕ : Y ÔzÕ XÔzÕ. In particular, ϕ ¦ Y ¥ ImÔϕÕ. Similarly, given a sieve X on X, we define its pull-back ϕ ¦ X as the sieve on Y given at a fat point z as the pre-image of XÔzÕ under the map ϕÔzÕ : Y ÔzÕ XÔzÕ. In other words, ϕ ¦ XÔzÕ consists of those rational points z Y such that the composition z Y X lies in XÔzÕ. The pull-back of a closed subsieve is again a closed subsieve: if X ⊆ X is a closed immersion and ϕ : Y X a morphism, then ϕ ¦ X¥ is the closed subsieve given by ϕ ¡1 Ô XÕ Y ¢ X X. X is an arbitrary morphism, then the pull-back of the sub-schemic sieve ImÔ X XÕ is the sub-schemic sieve ImÔY ¢ X X Y Õ of the base change. If X and Y are sieves on an V -scheme X, then we define their intersection X Y and union X Y as the sieves given respectively by point-wise intersection and union, that is to say, ÔX YÕÔzÕ XÔzÕ YÔzÕ ⊆ XÔzÕ and ÔX YÕÔzÕ XÔzÕ YÔzÕ ⊆ XÔzÕ. One easily checks that they are again sieves, that is to say, (contravariant) functors. We express this by saying that the sieves on X form a lattice. Clearly, the intersection of two closed subsieves is again a closed subsieve: subschemes, but this is no longer true for their union. The disjoint union X Y is equal to the union of the push-forwards of X and Y under the immersions X X Y and Y X Y respectively. By the Zariski closure of X in X, denoted X, we mean the intersection of all closed subschemes Y ⊆ X such that X ⊆ Y ¥ . By Noetherianity, X is a sieve on its Zariski closure X, and the latter is the smallest closed subscheme on which X is a sieve. We say that X is Zariski dense in X if X X. For instance, given a morphism ϕ : Y X of V -schemes, the Zariski closure of ImÔϕÕ is the so-called scheme-theoretic image of ϕ, that is to say, the closed subscheme of X given by the kernel of the induced morphism O X ϕ ¦ O Y . When ImÔϕÕ is Zariski dense, one says that ϕ is dominant. XÔwÕ is injective, for any sieve X. Proof. This is a form of duality: dominant morphisms are epimorphisms and so under a contravariant functor they become monomorphisms. More explicitly, since X ⊆ X ¥ , for some V -scheme X, it suffices to show injectivity for the latter, that is to say, we may assume X is representable, and then since the problem is local, we may assume X is affine with coordinate ring A. Let a, a ½ È XÔvÕ have the same image in XÔwÕ. If R S is the homomorphism corresponding to ϕ, then dominance means that this homomorphism is injective. Since, by assumption, the two homomorphisms A R induced respectively by a and a ½ give rise to the same homomorphism A S when composed with the injection R ⊆ S, they must already be equal, as we needed to show. Example. Suppose V is the spectrum of an algebraically closed field κ. Zariski closure does not commute with taking κ-rational points, that is to say, the κ-rational points of the Zariski closure of a sieve X in X may be bigger than the Zariski closure of XÔκÕ (in the usual Zariski topology) in XÔκÕ. For instance, as we will see shortly, the cone CÔOÕ of the origin O on the affine line A 1 κ has Zariski closure equal to A 1 κ , whereas its κ-rational points consist just of the origin. Strictly speaking, a sieve is a pair ÔX, XÕ consisting of a sub-presheaf X of an ambient space X, but we will often treat sieves as abstract objects, that is to say, disregarding their ambient space. This allows us, for instance, to view the same sieve as already defined on a smaller subscheme. To give a more formal treatment, let us call a germ of a sieve any equivalence class of pairs ÔY, Y Õ, where we call two such pairs ÔY, Y Õ and ÔY ½ , Y ½ Õ equivalent if there exists a pair ÔX, XÕ and locally closed immersions ϕ : Y X and ϕ : Y ½ X ½ such that ϕ ¦ Y X ϕ ½ ¦ Y ½ . In particular, Y is then also a sieve on the intersection Y Y ½ , viewed as a locally closed subscheme of X. Therefore, we may always assume, if necessary, that Y is Zariski dense in Y , and then it is also a Zariski dense sieve on any open subsieve of Y containing it. Henceforth, we will often confuse a sieve with the germ it determines. Complete sieves. The complement of a sieve X ⊆ X ¥ is in general not a sieve, as wit- nessed by any closed subsieve. Let us call a sieve X on X complete if XÔzÕ XÔjÕ ¡1 ÔXÔzÕÕ, for every morphism j : z z of fat points, where XÔjÕ : XÔzÕ XÔzÕ is the map induced functorially by j. More generally, if Y ⊆ X is an inclusion of sieves on X, then we say that Y is relatively complete in X, if YÔzÕ XÔjÕ ¡1 ÔYÔzÕÕ XÔzÕ, for all morphisms j. If V is the spectrum of an algebraically closed field κ, then V itself is a fat point, and any fat point z admits a unique morphism V z (given by the residue field of z). We let ρ z : XÔzÕ XÔκÕ be the induced map. One easily verifies that Y is relatively complete in z ÔYÔκÕÕ, for every fat point z. In fact, for V arbitrary, we have a similar criterion in that we only have to check the condition for every morphisms of fat points j : z z in which z has length one (necessarily therefore the spectrum of a field extension of the residue field of z). As we shall see in Proposition 7.1 below, open subsieves over an algebraically closed field are complete, and their complements are again sieves. The second property in fact follows from the first, as one easily verifies: Assume V is the spectrum of an algebraically closed field κ. Given a κ-scheme X and a subset V ⊆ XÔκÕ, we define the cone CÔV Õ over V to be the sieve given by CÔV ÕÔzÕ : ρ ¡1 z ÔV Õ for every fat point κ. By our previous discussion on complete sieves over an algebraically closed field, it follows that CÔV Õ is complete, and its complement is the cone CÔ¡V Õ. More generally, for any sieve X, the intersection X CÔV Õ is relatively complete in X, and its complement in X is equal to X CÔ¡V Õ. We have the following converse: 2.6. Lemma. Over an algebraically closed field κ, a subsieve Y ⊆ X is relatively complete if and only if it is the intersection of X and a cone. Proof. Let Y be relatively complete in X, and let V : YÔκÕ. Then Y CÔV Õ X, since both have the same z-rational points, for any fat point z. Given a sieve X, we define its completion as the cone Ô X : CÔXÔκÕÕ. By functoriality, X is contained in Ô X, and Ô X is the smallest complete sieve containing X. The category of sieves. By definition, a morphism between sieves is a natural transformation, giving rise to the category of V -sieves, denoted Ë Ú V . If V is Jacobson, 2 then the category of V -schemes fully embeds in the category Ë Ú V , by the following result: 2.7. Proposition. Assume V is Jacobson. Let X be a sieve on a V -scheme X, and let Y be a V -scheme. Any morphism s : Y ¥ X is induced by a morphism ϕ : Y X, that is to say, s ϕ ¥ and ImÔϕÕ ⊆ X. Proof. Assume first that Y z is a fat point, and X X ¥ is representable. Let q : sÔzÕÔ1 z Õ, where 1 z is the identity morphism on z viewed as a z-rational point on z. Hence q È XÔzÕ, that is to say, a V -morphism z X. We have to show that s q ¥ , so let w be any fat point, and a : w z a w-rational point on z. By definition of q ¥ , we have qÔwÕÔaÕ qa. On the other hand, functoriality yields a commutative diagram We trace the image of 1 z in XÔwÕ through this diagram. The top arrow sends it to a and hence its image in XÔwÕ is sÔwÕÔaÕ. On the other hand, sÔzÕÔ1 z Õ q and its image under X ¥ ÔaÕ is qa, showing that sÔwÕÔaÕ qa qÔwÕÔaÕ, whence s q ¥ . Assume next that Y is arbitrary. Let Z ⊆ Y be a closed subscheme of dimension zero. There exist finitely many fat points z 1 , . . . , z s ⊆ Y such that Z z 1 ¤ ¤ ¤ z s . By what we just proved, there exists a morphism q Z : Z X such that s Z ¥ q ¥ Z , for each zero-dimensional closed subscheme Z ⊆ Y . By Lemma 2.8 below, the direct limit of all such closed subschemes Z is Y , and hence, by the universal property of direct limits, we get a morphism ϕ : Y X, such that q Z ϕ Z . Checking at every fat point, we see that ϕ ¥ s. Finally, assume X is an arbitrary sieve. By what we just proved, the composition Y ¥ X ⊆ X ¥ is induced by a morphism ϕ : Y X. Since the image of s ϕ ¥ at any fat point z lies inside XÔzÕ, we must have ImÔϕÕ ⊆ X. 2.8. Lemma. If V is Jacobson, then any V -scheme is the direct limit of its zero-dimensional closed subschemes. Proof. I will only consider the case that V is the spectrum of a field κ. Reasoning on a finite open affine covering, we may further assume that X Spec A is an affine κ-scheme. Let à be the inverse limit of all residue rings Aßn of finite length (we say that n has finite colength). We have to show that A Ã. The inclusion A ⊆ à follows from the fact that the intersection of all ideals of finite colength is zero. Before we give the proof, we establish a preservation result under finite maps: if A à and A B is a finite homomorphism, then also B B. Indeed, if β 1 , . . . , β s generate B as an A-module, then they also generate every BßnB as a Aßn-module, where Aßn has finite length. Since the nB are cofinal in the set of all ideals of B of finite colength, B is generated by the β i as an Ã-module, and hence B B. This argument also shows that 2 A scheme is called Jacobson if it admits an open covering by affine Jacobson schemes; an affine scheme Spec A is Jacobson, if A is, meaning that any radical ideal is the intersection of the maximal ideals containing it. ÔAßIÕ is equal to ÃßI Ã, for any ideal I ⊆ A. By Noether Normalization, A is a finite extension of a polynomial ring over κ in the same number of variables as the dimension d of A. Hence, it will suffice to show the the identity A à for polynomial rings. We induct on d, where the case d 0 is trivial. Let A κÖx× with x a d-tuple of variables, and let o be a complete discrete valuation ring containing A and having nonzero center p ⊆ A (recall that p is the prime ideal of elements in A of positive value). Assume first that d 1, so that p is a maximal ideal. Since à is then contained in the completion Ô A p (as the latter is the inverse limit of all Aßp n ), and since by the universal property of completion Ô A p ⊆ o, we showed à ⊆ o. Since the intersection of all these complete discrete valuation rings is equal to the normalization of A, whence to A, we showed A à in this case. As mentioned above, Noether Normalization then proves the result for all one-dimensional algebras, and so we may assume d 1. If p is a maximal ideal, the previous argument yields again à ⊆ o. So assume p is a non-maximal, nonzero prime ideal. Since Aßp n has dimension less than d, induction plus preservation under homomorphic images then yields Aßp n Ãßp n Ã. In particular, the p-adic completion of à is equal to Ô A p , and hence is contained in o. Hence we showed that à ⊆ o, for any such discrete valuation ring o, so that the same argument as above yields A Ã. 2.9. Corollary. Suppose V is Jacobson. For any sieve X, we have an isomorphism of sieves Proof. Let z be a fat point and s : z ¥ X a morphism of sieves. By Proposition 2.7, this morphism is induced by a morphism a : z X of schemes, where X is some ambient space of X. Since s a ¥ , we have an inclusion ImÔaÕ ⊆ X, and hence a È XÔzÕ by Lemma 2.2. The converse follows along the same lines. Inspired by the result in Proposition 2.7, we call a natural transformation s : Y X between sieves algebraic, if there exists a morphism of V -schemes ϕ : Y X such that X and Y are sieves on respectively X and Y , and such that s is the restriction of ϕ ¥ : Y ¥ X ¥ ; we might also express this by saying that s extends to a morphism of schemes. Since the definition allows the ambient spaces to be dependent on s, we have actually defined a morphism between germs of sieves. It follows that if Y is Zariski dense in Y and X is a sieve on X (without any further restriction), then s extends to a morphism Ỹ X for some open Ỹ ⊆ Y on which Y is also a sieve, that is to say, such that Y ⊆ Ỹ ¥ . The composition of two algebraic natural transformations is again algebraic. Indeed, let s : Z Y and t : Y X be algebraic, extending respectively to morphisms ϕ : Z Y and ψ : Y ½ X. Since Y and Y ½ are both ambient spaces for Y, so is their intersection Y ¾ : Y Y ½ , which is therefore locally closed in either. Hence the restriction of ϕ to ϕ ¡1 ÔY ¾ Õ (respectively, the restriction of ψ to Y ¾ ) is a morphism extending s (respectively t), and therefore, the composition ϕ ¡1 ÔY ¾ Õ X extends t ¥ s. We can therefore define the explicit category of sieves, denoted Ë Ú V , as the subcategory of all sieves in which the morphisms are only the algebraic ones. A note of caution: not every morphism of sieves is algebraic, and we will discuss some examples later. Moreover, even if it is, one cannot always extend it to any ambient space of the source sieve. An example is in order: 2.10. Example. Let H ⊆ A 2 κ be the hyperbola with equation xy 1 over a field κ, and let L ¦ be the punctured line, that is to say, the affine line minus the origin. Note that this is an affine scheme with coordinate ring κÖx, 1ßx×. The projection A 2 κ A 1 κ onto the first coordinate induces an isomorphism H L ¦ . Its inverse induces an isomorphism L ¥ ¦ H ¥ , which is trivially algebraic, as both sieves are representable. However, although L ¥ ¦ is an open subsieve on A 1 κ , the above isomorphism does not extend (since 1ßx is not a polynomial). We may generalize the definitions of pull-back and push-forward along a morphism of sieves as follows. Let s : Y X be a morphism of sieves. Given a subsieve Y ½ ⊆ Y, we define its push-forward s ¦ Y ½ as the presheaf defined at each fat point z as the image of Y ½ ÔzÕ under sÔzÕ. Similarly, given a subsieve X ½ ⊆ X, we define its pull-back s ¦ X ½ as the presheaf defined at each fat point z as the pre-image of X ½ ÔzÕ under sÔzÕ. In view of Proposition 2.7, we will henceforth assume that the base space V is Jacobson. Section rings. Given a sieve X on X, we define its global section ring as where by the latter, we actually mean the collection of morphisms X ÔA 1 V Õ ¥ , but for notational simplicity, we will identify a scheme with the functor it represents if there is no danger for confusion. For each fat point z Spec R, we have a natural bijection for some affine open Spec λ ⊆ V , and hence induces a homomorphism λÖy× R, which, again for notational simplicity, we denote again by a. Now, set Ψ z ÔaÕ : aÔyÕ È R. This identification endows A 1 V ÔzÕ with a ring structure, and by transfer, then makes H 0 ÔXÕ into a ring. Indeed, given sub-schemic morphisms s, t : X A 1 V , we define their sum s t (respectively, their product st) as the morphism which at a fat point z maps a È XÔzÕ to sÔzÕÔaÕ tÔzÕÔaÕ Ψ ¡1 z Ψ z ÔsÔzÕÔaÕÕ Ψ z ÔtÔzÕÔaÕÕ änd a similar formula for sÔzÕÔaÕ¤tÔzÕÔaÕ. The functoriality of s t and st is easy, showing that they are again global sections. As we shall see below, algebraic morphisms will play a key role, and so we define the ring of algebraic sections H geom 0 ÔXÕ as the subset of H 0 ÔXÕ consisting of all algebraic morphisms X A 1 V . To see that this is closed under sums and products, let s and s ½ be two algebraic sections. By definition, there exist morphisms X A 1 V and X ½ A 1 V inducing s and s ½ respectively, where X and X ½ are ambient spaces for X. In particular, the locally closed subscheme X ¾ : X X ½ is then also an ambient space for X. Hence s t and st extend to the sum and product of the restrictions to X ¾ of X A 1 V and X ½ A 1 V , proving that they are also algebraic. By Theorem 3.7, every global section on a schemic motif is algebraic. ÔXÕ is a contravariant functor on the explicit category Ë Ú V . In particular, if two sieves are isomorphic, then they have the same global section ring, and if they are algebraically isomorphic, then they have the same algebraic section ring. Proof. If s : X Y is a morphism of sieves, then pulling-back induces a O V -algebra homomorphism H 0 ÔYÕ H 0 ÔXÕ given by t t ¥ s, for t : Y A 1 V . One easily verifies that this constitutes a contravariant functor. Since the pull-back of a algebraic morphism under an algebraic morphism is easily seen to be algebraic again, we get an induced homomorphism H geom 0 ÔYÕ H geom 0 ÔXÕ. 3.2. Proposition. The global section ring of a representable functor X ¥ is equal to the ring of global sections H 0 ÔXÕ : ΓÔO X , XÕ of the corresponding scheme, and this is also its algebraic section ring. V corresponds by Proposition 2.7 to a morphism of Vschemes X A 1 V , and it is well-known that the collection of all these is precisely the ring of global sections on X (see, for instance, [7, II. Exercise 2.4]). In particular, if X is a sieve on an affine scheme Spec A, then H 0 ÔXÕ and H geom 0 ÔXÕ are A-algebras. Corollary. The algebraic section ring H geom 0 ÔXÕ of a sieve X is the inverse limit of all H 0 ÔXÕ, where X runs over all ambient spaces of X. A 1 V is an algebraic section, then it extends to a morphism X A 1 V , where X is some ambient space of X, and hence by the argument in the above proof, it is the image of an element in H 0 ÔXÕ under the homomorphism H 0 ÔXÕ H geom 0 ÔXÕ induced by the inclusion X ⊆ X ¥ . If X ½ ⊆ X is a locally closed subscheme which is also an ambient space for X, then s extends to a morphism with domain X ½ , and this is therefore necessarily the restriction of the global section in H 0 ÔXÕ determined by s. This shows that the H 0 ÔXÕ form an inverse system as X varies over the germ of X, with limit equal to H geom 0 ÔXÕ. Example. If X is Zariski dense in X, then the only ambient spaces of X inside X are open, so that we may think of the ring of algebraic sections as a sort of stalk: where U runs over all open subschemes on which X is a sieve. In particular, if V is the spectrum of an algebraically closed field κ and XÔκÕ XÔκÕ, then there are no proper opens in X on which X is a sieve, and hence H geom 0 ÔXÕ H 0 ÔXÕ. This holds automatically if X is for instance a fat point. 3.5. Theorem. If X and Y are sieves on a common V -scheme, then the natural commutative diagram Similar properties hold for the algebraic section ring. Proof. The construction of the commutative square (6) and the verification that it is commutative, follows easily from Lemma 3. p X ÔsÕ p Y ÔtÕ t X Y . We then define u È H 0 ÔX YÕ as follows. Given a fat point z, let uÔzÕ be the map sending a È XÔzÕ YÔzÕ to sÔzÕÔaÕ if a È XÔzÕ, and to tÔzÕÔaÕ if a È YÔzÕ. This is well-defined, since sÔzÕ and tÔzÕ agree on XÔzÕ YÔzÕ by assumption. It is now easy to verify that this defines a morphism of sieves u : X Y A 1 V , that is to say, a global section of X Y, and that i X ÔuÕ s and i Y ÔuÕ t. Moreover, if s and t are algebraic, then so is u. Remark. To formulate a version for more than two sieves, we resort to the language of sheaf theory (and, shortly, we will put everything in this context anyway): given a union X X 1 ¤ ¤ ¤ X s , we have an exact sequence (7) 0 where δ ij : H 0 ÔX i Õ H 0 ÔX i X j Õ is the restriction homomorphism on the global sections induced by the inclusion X i X j ⊆ X i . The corresponding exact sequence for algebraic sections is Proof. Let us prove the second assertion first. Let ϕ : Y X be a morphism of Vschemes, so that X ImÔϕÕ. Replacing X by the Zariski closure of ImÔϕÕ, we may assume that ϕ is dominant. Our objective is to show that we have an equality (9) H geom 0 ÔImÔϕÕÕ H 0 ÔImÔϕÕÕ. Assume first that Y y is a fat point, and hence, since ϕ is dominant, so is then X x. Consider the induced homomorphism of Artinian local rings R ⊆ S, which is injective precisely because ϕ is dominant. Let s : ImÔϕÕ A 1 V be a global section. By Proposition 2.7, we can find q È S which, when viewed as a global section y A 1 V , extends the composition s ¥ ϕ ¥ . Let g 1 : S S R S and g 2 : S S R S be given by respectively a a 1 and a 1 a. Since g 1 and g 2 agree on R, the two corresponding rational points y ¢ x y y have the same image under sÔy ¢ x yÕ and hence, g 1 ÔqÕ g 2 ÔqÕ. By Lemma 3.10 below, we get q È R. By Corollary 3.3, this proves (9) whenever the domain of ϕ is a fat point. If the domain is a zero-dimensional scheme Z, then it is a finite disjoint union y 1 ¤ ¤ ¤ y s of fat points. By an induction argument, we may assume s 2, so that ImÔϕÕ ImÔ ϕ y1 Õ ImÔ ϕ y2 Õ. In particular, H 0 ÔImÔϕÕÕ and H geom 0 ÔImÔϕÕÕ satisfy each the same Cartesian square (6) by what we just proved for fat points (instead of induction, we may alternatively use (7) and ( 8)). By uniqueness, they must therefore be equal, showing that (9) holds whenever the domain is zero-dimensional. For Y arbitrary, we may write it as the direct limit of all its zero-dimensional closed subschemes by Lemma 2.8. Let ØZ : ImÔ ϕ Z ÕÙ be the collection of all sub-schemic motives, where Z ⊆ Y varies over all zero-dimensional closed subschemes of Y . Since the Z form a direct system, H 0 ÔImÔϕÕÕ is the inverse limit of all H 0 ÔZÕ H geom 0 ÔZÕ, by Lemma 3.9 below, and by the same argument, this direct limit is equal to H geom 0 ÔImÔϕÕÕ, proving (9). To prove the first assertion, we may assume, without loss of generality, that Z Z ¥ is representable, and then, since the problem is local, that X Spec A and Z are affine. Hence Z ⊆ A n λ is a closed subscheme for some n and some open Spec λ ⊆ V . Let s i be the composition of s with the morphism induced by the projection Replacing X by an open subscheme if necessary as per Corollary 3.3, we can find q i È A such that, viewed as a global section A 1 λ , it extends s i . Therefore, the morphism X A n λ given by Ôq 1 , . . . , q n Õ is an extension of s, as we wanted to show. 3.8. Remark. By the above proof, we actually showed that if the domain of ϕ is zerodimensional, then H 0 ÔImÔϕÕÕ is equal to the global section ring of the Zariski closure of ImÔϕÕ. However, this is no longer true in the general case, as can be seen from Corollary 3.3. 3.9. Lemma. Let ØZÙ be a direct system of sieves on some scheme X and let X be their direct limit. Then H 0 ÔXÕ is the inverse limit of all H 0 ÔZÕ. Proof. Contravariance turns a direct limit into an inverse limit, and the rest is now an easy consequence of the universal property of inverse limits: 3.10. Lemma. Let R ⊆ S be an injective homomorphism of rings. Then the tensor square is Cartesian, that is to say, if q 1 1 q in S R S for some q È S, then in fact q È R. Proof. Let for simplicity assume that R and S are algebras over some field κ (since we only need the result for R and S Artinian, this already covers any equicharacteristic situation). Let T : S κ S be the tensor product over κ, and let n be the ideal in T generated by all expressions of the form r 1 ¡ 1 r for r È R. Hence S R S T ßn. If q È S satisfies q 1 1 q in S R S, then viewed as an element in T , the tensor q 1 ¡ 1 q lies in n. The canonical surjection S SßR induces a homomorphism of tensor products T ÔSßRÕ κ ÔSßRÕ. Under this homomorphism, n is sent to the zero ideal, whence so is in particular q 1 ¡ 1 q. If the image of q in SßR were non-zero, then we can find a basis of SßR containing q. Hence q 1 and 1 q are two independent basis vectors of ÔSßRÕ κ ÔSßRÕ, contradicting that they are equal in the latter ring. Hence q È R, as we wanted to show. By Lemma 3.1 and properties of tensor products, we have for any two sieves X and Y, a canonical homomorphism (11) and a similar formula for algebraic sections. If X and Y are both representable, then this is an isomorphism, but not so in general. The topos of sieves. The Zariski topology on a V -scheme X induces a topos on each of its sieves X. More precisely, the admissible opens on X are the sieves of the form X U ¥ , where U ⊆ X runs over all opens of X; and the admissible coverings are all collections of admissible opens U i ⊆ X such that their union (as sieves) is equal to X, that is to say, such that the corresponding opens U i ⊆ X cover some ambient space of X. For simplicity, we will simply write X U for X U ¥ . In particular, since X is quasi-compact, any admissible covering contains a finite admissible subcovering. The collection of admissible opens does not depend on the ambient space X, for if X ½ ⊆ X is a locally closed subscheme on which X is also a sieve, then since its topology is induced by that of X, it induces the same admissible opens on X, and the same admissible coverings. Without going into details, we claim that the collection of admissible opens and admissible coverings yields a Grothendieck topology on X, turning Ë Ú V into a Grothendieck site. Nonetheless, since for each fat point z, this induces a topological space on XÔzÕ, we will just pretend that we are working in a genuine topological space, and borrow the usual topological jargon. Remark. Suppose V is the spectrum of a field κ. Unless z is the geometric point V itself, the topological space XÔzÕ is not separated: two z-rational points a, b È XÔzÕ are inseparable if and only if they have the same center, that is to say, if and only if ρÔκÕÔaÕ ρÔκÕÔbÕ, where ρÔκÕ : XÔzÕ XÔκÕ is the canonical map. Given an open U ⊆ X with corresponding open sieve U : X U , then, as we shall prove shortly in Theorem 9.3 below, we have UÔzÕ ρÔκÕ ¡1 ÔUÔκÕÕ. Therefore, if ρÔκÕÔaÕ ρÔκÕÔbÕ È U ÔκÕ, then a, b È UÔzÕ. Proof. By assumption, s extends to a morphism ϕ : Y X, where Y is a sieve on Y and To make Ë Ú V into a topos, we need to define structure sheafs for a given sieve X. We define presheaves O X and O geom X on X, by associating to an open U : Proof. This is essentially the content of Theorem 3.5 (see also Remark 3.6), applied to a So we are justified in calling O X the structure sheaf of the sieve X on a V -scheme X, and O geom X its algebraic structure sheaf. Stalks. Let X be a sieve with ambient space X. A closed point P È X is called a point on X, if the closed immersion i P : P ⊆ X, viewed as a P -rational point, belongs to XÔP Õ, or equivalently, if P ¥ ⊆ X. We define the stalk at a point P È X as usual as the respective direct limits where U runs over all admissible opens of X such that P È U. Clearly, if X X ¥ is representable, then O X ¥ ,P O geom X ¥ ,P is just the local ring O X,P at the closed point P È X by Proposition 3.2. In fact, we have: Proof. One inclusion is immediate, so let s È O geom X,P . Hence there exists an open U ⊆ X containing P such that s : X U A 1 V is an algebraic section. Since X U is then Zariski dense in U , there exists an open ambient space Ũ ⊆ U of X U and a morphism Ũ A 1 V extending s. This morphism corresponds to a global section of Ũ and hence is an element in O Ũ,P O X,P , since Ũ is open in X containing P . More generally, if X is a sieve on X and P a point on X, then O geom X,P OX ,P . A 1 V of a sieve X is a unit if and only if the image of sÔP Õ does not contain zero, for any point P È X. If V is the spectrum of an algebraically closed field κ, then this is equivalent with the image of sÔκÕ not containing zero. Proof. One direction is clear, so assume that the image of sÔP Õ does not contain zero, for any closed point P . Let z be a fat point, and let P be its center. It follows from the commutative diagram sÔzÕ where kÔP Õ is the residue field of P , that the image of sÔzÕ has empty intersection with the maximal ideal of the coordinate ring R of z, since π is just the residue map. Hence, for each a È XÔzÕ, its image sÔzÕÔaÕ is a unit in R, and hence we can define tÔzÕÔaÕ to be its inverse. So remains to check that t is a morphism of sieves X A 1 V , that is to say, a global section, and this is easy. 3.16. Proposition. For each point P on a sieve X, the stalk O X,P is a local ring. Proof. Let X be an ambient space of X. We have to show that given two non-units s, t È O X,P , their sum is a non-unit as well. Shrinking X if necessary, we may assume that s, t È H 0 ÔXÕ. I claim that sÔP ÕÔi P Õ and tÔi P ÕÔPÕ are both equal to zero, where i P : P ⊆ X is the closed immersion. Indeed, suppose not, say, sÔP ÕÔi P Õ 0, so that there exists an open U ⊆ X containing P such that sÔQÕ does not vanish on U ÔQÕ for any closed point Q È U . By Lemma 3.15, this implies that s is a unit in H 0 ÔX U Õ whence in O X,P , contradiction. Hence sÔP Õ tÔP Õ also vanishes at i P and hence cannot be a unit in O X,P . Note that we in fact proved that the unique maximal ideal consists of all sections s È O X,P such that sÔP ÕÔi P Õ 0. As before, V is a fixed Noetherian, Jacobson scheme. A motivic3 site Å over V is a subcategory of Ë Ú V which is closed under products, and such that for any V -scheme X, the restriction Å X (that is to say, the set of all Å-sieves on X) forms a lattice. In other words, if X, Y È Å are both sieves on a common scheme X, then X Y and X Y belong again to Å (including the minimum given by the empty set and the maximum given by X). We call Å an explicit motivic site, if all morphisms are algebraic. If Å is an arbitrary motivic site, then we let Å be the corresponding explicit motivic site, obtained by only taking algebraic morphisms. Given a motivic site Å, as the sieves form locally a lattice (on each V -scheme), we can define its associate Grothendieck ring GrÔÅÕ as the free Abelian group on symbols ÜXÝ, where X runs over all Å-sieves, modulo the scissor relations ÜXÝ ÜYÝ ¡ ÜX YÝ ¡ ÜX YÝ for any two Å-sieves X and Y on a common V -scheme, and the isomorphism relations for any two Å-sieves X and Y that are Å-isomorphic. We denote the image of an Åsieve X in GrÔÅÕ by ÖX×. In particular, since each representable functor is in Å, we may associate to any V -scheme X its class ÖX× : ÖX ¥ × in GrÔÅÕ. We define a multiplication on GrÔÅÕ by the fiber product (one easily checks that this is well-defined): ÖX× ¤ ÖY× : ÖX ¢ Y×. Since a motivic site has the same objects as its explicit counterpart, we get a canonical surjective homomorphism GrÔÅÕ GrÔÅÕ, which, however, need not be injective, since there are more isomorphism relations in the latter Grothendieck ring. On occasion, we will encounter variants which are supported only on a subcategory of the category of all V -schemes (that is to say, we only require the restriction of the site to one of the schemes in the subcategory to be a lattice), and we can still associate a Grothendieck ring to it. We will refer to this as a partial motivic site. Most motivic sites Å will also have additional properties, like for instance being stable under push-forwards along closed immersions, meaning that if i : Y ⊆ X is a closed subscheme and Y a motif in Å, then so is i ¦ Y. If this is the case, then Å is also closed under disjoint unions: given motives X and X ½ on X and X ½ respectively, then their disjoint union X X ½ is the union of the push-forwards i ¦ X and i ½ ¦ X ½ , where i : X X X ½ and i ½ : X ½ X X ½ are the canonical closed immersions. V plays a pivotal role in what follows; we call it the Lefschetz class and denote it by L. On occasion, we will need to invert this class, and therefore consider localizations of the form GrÔÅÕ L . To connect the theory of motivic sites to the classical construction, we must describe motivic sites whose Grothendieck ring admits a natural homomorphism into the classical Grothendieck ring GrÔÎ Ö V Õ (obviously, this utterly fails for the motivic site of all sieves). We will first introduce the various motives of interest in the next few sections, before we settle this issue in Theorem 7.4 below. The smallest motivic site on V is obtained by taking for sieves on a scheme X only the empty sieve and the whole sieve X ¥ . The resulting Gro- thendieck ring has no non-trivial scissor relations and so we just get the free Abelian ring on isomorphism classes of V -schemes. To define larger sites, we want to include at least closed subsieves of a scheme X. Any object in the lattice generated by the closed subsieves of X will be called a schemic motif on X. Since closed subsieves are already closed under intersection, a schemic motif on X is a sieve of the form (13) where the X i are closed subschemes of X. Let us call a V -scheme X schemically irreducible if X ¥ cannot be written as a finite union of proper closed subsieves. In particular, by an easy Noetherian argument, any schemic motif is the union of finitely many schemically irreducible closed subsieves. We call a representation (13) a schemic decomposition, if it is irredundant, meaning that there are no closed subscheme relations among any two X i , and the X i are schemically irreducible. Assume ( 13) is a schemic decomposition, and t be a second schemic decomposition. Hence, for a fixed i, we have Reversing the roles of the two representations, the same argument yields some i ½ such that Y j ⊆ X i ½. Since X i ⊆ Y j ⊆ X i ½ , irredundancy implies that these are equalities. Hence, we proved: We call the X i in the (unique) schemic decomposition (13) the schemic irreducible components of X. If X has dimension zero, then it is a finite disjoint sum of fat points, its schemic irreducible components. More generally, any schemic motif on X is a disjoint sum of closed subsieves given by fat points, and hence is itself a closed subsieve. To give a purely scheme-theoretic characterization of being schemically irreducible, recall that a point x È X is called associated, if O X,x has depth zero, that is to say, if every element in O X,x is either a unit or a zero-divisor. Any minimal point is associated, and the remaining ones, which are also finite in number, are called embedded. The closure of an associated point is called a primary component. We say that X is strongly connected, if the intersection of all primary components is non-empty, that is to say, if there exists a (closed) point generalizing to each associated point (in the affine case X Spec A, this means that the associated primes generate a proper ideal). For instance, if X is the union of two parallel lines and one intersecting line, then it is connected but not strongly. For an example with embedded points, take the affine line with two (embedded) double points given by the ideal Ôx 2 , xyÔy ¡ 1ÕÕ. A (reduced) example where any two primary, but not all three, components meet, is given by the 'triangle' xyÔx ¡ y ¡ 1Õ. Proof. It is easier to work with the contrapositives of these statements, and we will show that their negations are then also equivalent with the existence of finitely many non-zero ideal sheafs I 1 , . . . , I s ⊆ O X with the property that for each closed point x, there is some n such that I n O X,x 0. To prove the equivalence of this with being schemically reducible, assume s for some proper closed subschemes X n X. Let I n be the ideal sheaf of X n and let x be an arbitrary closed point. For each m, the closed immersion J m x X ⊆ X is a rational point on X along the m-th jet, whence must belong to one of the X n ÔJ m x XÕ, that is to say, J m x X is a closed subscheme of X n by Lemma 2.7. Since there are only finitely many possibilities, there is a single n such that each J m x X is a closed subscheme of X n . By Lemma 2.7, this means that I n O X,x is contained in any power of the maximal ideal m x , and hence by Krull's intersection theorem must be zero. Conversely, suppose there are non-zero ideal sheaves I 1 , . . . , I s ⊆ O X such that at each closed point, at least one vanishes. Let X n be the closed subscheme defined by I n , let z be a fat point, and let a : z X be a z-rational point. Let x be the center of z, a closed point of X, and let a x : O X,x R be the induced local homomorphism on the stalks, where R is the coordinate ring of z. By assumption, there is some n such that I n O X,x 0, whence so is its image under a x . By Lemma 2.2, this implies that a È X n ÔzÕ. Since this holds for any rational point, X ¥ is the union of all X ¥ n . We now prove the equivalence of the above condition with not being strongly connected. By (global) primary decomposition (see for instance [6, IV §3.2]), there exist (primary) closed subschemes Y n ⊆ X and an embedding O X O Y1 ¤ ¤ ¤ O Ys , such that the underlying sets of the Y n are the primary components of X. Let J n be the ideal sheaf of Y n , so that the above embeddability amounts to J 1 ¤ ¤ ¤ J s 0. If X is not strongly connected, then the intersection of all Y n is empty, which means that ( 14) Let I j be the intersection of all J m with m j, and let X j be the closed subscheme given by I j . Let x be any closed point. By (14), we may assume after renumbering that the maximal ideal of x does not contain J 1 , that is to say, J 1 O X,x O X,x . Therefore, , whence is zero, since I 1 J 1 0. Conversely, assume there are non-zero ideal sheafs I 1 , . . . , I s ⊆ O X such that I n O X,x 0, for each closed point x and for some n depending on x. This is equivalent with the sum of all AnnÔI n Õ being the unit ideal. Since any annihilator ideal is contained in some associated prime, the sum of all associated primes must also be the unit ideal, and hence the intersection of all primary components is empty. From the proof we learn that Spec A is schemic reducible if and only if there exist finitely many proper ideals whose sum is the unit ideal and whose intersection is the zero ideal. We can even describe an algorithm which calculates its schemic irreducible components. Let 0 g 1 ¤ ¤ ¤ g n be a primary decomposition of the zero ideal in A, and assume g 1 ¤ ¤ ¤ g s 1, for some s n. Then the schemic irreducible components of X are among the schemic irreducible components of the closed subschemes X i SpecÔAß AnnÔg i Õ for i 1, . . . , s. That the X i can themselves be schemic reducible, whence require further decomposition, is illustrated by the 'square' with equation xÔx ¡ 1ÕyÔy ¡ 1Õ 0. A sufficient condition for the X i to be already schemically irreducible is that s n and no fewer g i generate the unit ideal. By Noetherian induction, this has to happen eventually. In the same vein, we have: 5.3. Proposition. Suppose V is Jacobson. Let X be a V -scheme, and let X 1 , . . . , X s ⊆ X be closed subschemes with respective ideals of definition I 1 , . . . , s is the closed subscheme with ideal of definition I 1 ¤ ¤ ¤ I s . In particular, X is equal to its own Zariski closure if and only if Proof. Let Y be the closed subscheme with ideal of definition I : To prove the converse, suppose Z ⊆ X is a closed subscheme such that X ⊆ Z ¥ . We have to show that Y ⊆ Z, so suppose not. This mean that J O Y 0, where J is the ideal of definition of Z. By the Jacobson condition, there is a closed point x È Y such that J O Y,x 0, and then by Krull's Intersection Theorem, some n such that J ÔO Y,x ßm n x Õ 0, where m x is the maximal ideal corresponding to x. We may write m n x n 1 ¤ ¤ ¤ n t as a finite intersection of irreducible ideal sheafs, 4 and then for at least one, say for j 1, we must have J ÔO Y,x ßn 1 Õ 0. Let z be the fat point with coordinate ring R : O Y,x ßn 1 . By Lemma 2.8, this means that the z-rational point i given by the inclusion z ⊆ Y does not factor through Z. On the other hand, by the same Lemma, we have IR 0. Since the zero ideal is irreducible, at least one of the I j R must vanish, showing that i lies in XÔzÕ, whence by assumption in ZÔzÕ, contradiction. The last assertion is now immediate from the previous discussion. Proof. By an induction argument, we may reduce to the case that X Y ¥ Z ¥ where Y, Z ⊆ X are closed subschemes (alternatively, use (7)). In view of the local nature of the problem, we may furthermore reduce to the case that X Spec A is affine, so that Y and Z are defined by some ideals I, J ⊆ A. In particular, the Cartesian square ( 6) becomes ( 15) However, it is easy to check that putting AßÔI JÕ in the left top corner of this square also yields a Cartesian square, and hence, by uniqueness, we must have H 0 ÔXÕ AßÔI JÕ. By Proposition 5.3, the Zariski closure of X is SpecÔAßÔI JÕÕ, proving the assertion. 5.5. Definition. We define the schemic motivic site over V , denoted Ë V , as the full subcategory of Ë Ú V consisting of all schemic motives. By Proposition 2.7, the category of V -schemes fully embeds in Ë V , and the image of this embedding is precisely the full subcategory of representable schemic motives. In fact, by Theorem 3.7, all morphisms in Ë V are algebraic, so that Ë V is an explicit motivic site. Proof. For any two V -schemes X and X ½ , we have ÖX× ÖX ½ × ÖX X ½ ×, where X X ½ denotes their disjoint union. So remains to verify that any element in GrÔË V Õ is a linear combination of classes of schemes. This reduces the problem to the class of a single schemic motif X on a V -scheme X. Hence there exist closed subschemes For each non-empty I ⊆ Ø1, . . . , nÙ, let X I be the closed subscheme obtained by intersecting all X i with i È I, and let I denote the cardinality of I. A well-known argument deduces from the scissor relations the equality (16) ÖX× ô À I⊆Ø1,...,nÙ Ô¡1Õ I ÖX I × 4 An ideal is called irreducible if it cannot be written as a finite intersection of strictly larger ideals. in GrÔË V Õ, proving the claim. 5.7. Theorem. Suppose V is Jacobson. The schemic Grothendieck ring GrÔË V Õ is freely generated, as a group, by the classes of strongly connected V -schemes. Proof. Let Γ be the free Abelian group generated by isomorphism classes ÜXÝ of strongly connected V -schemes X, and let Γ ½ be the free Abelian group generated by isomorphism classes ÜXÝ of schemic motives X. I claim that the composition Γ ⊆ Γ ½ ։ GrÔË V Õ admits an additive inverse δ : GrÔË V Õ Γ. To construct δ, we will first define an additive morphism δ ½ : Γ ½ Γ which is the identity on Γ, and then argue that it vanishes on each scissor relation, inducing therefore a morphism δ : GrÔË V Õ Γ. It suffices, by linearity, to define δ ½ on an isomorphism class of a schemic motif (17) given by a schemic decomposition with X i ⊆ X strongly connected closed subschemes. By Noetherian induction, we may furthermore assume that δ ½ has been defined on the isomorphism class of any schemic motif on a proper closed subscheme of X. In particular, δ ½ ÜX I Ý has already been defined, where we borrow the notation from (16). We therefore set (18) δ ½ ÜXÝ : ô À I⊆Ø1,...,nÙ Ô¡1Õ I δ ½ ÜX I Ý By Proposition 5.1, the schemic decomposition is unique, and hence δ ½ is well-defined. Moreover, if n 1, so that X X ¥ 1 is a schemically irreducible closed subsieve, whence strongly connected by Proposition 5.2, then its image under δ ½ is just ÜX 1 Ý, showing that δ ½ is the identity on Γ. Next, let us show that even if (17) is not irredundant, (18) still holds in Γ. We can go from such an arbitrary representation to the schemic decomposition in finitely many steps, by adding or omitting at each step one strongly connected closed subsieve contained in X. So assume (18) holds for a representation (17), and we now have to show that it also holds for the representation adding a X ¥ 0 ⊆ X, with X 0 strongly connected. Since X 0 is schemically irreducible by Proposition 5.2, it must be a closed subscheme of one of the others, say, of X 1 . Let J range over all non-empty subsets of Ø0, . . . , nÙ. We arrange the subsets containing 0 in pairs ØJ , J ¡ Ù, so that J and J ¡ differ only as to whether they contain 1 or not. Since X 0 ⊆ X 1 , we get X J X J¡ , and as J ¡ has one element less than J , the two terms indexed by this pair in the sum (18) cancel each other out. So, in that sum, only subsets J not containing 0 contribute, which is just the value for the representation without X ¥ 0 . We also have to consider the converse case, where instead we omit one, but the argument is the same. We can now show that (18) is still valid even if the X i in (17) are not strongly connected. Again we may reduce the problem to adding or omitting a single closed subsieve X ¥ m be a schemic decomposition for X 0 . We have to show that the value of the sum in (18) for the representation n is the same as that for the representation (19) The first sum is given by (20) ô À I⊆Ø1,...,nÙ for each subset I. Substituting this is in (20) yields the sum corresponding to representation (19) (note that I J ranges over all non-empty subsets of Ø1, . . . , nÙ Ø1, . . . , mÙ, as required). To obtain the induced map δ, we must show next that δ ½ vanishes on any scissor relation. t be a second schemic motif on X, again assumed to be given by its schemic decomposition. Put Z ij : X i Y j , so that X Y is the union of the X ¥ i and Y ¥ j , whereas X Y is the union of the closed subsieves Z ¥ ij . By our previous argument, we may use these respective representations to calculate δ ½ of the scissor relation Comparing the various sums given by the respective right hand sides of (18), this reduces to the following combinatorial assertion. Given finite subsets I, J, let us call a subset N ⊆ I ¢ J dominant, if its projections onto the first and second coordinates are both surjective; then We leave the details to the reader. In conclusion, we have constructed an (additive) map δ : GrÔË V Õ Γ which is the identity on Γ. On the other hand, it follows from ( 16) that ÖδÖX×× ÖX×, showing that δ is an isomorphism. Immediately from this we get: 5.8. Corollary. Suppose V is Jacobson. If X and Y are strongly connected V -schemes, then ÖX× ÖY × in GrÔË V Õ if and only if X Y . To allow for additional relations, we want to include also open subsieves, or more generally, locally closed subsieves. For applications, it is more appropriate to put this in a larger context. Our point of departure is: 6.1. Lemma. For any V -scheme, the set of its sub-schemic sieves forms a lattice. Moreover, the product of two sub-schemic sieves is again sub-schemic. Proof. Recall that a sub-schemic sieve is just an image sieve. If ϕ : Y X and ψ : Z X are morphisms of V -schemes, then ImÔϕÕ ImÔψÕ ImÔϕ ¢ X ψÕ, where ϕ ¢ X ψ : Y ¢ X Z X is the total morphism in the commutative square given by base change. Likewise ImÔϕÕ ImÔψÕ ImÔϕ ψÕ, where ϕ ψ : Y Z X is the disjoint union of the two morphisms. showing that the latter is again sub-schemic. 6.2. Definition. We define the sub-schemic motivic site ×Ù Ë V as the full subcategory of Ë Ú V with objects the sub-schemic sieves. By Theorem 3.7, any morphism in this category is algebraic, so that ×Ù Ë V is again an explicit motivic site. Instead, we could have opted for a smaller site to take care of open coverings: define the motivic constructible site ÓÒ V by taking on each scheme the lattice generated by locally closed subsieves (note that this is again an explicit motivic site). We have a natural homomorphism of Grothendieck rings GrÔ ÓÒ V Õ GrÔ×Ù Ë V Õ, but I do not know whether it is injective and/or surjective. 6.3. Example. The constructible site is strictly smaller than the sub-schemic one, as illustrated by the following example: let ϕ : l 4 l 2 be the morphism corresponding to the homomorphism R 2 : κÖξ×ßξ 2 κÖξ× R 4 : κÖξ×ßξ 4 κÖξ× given by ξ ξ 2 . Given a fat point z Spec R, the z-rational points of l 2 are in one-one correspondence with the elements in R whose square is zero, whereas ImÔϕÕÔzÕ is the subset of all those that are themselves a square, in general a proper subset. As l 2 is a fat point, it has no non-trivial locally closed subsieves, showing that ImÔϕÕ is sub-schemic but not constructible. Moreover, the Zariski closure of ImÔϕÕ is l 2 . Proof. Let v w be a morphism of fat points. We have to show that any w-rational point a : w X whose image under XÔwÕ XÔvÕ belongs to Y ÔvÕ, itself already belongs to Y ÔwÕ. The condition that needs to be checked is that if the composition v w a X factors through Y , then so does a. Let x È X be the center of a. Since x is then also T , where T is the coordinate ring of w, whence a homomorphism O Y,x T , we get the desired factorization w Y . This will, among other things, allow us often to reduce the calculation of rational points to the affine case. Let X X 1 ¤ ¤ ¤ X n be an open cover. By Lemma 6.4, we get n . An easy argument on scissor relations, with notation as in (16), yields: 5 In particular, the class of a schemic motif lies in the subring generated by classes of affine schemes. 6.6. Example. As an example, let us calculate the class of the projective line P 1 V . It admits an open covering X 1 X 2 where X 1 and X 2 are obtained by removing respectively the origin and the point at infinity. Since X 1 X 2 A 1 V , we have where L ¦ denotes the class of the punctured line L ¦ X 1 X 2 , the affine line with the origin removed. One would be tempted to think that L ¦ is just L ¡ 1, but this is false, as 5 By assumption, all V -schemes are separated, and hence the intersection of affines is again affine. we shall see shortly. However, X 1 X 2 is an affine scheme, by Example 2.10, isomorphic to the hyperbola H ⊆ A 2 V with equation xy ¡ 1 0 under the projection A 2 A 1 V onto the first coordinate. In other words, we have V × 2L ¡ ÖH×. Let Y ⊆ X be a closed subscheme. Recall that the n-th jet of X along Y , denoted J n Y X, is the closed subscheme with ideal sheaf I n Y , where X Y of X along Y is then the ringed space whose underlying set is equal to the underlying set of Y and whose sheaf of rings is the inverse limit of the sheaves O J n Y X . In particular, if X Spec A is affine and I the ideal of definition of Y , then the ring of global sections of Ô X Y is equal to the I-adic completion Ô A of A (see, for instance, [7, II. §9]). We define the completion sieve along Y to be the sieve Mor V Ô¤, Ô X Y Õ represented by the formal completion Ô X Y of X at Y , that is to say, for each fat point z, it gives the subset of all z-rational points z X that factor through Ô X Y . We will simply denote it by Ô X ¥ Y and call any such presheaf again pro-representable. 67.1. Proposition. For a closed subscheme Y ⊆ X, the completion sieve Ô Y now follows from Lemma 2.5. The proof of the first assertion actually gives a stronger statement, which we formalize as follows. A sieve X on X is called a formal motif on X, if for each fat point z, there exists a sub-schemic subsieve Y z ⊆ X such that Y z ÔzÕ XÔzÕ (we call the Y z the sub-schemic approximations of X, in spite of the fact that they are not unique). A sub-schemic motif is a trivial example of a formal motif; the proof of Proposition 7.1 shows that completion sieves are formal too, whose approximations, in fact, can be taken to be schemic. More generally, we have 7.2. Lemma. If a sieve X on a V -scheme X has formal approximations in the sense that for each fat point z, there exists a formal subsieve Y z ⊆ X with the same z-rational points, then X itself is formal. X Y is no longer a scheme, but only a locally ringed space with values in the category of O Valgebras, and so for a (formal) scheme Z, the set Mor V ÔZ, Ô X Y Õ is to be understood as the set of morphisms Z Ô X Y of locally ringed spaces with values in the category of O V -algebras. Proof. By assumption, there exists a sub-schemic approximation Z z ⊆ Y z with the same z-rational points, and it is now easy to check that the Z z form a sub-schemic approximation of X. 7.3. Lemma. If a formal motif X has no z-rational points, for some fat point z, then X itself is empty. Proof. Suppose first that X ImÔϕÕ is sub-schemic, given by a morphism ϕ : Y X. In order for ϕÔzÕ to be empty, Y ÔzÕ has to be empty, whence Y has to be the empty scheme by Lemma 2.8, proving that ImÔϕÕ is the empty motif. Suppose now that X is merely formal, and let w be an arbitrary fat point. Let Y ⊆ X be a sub-schemic approximation with the same w-rational points. Since YÔzÕ ⊆ XÔzÕ À, we get Y À, by the subschemic case, showing that XÔwÕ À. Since this holds for all fat points w, the assertion follows. Using Lemma 6.1, one easily verifies that the formal motives on X form again a lattice, and the product of two formal motives is again a formal motif, leading to the formal motivic site ÓÖÑ V , and its corresponding Grothendieck ring GrÔ ÓÖÑ V Õ. Proof. Only the last of these homomorphisms requires an explanation. Given a formal motif X, we associate to it the class of XÔκÕ in the classical Grothendieck ring GrÔÎ Ö κ Õ. Note that by definition, XÔκÕ ImÔϕÕÔκÕ, for some morphism ϕ : Y X of κ-schemes. In particular, by Chevalley's theorem, XÔκÕ is a constructible subset of XÔκÕ and hence its class in GrÔÎ Ö κ Õ is well-defined. Clearly, this map is compatible with intersections, unions, and products, so that in order for this map to factor through GrÔ ÓÖÑ κ Õ, we only have to show that it respects isomorphisms. So assume s : X Y is an isomorphism of formal motives. Let Z ⊆ X be a sub-schemic approximation of X with the same κ-rational points. Its push-forward s ¦ Z is isomorphic with Z. By Theorem 3.7, the restriction s Z extends to a morphism ϕ : X Y , where X and Y are some ambient spaces of Z and Y respectively. Since ϕÔκÕ : XÔκÕ Y ÔκÕ maps ZÔκÕ bijectively onto s ¦ ZÔκÕ, these two constructible subsets are isomorphic in the Zariski topology. However, by definition of push-forward, s ¦ ZÔκÕ is the image of ZÔκÕ XÔκÕ under sÔκÕ, that is to say, is equal to YÔκÕ, as we needed to show. 7.5. Theorem. Assume V is Jacobson, and let s : Y X be a morphism of sieves. If Y and X are sub-schemic (respectively, formal) motives, then so is the graph of s. Moreover, the pull-back or the push-forward of a sub-schemic (respectively, formal) submotif is again of that form. Proof. Let X be an ambient spaces of X. Since the graph of the composition Y X ⊆ X ¥ is equal to the intersection of the graph ΓÔsÕ of s with Y ¢X, we may assume from the start that X X ¥ . Assume first that Y is sub-schemic. By Theorem 3.7, the morphism s extends to a morphism ϕ : Y X of V -schemes. Let Z ⊆ Y ¢ V X be the graph of this morphism, which therefore is a closed subscheme. Since ΓÔsÕ is equal to the intersection Suppose next that Y is merely a formal motif, and, for each fat point z, let Z z ⊆ Y be a sub-schemic approximation. Since ΓÔsÕ contains the graph of the restriction of s to Z z , and since they have the same z-rational points, the assertion follows from what we just proved for sub-schemic motives. Let Y ½ ⊆ Y be a sub-schemic or formal submotif. Since the push-forward s ¦ Y is the image of the restriction of s to Y ½ , we may reduce the problem to showing that ImÔsÕ is respectively sub-schemic or formal. The formal case follows easily, as in the previous argument, from the sub-schemic one. So assume once more that Y is sub-schemic, say of the form, ImÔψÕ with ψ : Z Y a morphism of V -schemes. With ϕ as above, one easily verifies that ImÔsÕ ImÔϕ ¥ ψÕ. To prove the same for the pull-back, simply observe that the pull-back s ¦ X ½ of a submotif X ½ ⊆ X is equal to the image of ΓÔsÕ ÔY ¢ X ½ Õ under the morphism induced by the projection Y ¢ X Y . The result then follows from our previous observations. 7.6. Corollary. Given an irreducible closed subscheme Y ⊆ X with ideal of definition I Y , let X be the closed subscheme given by the intersection of all powers I n Y (e.g., if X is integral, this is just X itself, by Krull's Intersection Theorem). We have isomorphisms of O V -algebras where y is the generic point of Y . Proof. Let X : Ô X ¥ Y be the formal motif determined by the formal completion along Y . We leave it to the reader to verify that the Zariski closure of X is equal to X. Replacing X by X, we may therefore assume that X is Zariski dense. An open subscheme U ⊆ X is an ambient space of X if and only if Y ⊆ U , since XÔzÕ ⊆ Y ÔzÕ, for each fat point z. By Corollary 3.3, the ring of algebraic sections H geom 0 ÔXÕ is therefore the direct limit of all H 0 ÔUÕ, where U ⊆ X runs over all opens containing Y . The latter condition is equivalent with y È U , and hence this direct limit is just O X,y . On the other hand, since the jets J n Y X are approximations of X, the inverse limit of the H 0 ÔJ n Y XÕ is equal to H 0 ÔXÕ by Lemma 3.9. Since H 0 ÔJ n Y XÕ ΓÔO X ßI n Y , XÕ, the result follows. We will shortly generalize this in Proposition 7.8 below, but let us first construct from this an example of a non-algebraic morphism. Let f ÔxÕ be a power series in a single indeterminate which is not a polynomial. The homomorphism κÖt× κÖÖx×× given by t f induces a natural transformation of sieves s f : it cannot extend to a morphism of schemes. Its graph, in accordance with Theorem 7.5, is the formal motif with approximations the graphs of the algebraic morphisms given by the various truncations of f . We can also use this to give a counterexample to Proposition 3.12 for global sections. In general, given a global section s : X A 1 V of a formal motif X, define a sieve X s by letting X s ÔzÕ consist of all z-rational points a È XÔzÕ such that sÔzÕÔaÕ is a unit, for each fat point z. Since X s s ¦ L ¥ ¦ , where L ¦ is the affine line without the origin, it is a formal motif by Theorem 7.5. Applied to the global section s f above, Ô L ¥ s f is the intersection of the open subsieves given by the truncations of f , whence not an admissible open in X for the Zariski topos. In particular, s f is not continuous. Put differently, the submotives of the form X s form in general a basis for a Grothendieck topology which is stronger than the Zariski one. 7.8. Proposition. For each n, the class of projective n-space in GrÔ ÓÖÑ V Õ is given by the formula Proof. Let Ôx 0 : ¤ ¤ ¤ : x n Õ be the homogeneous coordinates of P n V , and let X i be the basic open given as the complement of the x i -hyperplane. Hence every X i is isomorphic with A n V and their union is equal to P n V . Therefore, by Lemma 6.5, we have (23) By the binomial theorem, t n 1 ¡ Ôt ¡ uÕgÔt, uÕ Ôt ¡ Ôt ¡ uÕÕ n 1 u n 1 , and hence gÔt, uÕ as we wanted to show. We conclude this section with a characterization of the complete sieves among the formal motives over an algebraically closed field κ. 7.9. Theorem. Let κ be an algebraically closed field. A sieve X on a κ-scheme X is a complete formal motif if and only if it is the cone CÔF Õ over a constructible subset Proof. Suppose X is complete and formal. By Lemma 2.6, it is of the form CÔF Õ for some F ⊆ XÔκÕ. In fact, F XÔκÕ, and hence by definition of the form ImÔϕÕÔκÕ for some morphism ϕ : Y X. By Chevalley's theorem, F is constructible. To prove the converse, since cones are easily seen to commute with union and intersection, and since any constructible subset of XÔκÕ is an intersection and union of closed and open subsets, it suffices to prove that CÔF Õ is formal, whenever F is a closed or an open subset. The open case follows immediately from Lemma 6.4, and in the closed case, we have CÔF Õ Ô X ¥ F by Proposition 7.1 (note that the completion of a scheme along a subscheme only depends on the underlying variety of the subscheme, so that Ô X F is well-defined). Let V and W be two Noetherian Jacobson schemes. By a (schemic) adjunction over the pair ÔV, W Õ, we mean a pair of functors η : Ø W Ø V and ∇: Ë V Ë W , called respectively the left and right adjoint, such that we have, for each W -fat point z and each V -scheme X, an adjunction isomorphism (24) Θ z,X : XÔηÔzÕÕ Mor V ÔηÔzÕ, XÕ Mor W Ôz, ∇XÕ ∇XÔzÕ, which is functorial in both arguments. Whenever z and X are clear from the context, we may just denote this isomorphism by Θ, or even omit it altogether, thus identifying XÔηÔzÕÕ with ∇XÔzÕ. More generally, by an (arbitrary) adjunction we mean the same as above, except that the right adjoint now only takes values in the category of sieves, that is to say, is a functor Ë V Ë Ú W , where we identify the category of V -schemes with its image as a full subcategory of sieves. Of course, the morphisms on the right hand side of ( 24) are now to be taken in Ë Ú W , where the last equality is then given by Corollary 2.9 (note, however, that they are all algebraic by Proposition 2.7). If each ∇X is sub-schemic, or formal, then we call the adjunction respectively sub-schemic or formal. We can formulate the adjunction property as a representability question: given a functor η : Ø W Ø V and a V -sieve X, let ∇ η X be the presheaf over W associating to a fat W -point, the set XÔηÔzÕÕ. We have adjunction when all presheaves ∇ η X ¥ are sieves, where X varies over all V -schemes; the adjunction is (sub-)schemic or formal, if each ∇ η X ¥ is respectively a (sub-)schemic or formal motif. From this perspective, ∇ η is the right adjoint of η, and we simply call ∇ η the adjunction. We extend this to get a functor Given a sieve X on a V -scheme X, we define its adjoint ∇ η X as the presheaf over W given by ∇ η XÔzÕ : Θ z,X XÔηÔzÕÕ for any W -point z. It follows immediately from (24) that ∇ η X ¥ ∇X, and hence ∇ η X is a subsieve of ∇X. The adjunction isomorphism (24) then becomes (25) In particular, sub-schemic adjunctions preserve sub-schemic as well as formal motives, whereas formal adjunctions preserve formal motives. Proof. We verify (26) on a W -fat point z. Functoriality of adjunction implies that we have a one-one correspondence of diagrams It then follows from Theorem 7.5 that the adjoint of a sub-schemic motif is again subschemic, in case η is sub-schemic itself. Suppose next that X is formal, and, for each V -fat point w, let Y w ⊆ X be a sub-schemic approximation with the same w-rational points. For each W -fat point z, let Ỹz be defined as ∇ η ÔY ηÔzÕ Õ. By what we just proved, Ỹz ⊆ ∇ η X is a sub-schemic submotif, and one easily verifies that both sieves have the same z-rational points, proving the last assertion for sub-schemic adjunctions. The case of a formal adjunction then follows from Theorem 7.5 and Lemma 7.2. schemic, then this also induces homomorphisms of the corresponding (sub-)schemic Grothendieck rings. Proof. By Lemma 8.1, adjunction preserves motivic sites insofar it is (sub-)schemic or formal. As it is compatible with unions and intersections, it preserves scissor relations, and as it is functorial, it preserves isomorphisms as well as products. Before we describe some important instances in which we have adjunction, with applications discussed in § §9 and 11, we give an example of a formal adjunction. ΥÔzÕ : Υ r ÔzÕ be the fat point with coordinate ring κ m r ⊆ R, where ÔR, mÕ is the Artinian local ring corresponding to z. Note that we have a dominant morphism z ΥÔzÕ. For simplicity, let us take r 2. For fixed n, let l : l n be the n-th jet of a point on a line, with coordinate ring S : κÖξ×ßξ n . For each l, let w l be the fat point in A 2l κ with ideal of definition generated by all ξ l i and Q n , where Let Y l be the image sieve of the morphism ϕ l : w l l induced by ξ Q. I claim that ∇ Υ l is approximated by the Y l . To this end, fix a fat point z with coordinate ring ÔR, mÕ and let l be its length. A ΥÔzÕ-rational point a È lÔΥÔzÕÕ is completely determined by the image, denoted again a, of ξ in κ m 2 . Since a n 0, we must in particular have a È m 2 (note that m 2 is the maximal ideal of ΥÔzÕ), and hence can be written as Presumably, this argument should extend to any fat point other than l and any power r 2. To extend this to higher dimensional schemes, we face the problem that a rational point can be given by non-units. This forces us to be able to single out the field elements inside an Artinian local ring R. In characteristic p, this can be done: the elements of κ ⊆ R are precisely the p l -th powers. Using this, a slight modification of the above argument, then yields ∇ Υ A 1 κ as a formal motif: in the above, replace w l by A 1 w l and Y l by the image of the morphism A 1 κ given by ξ ξ p l 0 Q. It seems likely that we can again extend this argument to arbitrary schemes and arbitrary r 2. V be a morphism of Noetherian Jacobson schemes. Via f , any W -scheme Y becomes a V -scheme, and to make a notational distinction between these two scheme structures, we denote the latter by f ¦ Y . We will show that f ¦ constitutes a left adjoint, where the corresponding right adjoint is given by base change: given a V -scheme X, we set f ¦ X : W ¢ V X. V is a morphism of finite type of Noetherian Jacobson schemes, then f ¦ is a functor from Ø W to Ø V , and as such, it is the left adjoint of f ¦ . The corresponding adjunction associates to a V -sieve X on a V -scheme X, the W -sieve ∇ f¦ X on f ¦ X, inducing natural ring homomorphisms on the respective Grothendieck rings, namely ∇ f¦ : Proof. Let z be a W -fat point and let y be its center, that is to say, the closed point on W given as the image under the structure morphism z W . By the generalized Nullstellensatz ([5, Theorem 4.19]), the image x : f ÔyÕ is a closed point on V , and the residue field extension κÔxÕ ⊆ κÔyÕ is finite. As κÔyÕ ⊆ Rßm is also finite, f ¦ z is a V -fat point, proving the first assertion. The adjunction of f ¦ and f ¦ are well-known (and, in any case, easily checked; see, for instance [7, Chapter II.5], but note that left and right are switched there since they are formulated in the dual category of sheaves), proving that ∇ f¦ X f ¦ X. The last statement follows from Proposition 8.2. 8.6. Remark. Although f ¦ : Ø W Ø V is an embedding of categories, it is, however, not full: so are the closed subschemes in A 2 κ defined by the ideals Ôx 2 , y 3 Õ and Ôx 3 , y 2 Õ isomorphic as fat κ-points, but not as fat κÖx×-points. Nonetheless, Ø W is cofinal in Ø V , or, in the terminology of §12 below, both have the same universal point. 8.7. Diminution. Let f : W V be a finite and faithfully flat morphism of Noetherian Jacobson schemes. As opposed to the previous section, we will now consider f ¦ as a left adjoint. For technical reasons (see Remark 8.10 below for how to circumvent these), we make the following additional assumptions: ( ) V is of finite type over an algebraically closed field κ and f induces an isomorphism on the underlying varieties. The second condition implies that for any closed point x È V there is a unique closed point y È W lying above it, and hence the closed fiber f ¡1 ÔxÕ is a local scheme. Under these More precisely, for any W -scheme Y , there exists a V -scheme ∇ f ¦Y and a canonical morphism ρ Y : f ¦ Ô∇ f ¦Y Õ Y of W -schemes, such that, for any V -fat point z, the map sending a z-rational point a : z Proof. Since f is finite and flat, W is locally free over V . Since we may construct each ∇ f ¦Y locally and then, by the uniqueness of the universal property of adjoints, glue the pieces together, we may assume that Y Spec B, V Spec λ, and W Spec µ are affine, and that µ is free over λ (in all applications, we will already have global freeness anyway). Let α 1 , . . . , α l be a basis of µ over λ. Write B : µÖx×ßÔh 1 , . . . , h s ÕµÖx×, for some polynomials h i over µ. Let x be a new tuple of variables and define a generic tuple of arcs Applying (28) to the h i , we get polynomials hij over λ and we let A be the residue ring of λÖx× modulo the ideal generated by all these hij . I claim that X : Spec A represents ∇ f ¦Y . It follows from (28) that the map x Ù x yields a µ-algebra homomorphism B f ¦ A, where f ¦ A : A λ µ is the base change, and hence a µ-morphism ρ Y : f ¦ X Y . Fix a λ-fat point z, and a z-rational point a : z X. By base change, we get a To prove that the map a ΘÔaÕ establishes an adjunction isomorphism, we construct its converse. Given an f ¦ z-rational point b : f ¦ z Y , let B R λ µ be the corresponding µ-algebra homomorphism, where R is the coordinate ring of z. The latter homomorphism is uniquely determined by a tuple u in R λ µ such that all h i ÔuÕ 0. u ũ1 α 1 ¤ ¤ ¤ ũl α l yields a (unique) tuple ũ : Ôũ 1 , . . . , ũl Õ over R such that all hij ÔũÕ 0, determining, therefore, a λ-algebra homomorphism A R, whence a λ-morphism ΛÔbÕ : z X. So remains to verify that Λ and Θ are mutual inverses. Starting with the f ¦ z-rational point b, we get the z-rational point ΛÔbÕ, which in turn induces the f ¦ z-rational point ΘÔΛÔbÕÕ, given as the composition ρ Y ¥ f ¦ ΛÔbÕ. The latter corresponds by (29) to the µ-algebra showing that ΘÔΛÔbÕÕ b. If, on the other hand, we start with the z-rational point a, given by x ũ, we get the f ¦ z-rational point ΘÔaÕ, given by x u, where u is as in (29). Hence ΛÔΘÔaÕÕ is given by x ũ, that is to say, is equal to a, as we needed to show. To prove the last assertion, assume that Z is a closed subscheme of Y , so that its coordinate ring is of the form BßÔh s 1 , . . . , h t ÕB for some additional polynomials h i È µÖx×. Hence ∇ f ¦ Z is the closed subscheme of ∇ f ¦ Y given by the hij for s i t. Immediately from the above proof, by taking Y f ¦ X, we have the following result, which we will use in the next section: 8.9. Corollary. If f : W V is a finite and faithfully flat morphism satisfying ( ), then we have for each 8.10. Remark. Without assumption ( ), the pull-back of a V -fat point z is only a zerodimensional W -scheme, and hence a disjoint sum of W -fat points f ¦ z w 1 ¤ ¤ ¤ w s . We can then still make sense of Y Ôf ¦ zÕ, as the disjoint union Y Ôw 1 Õ ¤ ¤ ¤ Y Ôw s Õ, and the adjunction condition then becomes that this must be equal to Ô∇ f ¦ Y ÕÔzÕ. Since nowhere in the above proof we used that f ¦ R is local, we therefore can omit condition ( ) from the statement of Theorem 8.8. We have the following commutation rule for adjunctions in a Cartesian square: 8.11. Theorem (Projection Formula). Let f : W V be a finite and faithfully flat morphism of Noetherian Jacobson schemes satisfying ( ), let u : Ṽ V be a morphism of finite type, and let be the base change diagram, where W : W ¢ V Ṽ . We have an identity of adjunctions Proof. Note that f is again finite and faithfully flat, satisfying ( ), so that the diminution ∇ f ¦ makes sense. To prove the identity we have to check it on each W -sieve Y and each Ṽ -fat point z, becoming But one easily verifies that we have an equality of W -fat points concluding the proof of the theorem. 8.12. Frobenius transform. Assume for the remainder of this section that the base ring is a field κ of characteristic p 0. Let us denote the Frobenius homomorphism a a p on a κ-algebra A by F, or in case we need to specify the ring by F A , so that we have in particular a commutative diagram Due to the functorial nature, we can glue these together and hence obtain on any κ-scheme X a corresponding endomorphism F X . Diagram (31) implies that F A is not a κ-algebra homomorphism. To overcome this difficulty, we assume κ is perfect, so that F is an isomorphism on κ. To make (31) into a κ-algebra homomorphism, we must view the second copy of A with a different κ-algebra structure, namely, the one inherited from the composite homomorphism κ F κ A. Several notational devices have been proposed (see for instance [7, Chapter IV, Remark 2.4.1] or [17, Chapter 8.1.c]), but we will use the one already introduced in the previous section: the push-forward of A along F will be denoted F ¦ A. In other words, F ¦ A is A with its κ-action given by u ¤ a u p a. Since κ is perfect, A F ¦ A as rings, and in many instances, even as κ-algebras. In particular, (31) yields a κ-algebra homomorphism A F F ¦ A, called the κ-linear Frobenius. The image of the κ-linear Frobenius homomor- phism A F F ¦ A is the subring of A consisting of all p-th powers, and we will simply denote it by FA (rather than the more common A p , which might lead to confusions with Cartesian powers). Hence, pushing forward the inclusion homomorphism FA ⊆ A gives a factorization of the κ-algebra homomorphism F as A ։ F ¦ FA ⊆ F ¦ A, where the first homomorphism is an isomorphism if and only if A is reduced. For instance, if A κÖx×, then FA κÖx p ×, so that this factorization is given by the sequence of κ-algebra homomorphisms (32) κÖx× where h is obtained from h by replacing each coefficient with its (unique) p-th root. So, from this we can calculate F ¦ A for A of the form κÖx×ßÔf 1 , . . . , f s ÕκÖx× as F ¦ A κÖx×ßÔ f1 , . . . , fs ÕκÖx×, with fi σÔf i Õ as in (32). The κ-linear Frobenius A F ¦ A is then the induced homomorphism by the composite map g gÔx p Õ from (32). Similarly, viewing X as a κ-scheme via the composition X Spec κ F Spec κ, it will be denoted by F ¦ X, yielding a morphism of κ-schemes F X : F ¦ X X, called the κ-linear Frobenius. Its scheme-theoretic closure will be denoted by FX, so that we have a dominant morphism X FX, yielding a factorization (33) where the closed immersion F ¦ FX ⊆ X is the identity if and only if X is a variety. In particular, FX is the Zariski closure of ImÔF X Õ in X. We could view F κ as an automorphism of the base to get by Theorem 8.5 an adjunction pair ÔF ¦ κ , F κ¦ Õ. However, since X and F κ¦ F ¦ κ X are isomorphic as κ-schemes, this merely induces an action of the Frobenius. For the same reason, diminution does not induce any interesting endomorphism on the Grothendieck ring. Instead we take a relative point of view. To a morphism ϕ : Y X of κ-schemes, we can associate two commutative squares; the base change and the Frobenius square. Combined into a single commutative diagram of κ-morphisms, we have where F ¦ X Y : F ¦ X¢ X Y is the pull-back of Y along F X , called the Frobenius transform of Y in X, and where the canonical projection X Y is then also a closed immersion. We can calculate it explicitly in case X Spec A is affine and Y is defined by the ideal I ⊆ A. Traditionally, one denotes the ideal generated by the image of I under the Frobenius F A by I Öp× ; it is the ideal generated by all f p with f È I. With this notation, we have In particular, applying σ from (32) to the previous isomorphism in case X is affine space, we get: Fz constitutes a functor on Ø κ , which will play the role of left adjoint. However, in this case, the adjunction will only be sub-schemic, via the following right adjoint. For each κ-scheme Y , we define a sub-schemic motif F Y , called its Frobenius motif. In order to do this, we will work locally: show that it is a right adjoint locally, and then deduce its uniqueness and existence, as well as right adjointness, globally. So let Y be affine, say, a closed subscheme of A n κ , and let In particular, we get induced endomorphisms ∇ F on GrÔ×Ù Ë κ Õ and GrÔ ÓÖÑ κ Õ. Unraveling the definitions, the action of this adjunction on a motif Y on a scheme Y is given by Therefore, if a motif Y has an ambient space which is affine, we may take it to be an affine space A n κ , so that 8.15. Families of motives. Let s : Y X be a algebraic morphism of V -sieves (see Remark 8.17 below for the non-algebraic case). Hence, we can find ambient spaces Y and X of Y and X respectively, and a morphism ϕ : Y X of V -schemes extending s. We explain now how we may view s as a family of V -sieves, by associating to each V -morphism a : V X, a V -sieve Y a as follows. We may view Y as an X-sieve via ϕ, and, in accordance with previous notation, we denote this X-sieve by ϕ ¦ Y. Using a to as augmentation map, we define called the specialization of Y at a. By Theorem 8.5, this is a sieve on the base change To see that the specialization Y a is independent from the choice of ambient space Y , we simply observe that (35) 8.17. Remark. We can even apply this theory to a non-algebraic morphism s : Y X of formal motives. Indeed, let Z z ⊆ Y be sub-schemic approximations of Y. By Theorem 3.7, the restriction s Zz is algebraic. Hence, given a : V X, the specialization ÔZ z Õ a is sub-schemic by Proposition 8.16. Using (35), it is not hard to show that these specializations ÔZ z Õ a are approximations of Y a (as defined by the right hand side of ( 35)), showing that the latter is formal too. From now on, our base scheme will (almost always) be an algebraically closed field κ. Fix a fat point z and let j : z Spec κ be its structure morphism. Clearly, it is flat and finite and satisfies condition ( ), and so both augmentation and diminution with respect to j are well-defined. We define the arc functor of z, as a double adjunction7 In other words, given a motif X on a κ-scheme X, and a fat point w, we have Ô∇ z ÕÔwÕ XÔj ¦ j ¦ wÕ where j ¦ j ¦ w is the product z¢ κ w viewed as a fat point over κ, denoted henceforth simply by zw. Applied to a κ-scheme X, we get the so-called arc scheme ∇ z X, whose w-rational points are in one-one correspondence with the zw-rational points of X, and so we will identify henceforth XÔzwÕ Ô∇ z XÕÔwÕ. Moreover, we have by Corollary 8.9 a canonical morphism (36) ρ X : ∇ z X X. 9.1. Remark. In other words, ∇ z X is the Hilbert scheme classifying all maps from z to X. When z SpecÔκÖξ×ßξ n κÖξ×Õ, the resulting arc scheme is also known in the literature as a jet scheme (though we prefer to reserve the latter nomenclature for schemes of the form J n Y X). 9.2. Remark. By the argument in the proof of Theorem 8.8, for any fat point z, we may choose a basis ∆ Øα 0 , . . . , α l¡1 Ù of its coordinate ring ÔR, mÕ with some additional properties. In particular, unless noted explicitly, we will always assume that the first base element is 1 and that the remaining ones belong to m. Moreover, once the basis is fixed, we let x be the l-tuple of arc variables Ôx 0 , . . . , xl¡1 Õ, so that Ù x x0 x1 α 1 ¤ ¤ ¤ xl¡1 α l¡1 is the corresponding generic arc. It follows from (28) that f0 f Ôx 0 Õ, for any f È κÖx×. By [13, §2.1], we may choose ∆ so that, with a i : Ôα i , . . . , α l¡1 ÕR, we have a Jordan-Holder composition series 8 a l 0 a l¡1 a l¡2 ¤ ¤ ¤ a 1 m a 0 R. Given r È R, we expand it as in (29) in the basis as r r 0 r 1 α 1 ¤ ¤ ¤ r l¡1 α l¡1 , with r i È κ. I claim that r j 0 for j i whenever r È a i . Indeed, if not, let j i be minimal so that there exists a counterexample with r j 0. By minimality, r r j α j r j 1 α j 1 ¤ ¤ ¤ È a i showing that α j È a j 1 , since r j is invertible. However, this implies that a j a j 1 , contradiction. From this, it is now easy to see that the first m basis elements of ∆ form a basis of R m : Rßa m 1 . Therefore, calculating fm in (28) for f È κÖx× does not depend on whether we work over R or with R m , and hence, in particular, fm È κÖx 0 , . . . , xm × for every m l. With these observations, we can now prove the following important openness property of arcs: 9.3. Theorem. Given a κ-scheme X, a fat point z, and an open U ⊆ X, we have isomorphisms Proof. By the universal property of adjunction, whence of arcs, it suffices to verify (37) in case X Spec B ⊆ A n κ is affine and U SpecÔB f Õ is a basic open subset. Let A be the coordinate ring of ∇ z X. Since U is the closed subscheme of A 1 B given by g : f y ¡1 0, the corresponding arc scheme ∇ z U is the closed subscheme of A n A with coordinate ring A ½ : AÖỹ×ßÔg 0 , . . . , gl¡1 ÕAÖỹ×, where l is the length of z Spec R and the gi are given by ( 28), with ỹ a tuple of l variables. By Remark 9.2, we may calculate the gi using any basis α 0 1, . . . , α l¡1 of R, and so we may assume it has the properties discussed in that remark. In particular, by the last observation in that remark, each gi only depends on ỹ0 , . . . , ỹi . Clearly, g0 f0 ỹ0 ¡ 1. In particular, the A-subalgebra of A ½ generated by ỹ0 is just A f0 . We will prove by induction, that each ỹi belongs to this subalgebra, and hence A ½ A f0 , as we needed to prove. To verify the claim, we may assume by induction that ỹ0 , . . . ỹi¡1 belong to A f0 . The coefficient of α i in the expansion (38) is equal to gi , whence zero in A ½ . As observed in Remark 9.2, the choice of basis al- lows us to ignore all terms with α j for j i. Put differently, upon replacing R by RßÔα i 1 , . . . , α l¡1 ÕR, which does not effect the calculation of gi , we may assume that they are zero in (38). Hence, gi f0 ỹi terms involving only ỹ0 , . . . , ỹi¡1 8 Writing R as a homomorphic image of κÖy× so that y : Ôy 1 , . . . , yeÕ generates m, let aÔαÕ, for α È Z n 0 , be the ideal in R generated by all y β with β lexicographically larger than α. Then we may take ∆ to be all monomials y α such that y α Ê aÔαÕ, ordered lexicographically. 2x 00 x10 3ỹ 2 00 ỹ10 2x 00 x01 3ỹ 2 00 ỹ01 2x 00 x11 2x 10 x01 3ỹ 2 00 ỹ11 6ỹ 00 ỹ10 ỹ01 . Note that above the singular point x00 0 ỹ00 , the fiber consist of two 4-dimensional hyperplanes, whereas above any regular point, it is a 3-dimensional affine space, the expected value by Theorem 9.10 below. 9.5. Example. Another example is classical: let R 2 κÖξ×ßξ 2 κÖξ× be the ring of dual numbers and l 2 : SpecÔR 2 Õ the corresponding fat point. Then one verifies that a κrational point on ∇ l2 X is given by a κ-rational point P on X, and a tangent vector v to X at P , that is to say, an element in the kernel of the Jacobian matrix Jac X ÔPÕ. 9.6. Example. As a last example, we calculate ∇ ln l m , where l n is the n-th jet of the origin on the affine line, that is to say, SpecÔκÖξ×ßξ n κÖξ×Õ. With Ù we will expand Ù x m in the basis Ø1, ξ, . . . , ξ n¡1 Ù of κÖξ×ßξ n κÖξ×; the coefficients of this expansion then generate the ideal of definition of ∇ ln l m . A quick calculation shows that these generators are the polynomials g s Ôx 0 , . . . , xn¡1 Õ : for s 0, . . . , n ¡1, where the i j run over Ø0, . . . , n ¡1Ù. Note that g 0 xm 0 . One shows by induction that Ôx 0 , . . . , xs ÕκÖx× is the unique minimal prime ideal of ∇ ln l m , where s Ö n m × is the round-up of nßm, that is to say, the least integer greater than or equal to nßm. In particular, ∇ ln l m is irreducible (but not reduced) of dimension n ¡ Ö n m ×. Immediately from Theorems 8.5 and 8.8, we get: 9.7. Theorem. For each fat point z, the arc functor ∇ z induces a ring endomorphism on each of the motivic Grothendieck rings GrÔË κ Õ, GrÔ×Ù Ë κ Õ and GrÔ ÓÖÑ κ Õ. In case of complete formal motives, we even have: 9.8. Lemma. For any closed immersion Y ⊆ X of κ-schemes, and any fat point z, we have isomorphisms where ρ : ∇ z X X is the canonical map from (36). Proof. Let U : X ¡ Y . By Proposition 7.1, we have an equality (39) of sieves on X. By Theorem 7.5, we may pull back (39) under the map ρ : ∇ z X X, to get a relation where we used the openness of arcs (Theorem 9.3) for the last equality. On the other hand, taking arcs in identity (39), yields where one easily checks that arc functors commute with complements of complete sieves. Combining both identities and taking complements then proves the first isomorphism. To see the second isomorphism, we may assume, in view of the local nature of arcs, that X Spec A is affine. Let I ⊆ A be the ideal of definition of Y , so that the global sections of Ô X Y is the completion Ô A I of A with respect to I. Let AÖy×ßJ be the coordinate ring of the arc scheme ∇ z X, for some J ⊆ AÖy× and some tuple of variables y. By the first isomorphism, the global section ring of ∇ z Ô X Y is equal to the base change Ô A I Öy×ßJ Ô A I Öy×. The ideal defining ρ ¡1 ÔY Õ in ∇ z X is IÔAÖy×ßJÕ, and the completion of AÖy×ßJ with respect to this ideal is Ô It is easy to check that ∇ v ∇ w ∇ vw ∇ w ∇ w , so that all arc functors commute with one another. If κ has positive characteristic, we also have a Frobenius adjoint acting on the sub-schemic and formal Grothendieck rings, and we have the following commutation relation for any fat point z. Indeed, we verify this on an arbitrary motif X and a fat point w. The left hand side of (40) becomes whereas the right hand side becomes and these are both equal since an easy calculation shows that FÔzwÕ ÔFzÕÔFwÕ. Arcs and locally trivial fibrations. By adjunction, any morphism z z of fat points induces a natural transformation of arc functors ∇ z ∇ z. In particular, taking z to be the geometric point given by κ itself, we get a canonical morphism ∇ z X X, for any motif X, since ∇ κ ¤ is the identity functor. In case X X ¥ is representable, this is none other than the canonical morphism ρ X : ∇ z X X from (36). To formulate the key property of this morphism, we need a definition. We call a morphism Y X of κ-schemes a locally trivial fibration with fiber Z if for each (closed) point P È X, we can find an open U ⊆ X containing P such that the restriction of Y X to U is isomorphic with the projection U ¢ κ Z U . 9.9. Lemma. If f : Y X is a locally trivial fibration of κ-schemes with fiber Z, then Proof. By definition and compactness, there exists a finite open covering for i 1, . . . , n. In fact, for any non-empty subset I ⊆ Ø1, . . . , nÙ, we have an isomorphism f ¡1 ÔX I Õ X I ¢ κ Z, and hence, after taking classes in GrÔ×Ù Ë κ Õ, we get Öf ¡1 ÔX I Õ× ÖX I × ¤ ÖZ×. Since the f ¡1 ÔX i Õ form an open affine covering of Y and pre-images commute with intersection, a double application of Lemma 6.5 yields , where l and l are the respective lengths of z and z. In particular, Proof. Let R and R be the Artinian local coordinate rings of z and z respectively. Since arcs can be calculated locally (see the discussion following Lemma 6.4), we may assume X is the (affine) closed subscheme of A m κ with ideal of definition Ôf 1 , . . . , f s ÕκÖx×. Since the composition of locally trivial fibrations is again a locally trivial fibration, with general fiber the product of the fibers, we may reduce to the case that R RßαR with α an element in the socle of R, that is to say, such that αm 0, where m is the maximal ideal of R. Let ∆ be a basis of R as in Remark 9.2, with α l¡1 α (since α is a socle element, such a basis always exists). In particular, ∆ ¡ ØαÙ is a basis of R. We will use these bases to calculate both arc maps. To calculate a general fiber of the map s : ∇ z X ∇ zX , fix a fat point w with coordinate ring S, and a w-rational point b : w ∇ zX , given by a tuple ũ over S. The fiber sÔwÕ ¡1 Ô bÕ, is equal to the fiber of XÔzwÕ XÔzwÕ above ā, where ā : zw X is the zw-rational point corresponding to b, that is to say, the composition zw z ¢ ∇ zX X given by Theorem 8.8 (see Corollary 8.9). Being a rational point, ā corresponds therefore to a solution u in R κ S of the equations f 1 ¤ ¤ ¤ f s 0, where the relation with the tuple ũ is given by equation (29). Let x be the center of ā, that is to say, the closed point given as the image of ā under the canonical map XÔzwÕ XÔκÕ. Since X is smooth at x, the Jacobian Ôs ¢ nÕ-matrix Jac X : Ô f i ß x j Õ has rank m ¡ d at x. Replacing X by an affine local neighborhood of x and rearranging the variables if necessary, we may assume that the first Ôm ¡ dÕ ¢ Ôm ¡ dÕ-minor in Jac X is invertible on X. The surjection R R induces a surjection R κ S R κ S. The fiber above ā is therefore defined by the equations f j Ôu xl¡1 αÕ 0, for j 1, . . . , s. By Taylor expansion, this becomes (41) In fact, since u ũ0 mod mÔR κ SÕ and αm 0, we may replace each f j ß x i ÔuÕ in (41) by f j ß x i Ôũ 0 Õ. Hence, the fiber above ā is is the linear subspace of ÔR SÕ m defined as the kernel of the Jacobian Jac X Ôũ 0 Õ. In view of the shape of the Jacobian of X, we can find g ij È κÖx× such that for all i m ¡ d, by Kramer's rule. Therefore, viewing the parameter ũ0 as varying over XÔwÕ, the fiber of sÔwÕ is the constant space A d κ , as we needed to show. Applying this to ∇ z X X, (note that X ∇ κ X) we get a locally trivial fibration with fiber equal to A dÔl¡1Õ κ , so that the last assertion follows from Lemma 9.9. Calculations, like for instance Example 9.6, suggest that even for certain non-reduced schemes, there might be an underlying locally trivial fibration. Based on these examples, I would venture the following conjecture (here we write X red for the underlying reduced variety of a scheme X): 9.11. Question. Let z be a fat point of length l and X a d-dimensional κ-scheme. If the reduction of X is smooth, when is the induced reduction map Ô∇ z XÕ red X red a locally trivial fibration with fiber A m κ , for some m. 9.12. Remark. As we shall see in Example 10.2 below, m can be different from dÔl¡1Õ, the value that we get in the reduced case. In many cases, the answer seems to be affirmative, but there are exceptions, see Example 9.13 below. Moreover, as can be seen from Table (1) below, taking arcs does not commute with reduction, that is to say, Ô∇ z XÕ red is in general not equal to the arc space ∇ z ÔX red Õ of the reduction of X, nor even to the reduction of the latter arc space. 9.13. Example. The simplest instance to which Question 9.11 applies is when X itself is a fat point x. The expectation then is that (42) for some m (for expected values, see Example 10.2 below). Example 9.6 provides instances in which (42) holds. However, the following is a counterexample: let z : where C is the cuspidal curve with equation ξ 2 ¡ ζ 3 0 and O the origin, its unique singularity. Let us calculate its auto-arcs ∇ z z. As the monomials in ξ and ζ of degree at most two together with ξζ 2 form a basis of the coordinate ring R of z, its length is 7 and the generic arcs are Since the arc scheme ∇ z z lies above the origin, its reduction lies in the subvariety of A 14 κ defined by x0 ỹ0 0, and hence, we may put these two to zero in the generic arcs and work inside the affine space A 12 κ given by the remaining arc variables. From the fact that ξ 3 0 in R, the arc scheme is contained in the closed subscheme of A 12 κ by the coefficients of the expansion of 2 vanishes, the reduction lies in the subvariety given by x2 0, and so we may again put this variable equal to zero and work in the corresponding 11-dimensional affine space. The remaining equations come from the expansion of showing that the reduced arc space is the singular variety with equations x2 Note that the latter can be viewed as the tangent bundle of the cusp. More precisely, instead of the anticipated (42), we obtain the following modified form of the auto-arc variety κ , a singular 9-dimensional variety. However, I could not find such a form for values higher than 4. Locally constructible sieves. We say that a sieve X on a κ-scheme X is locally constructible, if XÔzÕ is constructible in XÔzÕ, for each fat point z, by which we mean that ∇ z XÔκÕ is constructible in the Zariski topology on the variety ∇ z XÔκÕ viewed as the space of closed points of ∇ z X. 9.14. Proposition. Any formal motif is locally constructible. Proof. This follows from Chevalley's theorem and Theorem 9.7 in case X is sub-schemic, since, for a morphism ϕ : Y X of κ-schemes, ImÔϕÕÔzÕ, as a subset of ∇ z X, is the image of the map ∇ z Y ÔκÕ ∇ z XÔκÕ. The formal case then follows from this, since there exists a sub-schemic motif Y ⊆ X such that YÔzÕ XÔzÕ. In this section, we assume κ is an algebraically closed field. The dimension of an arc scheme ∇ z X is a subtle invariant depending on z and X, and not just on their respective length l and dimension d; see Table (1) below. The underlying cause for this phenomenon is the fact that taking reduction does not commute with taking arcs. To exemplify this behavior, we list, for small lengths, some defining equations of arcs and their reductions for three different closed subschemes X with the same underlying one-dimensional variety, the union of two lines in the plane. Here l l denotes the closed point with coordinate ring κÖξ×ßξ l κÖξ×, that is to say, the l-th germ of the origin on the affine line. TABLE 1. Dimension δ and equations of arcs and their reductions. As substantiated by the data in this table, we have the following general estimate: 10.1. Lemma. The dimension of ∇ z X is at least dl, where d is the dimension of X and l the length of z. If X is a variety, then this is an equality. Proof. Assume first that X is an irreducible variety, so that it contains a dense open subset U which is non-singular. By Theorem 9.3, the pull-back Moreover, by Theorem 9.10 the dimension of ∇ z U is equal to dl, whence, by density, so is that of ∇ z X. If X is only reduced, then we may repeat this argument on an irreducible component of X, and using once more the openness of arcs, conclude that ∇ z X has dimension dl. For X arbitrary, let V : X red be the variety underlying X. The closed immersion V ⊆ X yields a closed immersion ∇ z V ⊆ ∇ z X by Corollary 8.9. The result now follows from the reduced case applied to V . We will call the difference dimÔ∇ z XÕ ¡dl the defect of X at z. Varieties therefore have no defect. The bound given by Lemma 10.1 is far from optimal, as can be seen by taking the arc scheme of a fat point (see, for instance, Example 9.6). The growth of the dimension of auto-arcs (see Example 9.13), that is to say, the function δÔzÕ : dimÔ∇ z zÕ for z a fat point, is still quite puzzling. By Example 9.6, we have δÔl n Õ n ¡ 1. However, the next example shows that δÔzÕ can be bigger than ℓÔzÕ. x : be the generic arcs, so that ∇ on o n is the closed subscheme of A on κ given by the coefficients of the monomials Ù x i Ù y n¡i , for i 0, . . . , , n. Since the arc scheme ∇ on o n lies above the origin, its defining equations contain the ideal Ôx 00 , ỹ00 Õ n . To calculate its dimension, we may take its reduction, which means that we may put x00 and ỹ00 equal to zero in (43). However, any monomial of degree n in the generic arcs is then identical zero, showing that the reduction of the arc scheme is given by x00 ỹ00 0, and hence, its dimension is equal to One might be tempted to propose therefore that δÔzÕ is equal to the embedding dimension of z times its length minus one, but the next example disproves this. Namely, without proof, we state that δÔzÕ 7 for z the fat point in the plane with equations ξ 2 ξζ 2 ζ 3 0 (note that z has length 5 and embedding dimension 2, so that the expected value would be 2 ¢ 4 8). Note that the auto-arc space ∇ z z is often, but not always an affine space (see Question 9.11 and the example following it). It seems plausible that δÔJ n P Y Õ grows as a polynomial in n of degree d, for any ddimensional closed germ ÔY, P Õ. In particular, we expect the limit eÔY, P Õ : lim to exist. For instance, an easy extension of the above examples yields eÔA m κ , OÕ m. In view of Question 9.11, we would even expect that the auto-Igusa-zeta series is rational over the localization of the classical Grothendieck ring with respect to L, for any d-dimensional closed germ ÔY, P Õ, a phenomenon that we will study in §14 below under the name of motivic rationality (and where we also explain the choice of power of L). What about its motivic rationality over the localization GrÔ ÓÖÑ κ Õ L of the formal Grothendieck ring? Dimension of a motif. Given a formal motif X on a κ-scheme X, we define its dimension as the dimension of XÔκÕ. This is well-defined since XÔκÕ is a constructible subset of XÔκÕ by Proposition 9.14. If X X ¥ is representable, then its dimension is precisely the dimension of the scheme X. On the other hand, if X is the formal completion of X at a closed point, then X has dimension zero, whereas its global section ring has dimension equal to that of X at P by Corollary 7.6. Proof. Since dimension is determined by the κ-rational points, we may take, using Theorem 7.4, the image of this common class in GrÔÎ Ö κ Õ, where the result is known to hold. As we will work over G : GrÔ ÓÖÑ κ Õ L below, we extend the notion of dimension into an integer valued invariant on this localized Grothendieck ring by defining the dimension of ÖX× ¤ L ¡i to be dimÔXÕ ¡ i, for any formal motif X and any i È Z. In particular, if X has dimension d and z length l, then Ö∇ z X× ¤ L ¡dl has positive dimension, which is the reason behind the introduction of this power of the Lefschetz class in the formulas below. This also gives us the Kontsevich filtration by dimension on G. Namely, for each m È N, let Γ m ÔGÕ be the subgroup generated by all classes ÖX× ¤ L ¡i of dimension at most ¡m. This is a descending filtration and the completion of G with respect to this filtration will be denoted Ô G. However, since we define motivic filtration locally (see §15 below), we will not make use of it. We continue with the setup from §9: let j z : z Spec κ be the structure morphism of a fat point z over an algebraically closed field κ. Instead of looking at the double adjunction giving rise to the arc functor ∇ z , we consider here the diminution part only, that is to say, the right adjoint ∇ j ¦ z satisfying for each z-sieve Y on a z-scheme Y and each κ-fat point w, an isomorphism where this time, we have to view j ¦ z w zw as a z-fat point. By Theorem 8.8, we associate in particular to any z-scheme Y , a κ-scheme by Corollary 8.9. Apart from j z , we also have the residue field morphism π z : Spec κ z. To a z-scheme Y , we can therefore also associate the base change Ȳ : π ¦ z Y , called the closed fiber of Y . We can think of Y as a fat deformation of j ¦ z Ȳ . Indeed, since κ ¢ z κ κ, any κ-rational point of Y is also a κ-rational point on Ȳ , that is to say, Y ÔκÕ Ȳ ÔκÕ j ¦ z Ȳ ÔκÕ, showing that Y and j ¦ z Ȳ have the same underlying variety. 11.1. Example. For instance, if C is the curve x 2 ¡y 3 and l n the fat point with coordinate ring R n : κÖξ×ßξ n κÖξ×, then the l 3 -scheme X with coordinate ring R 3 Öx, y×ßÔx 2 ¡ y 3 ¡ ξ 2 ÕR 3 Öx, y× has closed fiber C, and XÔκÕ CÔκÕ. Note however that XÔl 3 Õ CÔl 3 Õ. In fact, truncation yields a map XÔl 3 Õ CÔl 2 Õ. Hence, by (44), we may likewise think of ∇ j ¦ z Y as a fat deformation of the arc space ∇ z Ȳ of its closed fiber, justifying the term deformed arc space for ∇ j ¦ z Y . This construction is compatible then with specializations in the following sense. Fix a κ-scheme Z. The base change j Z : Z ¢ z Z is again a finite, flat homomorphism satisfying condition ( ), thus allowing us to consider the diminution ∇ j ¦ Z , associating to any Z ¢ z-scheme Y , a Zscheme ∇ j ¦ Z Y , called the relative arc scheme of Y . The deformed arc space is then given by the special case when Z Spec κ. 11.2. Proposition. Let z be a κ-fat point and Z a κ-scheme. For every Z ¢ z-scheme Y , viewed as a family over Z in the sense of §8.15, and for any κ-rational point a on Z, we have an isomorphism So, returning to Example 11.1, let X ⊆ A 3 l3 be the hypersurface with equation x 2 ¡ y 3 ¡ zξ 2 . As a family over A 1 l3 via projection on the last coordinate, its specializations X a are all isomorphic if a 0, whereas the special fiber X 0 C ¢ l 3 . The corresponding relative arc scheme X is given by Its specializations are again all isomorphic (to the third order Milnor fiber; see below) whereas the special fiber is isomorphic to ∇ l3 C. The closed subscheme relation defines a partial order relation on Ø κ , that is to say, we say that z z if and only if z is a closed subscheme of z (and not just isomorphic to one). We already discussed direct limits with respect to the induced ordering on disjoint unions of fat points in Lemma 2.8. Here we will investigate more closely the direct limit of fat points themselves. We will assume that such a direct system contains a least element. It follows that all fat points in the system must have the same center (to wit, the center of the least element). In other words, any fat point in the directed system has the same underlying closed point, and so we will call such a system a point system. We want to adjoin to the category of fat points its direct limits, but the problem is that the category of schemes is not closed under direct limits either. However, the category of locally ringed spaces is: if ÔX i , O Xi Õ form a direct system, then their direct limit is the topological space X : lim X i endowed with the structure sheaf O X : lim O Xi . Since we will assume that all fat points have the same underlying topological space, namely a single point, the construction simplifies: the direct limit of a point system is simply the one-point space with its unique stalk given as the inverse limit of all the coordinate rings of the fat points in the system. As already indicated in Footnote (6), a morphism in this setup will mean a morphism of locally ringed spaces with values in the category of κ-algebras. For example, if R is any κ-algebra and o the locally ringed space with underlying set the origin and (unique) stalk R, and if X Spec A is any affine scheme, then Mor κ Ôo, XÕ is in one-one correspondence with the set of κ-algebra homomorphisms Hom κ¡alg ÔA, RÕ. Let ⊆ Ø κ be a point system. Its direct limit lim , as a one-point locally ringed space, is called a limit point. Some examples of limit points are: (1) If is finite, the direct limit is just its maximum, whence a fat point. (2) Given a closed germ ÔY, P Õ, its formal completion Ô Y P is the direct limit of the jets J n P Y , whence a limit point. (3) The direct limit of all fat points with the same center is called the universal point and is denoted u κ , or just u. Any limit point admits a closed immersion into u. In particular, up to isomorphism, u does not depend on the underlying point. 12.1. Lemma. The stalk of the universal point u κ is isomorphic to the power series ring over κ in countably many indeterminates. Proof. Any fat point is a closed subscheme of some formal scheme Þ ÔA n κ Õ. Hence suffices to show that the inverse limit of the power series rings S n : κÖÖx 1 , . . . , x n ×× under the canonical projections S m S n given by modding out the variables x i for n i m is isomorphic to the power series ring κÖÖx×× in countably many indeterminates x Ôx 1 , x 2 , . . . Õ. To this end, let f n È S n be a compatible sequence in the inverse system. For each exponent ν Ôν 1 , ν 2 , . . . Õ in the direct sum N ÔNÕ of countably many copies of N, and each n, let a n,ν È κ be the coefficient of x ν : x ν1 1 ¤ ¤ ¤ x ν iÔνÕ iÔνÕ in f n , where iÔνÕ is the largest index for which ν i is non-zero. Compatibility means that there exists for each ν an element a ν È κ such that a ν a n,ν for all n iÔνÕ. Hence f : ν a ν x ν È κÖÖx×× is the limit of the sequence f n , proving the claim. 12.2. Remark. I would guess that, similarly, the direct limit of all disjoint unions of fat points is isomorphic to the affine space A ω κ of countable dimension. To make the limit points into a category, denoted Ý Ø κ , take morphisms to be direct limits of morphisms of fat points. More precisely, given point systems , ⊆ Ø κ with respective direct limits x and y, then a morphism (of locally ringed spaces) ϕ : x y is a morphism of limit points if for each z È there exists a v È such that ϕÔzÕ ⊆ v, or, dually, if the induced morphism lim O v lim O z has the property that for each z È , we can find a v È such that this morphism factors through O v O z . In this way, the category Ý Ø κ of limit points is an extension of the category Ø κ of fat points, which in a sense acts as its compactification. In particular, any limit point x admits a canonical structure morphism j x : x Spec κ. We also extend the partial order relation on Ø κ to one on Ý Ø κ as follows. Firstly, we say that z x for z a fat point and x lim a limit point, if z v for some fat point v È . It follows that there is a canonical embedding z ⊆ x which induces a surjection on the stalks, and which we therefore call a closed embedding in analogy with the scheme-theoretic concept. We then say for a limit point y lim that y x if for every z È we have z x. It follows that we have a canonical morphism of limit points y x which is again surjective on their stalks, and hence can rightly be called once more a closed embedding. One checks that this defines indeed a partial order on limit points extending the one on fat points. We call a limit point x bounded if it is the direct limit of fat points of embedding dimension at most n, for some n. Formal completions of closed germs are examples of bounded limit points, whereas u clearly is not. In fact, any bounded limit point arises in a similar, analytical way: 12.3. Proposition. The bounded limit points are in one-one correspondence with analytic germs. More precisely, the stalks of bounded limit points are precisely the complete Noetherian local rings with residue field κ. Proof. Let x be a bounded limit point, say, realized as the direct limit of fat points z ⊆ A n κ centered at the origin, for some fixed n. Let κÖx×ßa z be the coordinate ring of z, so that a z ⊆ κÖx× is m-primary, where m is the maximal ideal generated by the variables. Let I be the intersection of all a z κÖÖx××. I claim that x has stalk equal to S : κÖÖx××ßI. Indeed, by a theorem of Chevalley ([12, Exercise 8.7]), there exists for each i some ideal of definition a of one of the fat points in the direct system such that aS ⊆ m i S. In particular, the inverse limit is simply the m-adic completion of S, which is of course S itself. The converse is also obvious: given a complete Noetherian ring ÔS, mÕ with residue field κ, then by Cohen's structure theorem, it is of the form κÖÖx××ßI for some ideal I. One easily checks that it is the coordinate ring of the direct limit of the corresponding jets SpecÔSßm n Õ. Any limit point x lim defines a presheaf x ¥ by assigning to a fat point z the set of morphisms Mor Ý Øκ Ôz, xÕ. 12.4. Corollary. The presheaf x ¥ defined by a limit point x lim is the inverse limit of the representable functors v ¥ for v È . If x is moreover bounded, then x ¥ is a formal motif. Proof. Given a fat point z Spec R, we have to show that vÔzÕ for v È forms an inverse system with inverse limit equal to Mor Ý Øκ Ôz, xÕ. Let ÔS, mÕ be the stalk of x, that is to say, the inverse limit of the coordinate rings of the fat points belonging to . The first statement is immediate by functoriality, and for the second, note that since R has finite length, (46) More precisely, any κ-algebra homomorphism a : S R factors through Sßm l R, for l ℓÔRÕ. Moreover, if n is the embedding dimension of R, then there exists a complete, Noetherian residue ring S of S of embedding dimension at most n such that a factor as S Sßm l S R. By the same argument as in the proof of Proposition 12.3, there is some v Spec T È such that T Sßm l S, showing that a is already induced by the morphism z v. In fact, if x is bounded, then we may choose v independent from a, showing that x ¥ ÔzÕ vÔzÕ. Since all v È embed in the same affine space, x ¥ is a locally schemic sieve on this space, whence a formal motif. 12.5. Remark. The identification (46) shows that x ¥ is pro-representable in the sense of Footnote (6). Let x lim be a limit point. Given a presheaf X, the collection of all XÔvÕ for v È is an inverse system of sets given by the maps i v,w : XÔwÕ XÔvÕ if v w in , where i v,w is induced by the embedding v ⊆ w. We denote the inverse limit of this system simply by XÔxÕ. It follows from the definition of morphisms of limit points that X becomes a functor on the category of limit points. In other words, any presheaf on Ø κ extends to a presheaf on Ý Ø κ ; this principle will simply be called continuity. Since inverse limits commute with presheafs, one easily verifies that if s : X Y is a morphism of presheafs (=natural transformation), then for any limit point x, this induces a map XÔxÕ YÔxÕ, showing that extension by continuity is functorial. The extension of a (pro-)representable functor on Ø κ to Ý Ø κ will again be called (pro-)representable. If is a point system in Ø κ with limit x, and if z is any fat point with canonical morphism j z : z Spec κ, then the base change j ¦ z consisting of all zv for v È is again a point system, whose limit we simply denote by zx (the reader can check that this defines a product in the category Ý Ø κ ). Repeating this argument on the first factor then shows that we may even multiply any two limit points. However, this multiplication does no longer behave as well as before. For instance, since the base change j ¦ z Ô Ø κ Õ by any fat point z is equal to the whole category Ø κ , we get zu u. Spec κ be the structure morphism of the limit point x. Strictly speaking, as this is only a direct limit of finite, flat morphisms, the theory of diminution does not apply, and neither that of augmentation. Nonetheless, without going into details, one could develop the theory under this weaker condition, although we will only give an ad hoc argument in the case we need it. So, given a sieve X on Ý Ø κ , we define ∇ x X : ∇ j ¦ x ∇ ÔjxÕ¦ X at a limit point y as the set XÔxyÕ, where we view xy again as a limit point (over κ). 12.6. Lemma. For any limit point x and any κ-scheme X, the base change x ¦ X ¥ is pro- representable, by the so-called arc scheme ∇ x X along x. Proof. Let x be the direct limit of the directed subset ⊆ Ø κ . Suppose first that X is affine. Since the ∇ w X for w È form an inverse system of affine schemes, their inverse limit is a well-defined affine scheme X with coordinate ring the direct limit of the coordinate rings of the arc schemes along fat points in . By continuity, it suffices to verify that ∇ x X ¥ X¥ on Ø κ . To this end, fix a fat point z. From where we used the universal property of inverse limits in the third line, the claim now follows. The general case follows from this by the open nature of arc schemes (Theorem 9.3) and the fact that if X admits an open affine covering of cardinality N , then so does any arc scheme ∇ z X by base change. Y be a formal completion, viewed as the limit point of the germs J n O Y , and let X be a κ-scheme. By Lemma 12.6, we have an associated arc scheme ∇ Ô Y X. For each n, we have a canonical map ∇ Ô Y X ∇ J n O Y X, which in general is not surjective (it is so, by Theorem 9.10, when X is smooth). To study this image, we make the following definitions. Given a closed embedding v ⊆ w, the image sieve given by the canonical map ∇ w X ∇ v X is called the sieve of w-extendable arcs on X along v, and will be denoted ∇ wßv X. By construction, it is sub-schemic. Let ∇ n X : ∇ J n O Y X, and ∇ mßn X : ∇ n X is not of finite type, the corresponding image sieve, denoted ∇ Ô Y ßn X and called the n-th order Ô Y -extendable arcs on X, may fail to be sub-schemic. We do have: 13.1. Theorem. For each κ-scheme X, for each formal completion Ô Y of a closed subscheme Y ⊆ A n κ at a point, and for each n, the n-th order extendable arcs on X along this formal completion, ∇ Ô Y ßn X, is a formal motif. Proof. Without loss of generality, we may assume that Ô Y is the completion of Y at the origin. It is clear that ∇ Ô Y ßn X is the intersection of all ∇ mßn X, for m n. To show that is a formal motif, it suffices to show that its complement is locally sub-schemic. Since each ∇ mßn X is sub-schemic, this will follow if we can show that for each fat point z, there is some m z such that ∇ Ô Y ßn XÔzÕ ∇ mzßn XÔzÕ. Recall that for ÔR, mÕ a quotient of a power series ring κÖÖξ×× modulo an ideal generated by polynomials, we have uniform strong Artin Approximation, in the sense that for any polynomial system of equations f 1 ¤ ¤ ¤ f s 0 and every n, there exists some N , such any solution of f 1 ¤ ¤ ¤ f s 0 in Rßm N is congruent modulo m n to a solution in R: see for instance [15,Theorem 7.1.10], where the proof is only given for the power series ring itself, but immediately generalizes to any quotient by a polynomial ideal, whence in particular to the stalk of the formal completion Ô Y . This means that ∇ Ô Y ßn XÔκÕ ∇ mßn XÔκÕ, for some m n. To obtain a similar identity over an arbitrary fat point z, we apply the same result but replacing X by the arc scheme ∇ z X, yielding the existence of a m z n such that as required. 13.2. Remark. Since we may no longer have the required strong Artin Approximation estimate, I do not know whether this result generalizes to arbitrary limit points, that is to say, is ∇ yßx X a formal motif, for limit points x y. The first case to look at is when x is a fat point and y is bounded (but not a formal completion). Although we can work in greater generality, we will assume once more that our base scheme is an algebraically closed field κ. Motivic Igusa-zeta series. For any κ-scheme X and any closed germ ÔY, P Õ, we define the motivic Igusa-zeta series of X along the germ ÔY, P Õ as the formal power series Igu ÔY,PÕ in GrÔ ÓÖÑ κ Õ L ÖÖt××, where d is the dimension of X and j n P ÔY Õ the length of the n-th jet J n P Y (which is also equal to the Hilbert-Samuel function of O Y,P for large n). This definition generalizes the one in [1] or [3, §4], when we take the germ of a point on a line. X mot of a κ-scheme X along an arbitrary closed germ ÔY, P Õ is rational over GrÔ ÓÖÑ κ Õ L . More generally, given any formal motif X on a κ-scheme X, we define its Igusa-zeta series along the germ ÔY, P Õ as the formal power series In particular, whenever Question 9.11 holds affirmatively, the image of the motivic Igusa zeta series of the fat point would be rational over the classical Grothendieck ring. Skipping the easy calculations, we have for instance that where z is the fat point with coordinate ring κÖx, y×ßÔx 2 , y 2 Õ. Interestingly, this is also the Igusa zeta-series of the fat point with coordinate ring κÖx, y×ßÔx 2 , xy, y 2 Õ. Motivic Hilbert series. Given a motivic site Å, we let Å 0 be its restriction to the sub- category of zero-dimensional schemes, that is to say, the union of all Å Z , where Z runs over all zero-dimensional κ-schemes. As the product of two zero-dimensional schemes is again zero-dimensional, Å 0 is a partial motivic site, and hence has an associated Grothen- dieck ring Gr 0 ÔÅÕ : GrÔÅ 0 Õ, called the Grothendieck ring of Å in dimension zero. There is a natural homomorphism Gr 0 ÔÅÕ GrÔÅÕ, which in general will fail to be injective, as there are a priori more relations in the latter Grothendieck ring. In particular, applied to (sub-)schemic or formal motives, we get the corresponding Grothendieck rings in dimension zero Gr 0 ÔË κ Õ, Gr 0 Ô×Ù Ë κ Õ, and Gr 0 Ô ÓÖÑ κ Õ. Motivic Hilbert-Kunz series. Assume that κ has characteristic p. Recall that for a given closed subscheme Y ⊆ X, we defined in §8 its Frobenius transform in X as the pull-back We may also take the pull-back with respect to the powers F n X of the Frobenius, yielding the n-th Frobenius transform F n¦ X Y . If Y has dimension zero, then so does any of its Frobenius transforms, and so the following series, called the motivic Hilbert-Kunz series, ÖF n¦ X Y × t n is a well-defined series in Gr 0 ÔË κ ÕÖÖt××. Taking the length function ℓ yields the classical Hilbert-Kunz series, of which not too much is known (one expects it to be rational). Of course, we could also take Y to be of higher dimension, and get the corresponding motivic Hilbert-Kunz series in GrÔË κ ÕÖÖt××. P as a formal motif by means of its jets. However, this is not the only way to locally approximate it with schemic subsieves. Given a system of parameters ξ 1 , . . . , ξ e in O Y,P (that is to say, a tuple of length e dimÔO Y,P Õ generating an ideal primary to the maximal ideal), let y n be the fat point with coordinate ring B n : O Y,P ßÔξ n 1 , . . . , ξ n e ÕO Y,P , and j yn : y n Spec κ the canonical morphism. The reader can check that given a fat point z, there exists some n such that y n ÔzÕ Ô Y P ÔzÕ, that is to say, Ô Y P is the limit point corresponding to the direct system Øy n Ù n (see §12). Recall that by the Monomial Theorem, the element Ôξ 1 ¤ ¤ ¤ ξ e Õ n¡1 is a non-zero element in the socle of B n (meaning that the ideal it generates has length one). Let X ⊆ A W n Õ a ∇ yn X if a is the zero section; M n ÔXÕ otherwise. We define the associated Milnor series Mil ÔY,PÕ L ¡dℓÔynÕ ÖM n ÔXÕ×t n as a power series over GrÔ ÓÖÑ κ Õ L . When ÔY, P Õ is the germ of a point on a line, we get the schemic variant of the series introduced by Denef-Loeser et al., and by (48), this series can be viewed as a deformation of the motivic Igusa-zeta series. Therefore, in view of Conjecture 14.2, we expect the motivic Milnor series also to be rational, and in fact, as a rational function, to have degree zero. Assuming this to be true, we can calculate the limit of this series when t , and this conjectural limit, presumably in GrÔ ÓÖÑ κ Õ L , will be called the motivic Milnor fiber of X along the closed germ ÔY, P Õ. Motivic Hasse-Weil series. Another important generating series in algebraic geometry whose rationality-proven by Dwork in [4]-is postulated to be motivic, is the Hasse-Weil series of a scheme over a finite field F q : its general coefficient is the number of rational points over the finite extensions F q n . To turn this into an abstract counting principle, we use the inversion formula relating the number of degree n effective zero cycles on X to the number of rational points in an extension of degree n, and observe that the former cycles are in one-one correspondence with the rational points on the n-fold symmetric product X ÔnÕ of X (given as the quotient of X n modulo the action of the symmetric group on n-tuples). Therefore, following Kapranov [8], we propose the following motivic variant, the Motivic Hasse-Weil series: as a power series over GrÔ ÓÖÑ κ Õ L . Kapranov himself proved rationality of the image of this series over GrÔÎ Ö κ Õ L , as well as a functional equation, for certain smooth, projective irreducible curves, but the general case is still open. We know from work of Larsen and Lunts on smooth surfaces ( [10]), that, in general, this cannot hold over the Grothendieck ring itself: in [11], they show that rationality over the Grothendieck ring is equivalent with the complex surface having negative Kodaira dimension. It is therefore natural to conjecture the same properties for our motivic variant HW mot X . Motivic Poincare series. Given a closed germ ÔY, P Õ with formal completion Ô Y , viewed as a limit point, and a κ-scheme X, by Theorem 13.1, we can now define the motivic Poincaré series of X along ÔY, P Õ as the formal series Poin ÔY,PÕ where d is the dimension of X. Denef and Loeser proved in [2] that along the germ of a point on the line, the image of this series in the localized classical Grothendieck ring is rational, provided κ has characteristic zero. It is therefore natural to postulate: 14.6. Conjecture. For any closed germ ÔY, P Õ and any κ-scheme X, the associated motivic Poincaré series Poin ÔY,PÕ X mot is rational over GrÔ ÓÖÑ κ Õ L . Given X and a formal completion Ô Y , we may ask for each n, which are the fat points z containing the n-th jet j n : J n O Y such that ∇ Ô Y ßjn X ⊆ ∇ zßjn X, that is to say, when are Ô Y -extendable arcs also z-extendable? For instance, if Ô Y is the completion of the affine line, then by Theorem 9.10, we can extend along any jet of a non-singular germ ÔW, OÕ, since there exist closed immersions j n ⊆ J n O W ⊆ j d n , where d is the dimension of ÔW, OÕ. However, I do not know whether we can extend along the fat point given by, say, x 4 y 4 x 3 ¡ y 2 0. For which schemes X can every Ô Y -extendable arc be extended along any fat point? This is true if X is smooth, but are there any other cases? Unlike the Kontsevitch-Denef-Loeser motivic integration, we will only define integration on the (truncated) arc schemes. We will work over the localized Grothendieck ring G : GrÔ ÓÖÑ κ Õ L , for κ an algebraically closed field. Before we develop the theory, we discuss a naive approach. Motivic measure. We fix a fat point z. Our goal is to define a motivic measure µ z on formal motives. To this end, we define µ z ÔXÕ : Ö∇ z X× in G. In particular, this measure does not depend on the ambient space of X, only on its germ. Using Theorem 9.7, we can extend the motivic measure to an endomorphism on G. We would like to normalize this measure, with the ultimate goal-which, however, we do not discuss in this paper-to make the comparison between different fat points and take limits. One way to normalize is to make the value weightless (in the sense of dimension), by μz ÔXÕ : Ö∇ z X× L dimÔ∇zXÕ This, of course, is no longer additive. Below, however, we will normalize differently, by fixing an ambient space. Following integration theory practice, we would like to say that ÷ 1 X d z x : µ z ÔXÕ : Ö∇ z X× and extend this to arbitrary step functions. Here, a step function would be a formal, finite sum s g i 1 Xi with g i È G and X i formal motives. However, how to interpret this as a function? As usual, we should do this at each fat point w, and interpret 1 X ÔwÕ as the characteristic function on XÔwÕ of XÔwÕ, where X is an ambient space of X. Likewise, provided X is an ambient space for all X i , we let sÔwÕ be the function XÔwÕ G associating to a w-rational point a È XÔwÕ the sum of all g i for which a È X i ÔwÕ. However, the main obstruction is that this point-wise defined function is in general not functorial. The reason is the non-functorial nature of fibers, which in turn stems from the lack of complements in categories-note that the complement of any fiber is the union of the other fibers. To recover functoriality, we work over a subcategory of fat points: Flat and split points. More precisely, let Ø flat κ and Ø split κ be the respective categories of flat points and split points over κ, whose objects are fat points over κ and whose morphisms are respectively flat and split epimorphisms. Recall that a morphism ϕ : Y X is called a split epimorphism if there exists a morphism, also called a section, σ : X Y such that ϕσ is the identity on X. Any split epimorphism is (faithfully) flat, so that Ø split κ is a subcategory of Ø flat κ . Note that each structure morphism z Spec κ is a split epimorphism, and by base change, so is each projection map zw w. We will call a contravariant functor X from Ø flat κ (respectively, from Ø split κ ) to the category of sets a flat (respectively, a split) presheaf. If, moreover, we have an inclusion morphism X ⊆ X ¥ , where X ¥ is the restriction of the representable functor of a κ-scheme X, we call X a flat (respectively, a split) sieve. In particular, ordinary presheafs or sieves (that is to say, defined on Ø κ ) when restricted to Ø split κ are split-and to emphasize this, we may call them full sieves-, but as the next result shows, not every split sieve is the restriction of a full sieve: 15.1. Proposition. The complement of a schemic motif X ⊆ X ¥ is a flat sieve. The com- plement of a formal motif X ⊆ X ¥ is a split sieve. Proof. Any (full) schemic motif is the union of closed subsieves, and hence its complement is the intersection of complements of closed subsieves. Since the intersection of (flat or split) sieves is again a sieve, we only need to verify that the complement of a single closed subsieve Y ¥ ⊆ X ¥ is a flat sieve. The only thing to show is functoriality, so let w z be a flat morphism of fat points. We have to show that under the induced map XÔzÕ XÔwÕ any z-rational point a not in Y ÔzÕ is mapped to a point not in Y ÔwÕ. Since fat points are affine, we may replace X by an affine open, and so assume from the start that it is affine with coordinate ring A. Let I be the ideal defining Y , and let R and S be the coordinate rings of z and w respectively. The z-rational point a corresponds to a morphism A R; it does not belong to Y ÔzÕ if and only if the image IR of I under A R is non-zero. Suppose towards a contradiction that a È Y ÔwÕ, so that IS 0. Since R S is by assumption flat, whence faithfully flat, we must have IR IS R 0, contradiction. Assume next that X is a sub-schemic motif, that is to say, X ImÔϕÕ for some morphism ϕ : Y X. Let λ : w z be a split epimorphism of fat points and let a : z X be a z-rational point outside ImÔϕÕÔzÕ. We have to show that the image a ¥ λ of a in XÔwÕ showing that a lies in the image of ϕÔzÕ, contradiction. Lastly, assume that X is formal, so that there exists for each fat point z a sub-schemic motif Y z ⊆ X such that Y z ÔzÕ XÔzÕ. Let λ : w z be a split epimorphism. Since Y w ⊆ X, we have ¡XÔzÕ ⊆ ¡Y w ÔzÕ. By what we just proved, ¡Y w ÔzÕ, is sent under XÔλÕ : XÔzÕ XÔwÕ inside ¡Y w ÔwÕ, and by construction, the latter is equal to ¡XÔwÕ. A fortiori, ¡XÔzÕ is then sent inside ¡XÔwÕ, proving the assertion. 15.2. Remark. It is important to note that we may not apply this argument to an arbitrary split sieve, since a section of a split morphism is not split and hence does not induce a morphism on the rational points of the split sieve. The point in the above argument is that formal motives are presheafs on the full category of fat points, and hence any section does induce a map between their rational points. We call any Boolean combination of closed subsieves a flat-schemic motif. Our aim is to define motivic sites of schemic, sub-schemic and formal motives with respect to flat and/or split points, but without changing the corresponding Grothendieck ring. A priori, there might be more morphism, that is to say, natural transformations, in this restricted context, and to circumvent this issue, we only allow morphisms that extend to true motives. More precisely, we define the flat-schemic motivic site Ë flat κ , as the category with objects all flat-schemic motives, and with morphisms all natural transformations s : Y X of flat schemic motives which extend to a morphism of schemic motives in the sense that there are schemic motives X ⊆ X ½ and Y ⊆ Y ½ and a morphism of schemic motives s ½ : Y ½ X ½ whose restriction to Y is s. To not introduce unwanted isomorphisms, we moreover require that if s is injective, then so must its extension s ½ be. Likewise, we call any Boolean combination of sub-schemic (respectively, formal) motives a split-subschemic (respectively, a split-formal) motif, and we define the split-sub-schemic motivic site ×Ù Ë split κ (respectively, the split-formal motivic site ÓÖÑ split κ ), as the category with objects all split-sub-schemic (respectively, split-formal) motives, and as morphisms all natural transformations which extend to a morphism of sub-schemic (respectively, formal) motives, with injective morphisms extending to injective ones. All these sites satisfy the same properties as ordinary motivic sites, apart from being defined only over a restricted category, but have the additional property that their restriction to any scheme is a Boolean lattice. At any rate, we can define their corresponding Grothendieck rings. Moreover, for each fat point z, the arc sheaf of a flat schemic (respectively, split subschemic, or split formal) motif exists, and is again of that form. The induced action on the corresponding Grothendieck ring is equal to that of ∇ z Ô¤Õ. Proof. I will only give the argument for the case of most interest to us, the formal motives, and leave the remaining cases, with analogous proof, to the reader. Before we do this, let us first discuss briefly Boolean lattices. Let B be a Boolean lattice. Given a finite collection of subsets X 1 , . . . , X n È B, and an n-tuple ε Ôε 1 , . . . , ε n Õ with entries ¨1, let X ǫ be the subset given by the intersection of all X i with ε i 1 and all ¡X i with ε i ¡1. Then any element in the Boolean sublattice BÔX 1 , . . . , X n Õ of B generated by X 1 , . . . , X n is a disjoint union of the X ε . In particular, if all X i belong to a sublattice L ⊆ B, then any element in BÔX 1 , . . . , X n Õ is a disjoint union of sets of the form C ¡ D with D ⊆ C in L. We now define a map γ from the free Abelian group ZÖ ÓÖÑ split κ × to GrÔ ÓÖÑ κ Õ as follows. By the above argument, a typical element in ÓÖÑ split κ is a disjoint union of split formal motives of the form X ¡Y with Y ⊆ X (full) formal motives. We define its γ-value to be the element ÖX× ¡ ÖY×. This is well-defined, for if it is also equal to a difference of motives X ¡ Ỹ then one easily checks that X Ỹ X Y and X Ỹ X Y, so that ÖX× Ö Ỹ× Ö X× ÖY×. We extend this to disjoint sums by taking the sum of the disjoint components, and then extend by linearity, to the entire free Abelian group. It is not hard to verify that γ preserves all scissor relations. So we next check that it preserves also isomorphism relations. We may again reduce to an isomorphism of the form s : X ¡ Y X ¡ Ỹ with Y ⊆ X and Ỹ ⊆ X formal motives. By assumption, s extends to an injective morphism s ½ : X ½ X½ with X ½ and X½ formal motives. Upon replacing X and X ½ with their common intersection, we may assume that they are equal. Since s ½ is injective, it induces an isomorphism between X and its image, as well between Y and its image. Hence X ¡ Y s ½ ÔXÕ ¡ s ½ ÔYÕ and, since s ½ extends s, the latter must be equal to X ¡ Ỹ, yielding γÔX ¡ YÕ ÖX× ¡ ÖY× Ös ½ ÔXÕ× ¡ Ös ½ ÔYÕ× γÔ X ¡ ỸÕ as we wanted to show. Hence, γ induces a map GrÔ ÓÖÑ split κ Õ GrÔ ÓÖÑ κ Õ. By construction, it is surjective and the identity on GrÔ ÓÖÑ κ Õ (when viewing a full motif as a split motif), showing that it is a bijection. By construction it is also additive, and the reader readily verifies that it preserves products, thus showing that it is an isomorphism. For the last assertion, it suffices once more to verify this on a split formal motif of the form X ¡ Y and we set ∇ z ÔX ¡ YÕ : ∇ z X ¡ ∇ z Y Since the w-rational points of both sides are the same, to wit, XÔzwÕ ¡ YÔzwÕ, for any fat point w, this is well-defined, and the assertion then follows from the universal property of adjunction. 15.4. Remark. Consider the flat schemic motif l ¥ 3 ¡ l ¥ 2 . It has no κ-rational points, but is does have a l 3 -rational point, namely the identity morphism on l 3 . This example shows that the analogue of Lemma 7.3 does not hold for split formal motives. We do have: 15.5. Lemma. If X is a split formal motif and z a fat point such that ∇ z X is empty, then X too is empty. In particular, all arc maps are injective on each ambient space. Proof. We may again reduce to the case that X is of the form Y ¡ Z with Z ⊆ Y (full) formal motives. Let w be an arbitrary fat point. The closed immersion w ⊆ zw induces maps ZÔzwÕ ZÔwÕ and YÔzwÕ YÔwÕ. Since composing the closed immersion with the (split) projection zw w is the identity, the two above maps are surjective. Since ∇ z X is the empty motif, it has no w-rational points, that is to say, ZÔzwÕ YÔzwÕ. Surjectivity then yields that ZÔwÕ YÔwÕ, whence XÔwÕ . Since this holds for any fat point w, we see that X is the empty motif. To prove the last assertion, assume ∇ z X ∇ z Y for X, Y split formal motives on a scheme X. By what we just proved, X ¡ ÔX YÕ and Y ¡ ÔX YÕ are both empty, from which the claim now follows. From now on, we will work in the largest of these motivic sites, the category of splitformal motives ÓÖÑ split κ , and we view the class of any such motif as an element in the localized Grothendieck ring G : GrÔ ÓÖÑ κ Õ L . Let G be the constant presheaf with values in G, that is to say, the contravariant functor on the category of split points which associates to any fat point the set G and to any split epimorphism of fat points the identity on G. Given a morphism, that is to say, a natural transformation, s : X G, we define, for each g È G, the fiber s ¡1 ÔgÕ as the subfunctor of X given at each fat point z by the fiber sÔzÕ ¡1 ÔgÕ of sÔzÕ : XÔzÕ G at g. If both X and all fibers are split formal motifs, and s has only finitely many non-empty fibers, then we call s a formal invariant. 15.6. Corollary. The formal invariants on a split formal motif X form an algebra over G. Proof. Clearly, any multiple of a formal invariant by an element in G is again a formal invariant. Let s, t : X G be formal invariants. We have to show that s t and st are also formal invariants. Functoriality is easily verified, so we only need to show that the fibers are again split formal motives. Fix a fat point z, and an element g È G. A z-rational point a È XÔzÕ lies in Ôs tÕ ¡1 ÔgÕÔzÕ (respectively, in ÔstÕ ¡1 ÔgÕÔzÕ), if sÔzÕÔaÕ tÔzÕÔaÕ g (respectively, if sÔzÕÔaÕ ¤ tÔzÕÔaÕ g). Since sÔzÕ and tÔzÕ have finite image, their are only finitely many ways that g can be written as a sum p q (respectively, a product pq), with p in the image of sÔzÕ and q in the image of tÔzÕ. Hence, the rational point a lies in the intersection sÔzÕ ¡1 ÔpÕ ¨ tÔzÕ ¡1 ÔqÕ ¨, for one of these finitely many choices of p and q. Since a finite union of intersections of split formal motives is again split formal, the result follows. Motivic integrals. Let X be a κ-scheme, z a fat point, and s : X G a formal invariant with X a split formal motif on X. We define the (split) motivic integral of s on X along z as (50) ÷ s d z X : L ¡dl ô gÈG g ¤ Ö∇ z Ôs ¡1 ÔgÕÕ×, where d is the dimension of X and l the length of z. Note that the sum on the right hand side of (50) is finite by definition, so that s d z X is a well-defined element in G. At the reduced fat point, Spec κ, we drop the subscript in the measure, and so the this integral becomes ÷ s dX : L ¡d ô gÈG g ¤ Ös ¡1 ÔgÕ×. To a formal motif Y on X, we can associate two invariants. Firstly, the constant map, denoted again Y, which at each fat point is the constant map sending every rational point to ÖY×. Proof. Motivic integration is clearly preserved under multiplication by a constant g È G. To prove additivity, we may induct on the number of characteristic functions, and reduce to the case of a sum s h1 Z , that is to say, we have to prove (53) Let (51) be the fiber decomposition of s. Since the fiber decomposition of s h1 Z is then where Y is the union of the Y i , the left hand side of (53) is where d and l are respectively the dimension of X and the length of z. Grouping together the n 1 terms with coefficient h, and for each i, the two terms with coefficient g i , this sum becomes g i Ö∇ z Y i × hÖ∇ z Z×Õ, since ∇ z acts on the Grothendieck ring by Theorem 9.7, and since both Y i ¡ Z and Z ¡ Y are disjoint from Y i Z. However, this is just the right hand side of (53), and so we are done. Let s : X G be a formal invariant on a κ-scheme X. Given an open U ⊆ X, let s U denote the restriction of s to X U ¥ . It is easy to see that s U is a formal invariant on U . Let U 1 , . . . , U n be an open covering of X. For each non-empty subset I ⊆ Ø1, . . . , nÙ, let U I be the intersection of all U i with i È I. We have the following local formula for the motivic integral (here we call a scheme equidimensional if every non-empty open has the same dimension as the scheme): 15.8. Theorem. Let s : X G be a formal invariant on an equidimensional κ-scheme X, let z be a flat point, and let U 1 , . . . , U n be an open covering of X. Then we have an equality Since by assumption all non-empty U I have the same dimension as X (and, of course, the empty ones do not contribute), the result follows from (50). Relations among motivic series. Given an element α È Gr 0 ÔË κ Õ, we define ÷ s d α X : where α n 1 Öz 1 × ¤ ¤ ¤ n s Öz s × is the unique decomposition in classes of fat points given by Proposition 14.4. We then formally extend this over Gr 0 ÔË κ ÕÖÖt××, by treating t as a constant. In this sense, we get, for a closed germ ÔY, P Õ, and a κ-scheme X, the following identity of power series: Let Å be a motivic site over an algebraically closed field κ and let X be an κ-scheme. By assumption, Å X is a lattice, and so we can define its lattice group Λ X ÔÅÕ as the free Abelian group on Å-motives on X modulo the scissor relations ÜXÝ ÜYÝ ¡ ÜX YÝ ¡ ÜX YÝ for any two Å-motives X and Y on X. In other words, same definition as for the Grothendieck ring, but without the isomorphism relations. In particular, there is a natural linear map Λ X ÔÅÕ GrÔÅÕ. We will denote the class of a motif X again by ÖX×. For each n, consider the embedding Å X n Å X n 1 via the rule X X ¢ X ¥ . One verifies that this induces a well-defined linear map Λ n : Λ X n ÔÅÕ Λ n 1 : Λ X n 1 ÔÅÕ, where X n is the n-fold Cartesian power of X. Moreover, the Cartesian product defines a multiplication Λ m ¢Λ n Λ m n , for all m, n. Hence n Λ n is a graded ring, called the graded lattice ring of Å on X, and denoted Λ X ÔÅÕ. It admits a natural ring homomorphism into We can now state a combinatorial property of the split motivic integral: 16.1. Proposition. Over a κ-scheme X and a fat point z, we can define for each formal invariant s : X G on X and each g È Λ X Ô ÓÖÑ κ Õ, an integral g s d z X, such that if g is the class in Λ X Ô ÓÖÑ κ Õ of a formal motif Y on X, then Proof. By definition, g is a Z-linear combination of classes of formal motives on X, say, of the form g n 1 ÖY 1 × ¤ ¤ ¤ n s ÖY s ×. Define ÷ g s d z X : To show that this is well-defined, we have to verify this only for scissor relations, that is to say, we have to show that for Y, Y ½ formal motives on X. This is immediate from the easily proven identity of characteristic functions Using this, we can now show that the lattice rings are not very interesting invariants (and hence only by also taking isomorphism relations, do we get something significant): 16.2. Corollary. The natural map sending a formal motif on some Cartesian power of X to its class in Λ X Ô ÓÖÑ κ Õ is injective. Proof. Note that there are no non-trivial relations among classes of motives on different Cartesian powers of X, so after replacing X by one of its Cartesian powers, we may reduce to the case that X and Y are formal motives on X having the same class in Λ X Ô ÓÖÑ κ Õ. for any formal invariant s on X and any fat point z. Take s : 1 X . The left hand side of (55) is equal to L ¡dl Ö∇ z X× (as an element in G), whereas the right hand side is equal to L ¡dl Ö∇ z ÔX YÕ×, where d and l are respectively the dimension of X and the length of z. Using that ∇ z preserves scissor relations, we get ∇ z ÔX ¡ YÕ 0. Hence X ¡ Y À by Lemma 15.5, showing that Y ⊆ X. Replacing the role of X and Y then proves the other inclusion. V ¢ÔL ¦ Õ m ,where L ¦ is the affine line A 1 V minus a point. Since ÖL ¦ × L ¦ L¡ Ô L by Proposition 7.1, the class of such an intersection is equal to the product L n¡m ÔL ¡ Ô LÕ m . Since there are n 1 m ¨terms with I m in (23), the class of P n V is equal to gÔL, Ô LÕ, where gÔt, uÕ : V ¢ÔL ¦ Õ m , Y ÔFzÕ F Y ÔzÕ for any fat point z. More precisely, the canonical (dominant) morphism z Fz induces a map Y ÔFzÕ Y ÔzÕ. By Lemma 2.3 it is injective, and we want to show that its image is F Y ÔzÕ. Let Ôf 1 , . . . , f s ÕκÖx× be the ideal defining Y . By Corollary 8.13, the Frobenius transform F ¦ A n κ Y is given by the ideal Ôf 1 Ôx p Õ, . . . , f s Ôx p ÕÕκÖx×. An Fz-rational point a in Y corresponds to a κ-algebra homomorphism A FR, where R is the coordinate ring of z, and hence to a solution of f 1 Conversely, by reversing these arguments, we see that any such z-rational point is induced by a p-th power in R, and hence comes from a Fz-rational point. This concludes the proof of (34) when Y is affine, and proves in particular that F8.14. Theorem. The functors zFz and Y F Y constitute a sub-schemic adjunction. Conversely, by reversing these arguments, we see that any such z-rational point is induced by a p-th power in R, and hence comes from a Fz-rational point. This concludes the proof of (34) when Y is affine, and proves in particular that F8.14. Theorem. The functors z Conversely, by reversing these arguments, we see that any such z-rational point is induced by a p-th power in R, and hence comes from a Fz-rational point. This concludes the proof of (34) when Y is affine, and proves in particular that F Let ÔX, P Õ be a closed germ over κ. For t a single variable, we define the motivic Hilbert series as the series Proof. Note that many authors take instead the n 1-th power. We will refer to the objects in a motivic site as 'motives'. This nomenclature is to express the fact that a motif represents something geometrical which is not a scheme but ought to be something like a scheme, thus 'motivating' our geometric treatment of it. Note that Ô See §11 below for the corresponding single adjunction.