Birational invariants and A^1-connectedness

Birationa l in variants and A 1 -connect edness Aravind Asok ∗ Department of Mathematics Univ ersity of Southern California Los Angeles, CA 90089-253 2 asok@usc.edu Abstract W e study some aspects of the re lationship between A 1 -homo topy theory and birational ge- ometry . W e stud y the so-called A 1 -singular cha in comp lex and zero th A 1 -homo logy shea f o f smooth algebraic varieties over a ﬁeld k . W e exhibit some ways in which these objects are similar to their co unterpa rts in classical to pology and similar to their motivic coun terparts (the (V oev odsky) motive and zeroth Suslin homology sheaf). W e show that if k is i nﬁn ite the zeroth A 1 -homo logy sheaf is a b irational inv ariant of smo oth pr oper varieties, and we explain how these sheaves contr ol various co homolo gical inv ariants, e.g., unr amiﬁed ´ etale coho mology . In particular, we deduce a number of v anishing results for cohomo logy of A 1 -conne cted v arieties. Finally , we gi ve a partial conv erse to these vanishing statements by gi ving a characterization of A 1 -conne ctedness by means of v anishing of unramiﬁed in variants. Contents 1 Introduction 1 2 A 1 -homotopy , A 1 -homology and Suslin homology (sheav es) 4 3 Birational geometry and strictly A 1 -in variant sheav es 12 4 An unramiﬁed characterization of A 1 -connectedness 17 1 Introd uction In this paper , we continue to in vestigat e the relations hip between birational geometr y and connect- edness in the sense o f A 1 -homoto py theory that was initia ted in [AM11]. Dev eloping some ideas of [AM11, § 4], w e study cohomologic al conse quences of homotopical connec tivit y hypothes es an d, more speciﬁcally , v anish ing resu lts fo r v ario us typ es of co homologica l in v ariants such as u nramiﬁed ´ etale coho mology . For th e m ost part, ho wev er , the results of this paper are logicall y indepe ndent of [AM11]. ∗ Aravind Asok was p artially supported by National Science Founda tion A wards DMS-09008 13 and DMS-0966589. 1 2 1 Introduction In the conte xt of the Morel-V oe v odsky A 1 -homoto py theory of smooth sche mes over a ﬁeld k [MV99], on e may associate with any s mooth scheme X a Nis nev ich sheaf o f sets, d enoted π A 1 0 ( X ) , called the sheaf of A 1 -conne cted compone nts of X . A smooth scheme X is called A 1 -conne cted if π A 1 0 ( X ) is isomorp hic to the co nstant 1 -poi nt sheaf (see Deﬁnition 2.2 for more details). In [AM11, D eﬁnition 2.2.2 and Corollar y 2.4.4] it was sho wn that v arieties that are A 1 -conne cted are “nearl y ra tional in a strong sense. ” For example , if k has characteri stic 0 , then stably k -rational smooth proper varie ties or , more generally , Saltman’ s retract k -r ational smoot h pro per v arieties are A 1 -conne cted. Many sche mes of intere st are not A 1 -conne cted and provid ing an exp licit des cription of π A 1 0 ( X ) for an y arbitrary smooth va riety seems dif ﬁcult at the moment. For that reaso n, we seek to und er- stand auxiliary in v ariants that are contro lled by the sheaf of A 1 -conne cted compone nts. H ere, w e study the homological counterpar t of π A 1 0 ( X ) : the zeroth A 1 -homolo gy sheaf, denoted H A 1 0 ( X ) (see Deﬁnition 2.5). While in some ways A 1 -homolo gy is similar to the more famili ar Suslin ho- mology (see, e.g., [S V96]), the two theories are dif ferent . While the zero th Suslin homology shea f (see Deﬁnition 2.9) of a smooth proper k -scheme X can be describ ed usi ng the Chow group of 0 -c ycles on X , as w e explain , the zeroth A 1 -homolo gy is more closely related to R -equi vale nce classe s in X . Intuiti v ely speakin g, the sheaf π A 1 0 ( X ) formalizes the idea of algebraic path co mponents of X , where algebraic paths are interp reted as chain s of afﬁne lines . Indeed [AM11, Theorem 6.2.1] pro ves that if X is smooth and proper ov er k , then for any ﬁnitely gener ated separa ble exten sion L/k , the s et of se ctions π A 1 0 ( X )( L ) can be i dentiﬁed with t he set of R -equi v alence classes in X ( L ) in the sense of Manin. While this result does not identify the whole she af π A 1 0 ( X ) , it suggests that π A 1 0 ( X ) is a birati onal in varian t of smooth proper k -va rieties. By anal ogy with topol ogy , one might expect that the she af H A 1 0 ( X ) should be the free abelian group on the A 1 -conne cted compone nts of X . Mirrori ng the expec ted beha vior of π A 1 0 ( X ) abo ve, one might also expect that H A 1 0 ( X ) is a biratio nal in v ariant for smooth proper k -var ieties. Both of these statements are true prov ided that in the ﬁrst statement one interpret s the express ion “free abelia n group” correct ly , and in the second statement one restricts k appropr iately (the correspond - ing statemen t for Suslin homolo gy is well known) . Precisely , w e pro ve the follo wing results. Pro position 1 (See Proposition 3.5) . Suppo se k is a ﬁeld and X is a smooth k -scheme. The mor - phism H A 1 0 ( X ) → H A 1 0 ( π A 1 0 ( X )) , induced by the canon ical morphism X → π A 1 0 ( X ) , is an iso- morphism of Nisnev ich sheaves of abelian gr oups . Remark 2 . By Lemma 3.3, H A 1 0 ( X ) is the free strictly A 1 -in v arian t sheaf (see Deﬁnition 2.3) of abelia n g roups generate d by X . T hus, this th eorem says that H A 1 0 ( X ) is the free strictly A 1 -in v arian t sheaf of groups on π A 1 0 ( X ) . One particularl y useful cons equence of this result is that if the m or - phism H A 1 0 ( X ) → H A 1 0 (Sp ec k ) is not an isomorphism, then X is A 1 -disco nnected. Theor em 3 (See T heorem 3 .9) . Suppose k is an inﬁnite ﬁeld. If X and X ′ ar e st ably k -biratio nally equiva lent smoot h pr oper schemes, then H A 1 0 ( X ) ∼ = H A 1 0 ( X ′ ) . For any ﬁeld k , the sh eaf H A 1 0 (Sp ec k ) is iso morphic to Z (see Example 2.6) and th us coincid es with the zeroth Suslin homology of a point. For a genera l smooth k - scheme X , v arious classica l stable birational in v ariants are related to H A 1 0 ( X ) . Suppose n is an intege r coprime to the char - acteris tic of k , and L /k is a ﬁnitely generate d separable ﬁeld exte nsion. Consider the functor on 3 1 Introduction k -algebra s deﬁned by A 7→ H i ´ et ( A, µ ⊗ j n ) (we abus e terminology and write A instead of Sp ec A for notational con venienc e). Gi ven a discret e valu ation ν of L/k with associated v aluation ring A , one says that a class α ∈ H i ´ et ( L, µ ⊗ j n ) is unramiﬁed at ν if α lies in the image of the restriction map H i ´ et ( A, µ ⊗ j n ) → H i ´ et ( L, µ ⊗ j n ) . For any inte gers i, j , Colliot-Th ´ el ` ene and Ojangure n [CTO89] deﬁne the un ramiﬁed coho mology group H i ur ( L/k , µ ⊗ j n ) as the subgro up of H i ´ et ( L, µ ⊗ j n ) consisting of those classe s α that are unr amiﬁed at ev ery discr ete valua tion of L tri vial on k . Colliot-Th ´ e l ` ene and Ojanguren als o pro ved that the gro ups H i ur ( L/k , µ ⊗ j n ) are st able bira tional in varian ts of smoo th proper varieti es (see [CTO89 , Proposi tion 1.2]). Fo r anot her point of vie w on these state ments see [CT95, The orem 4.1.1]. The n ext re sult demonst rates that the groups H i ur ( L/k , µ ⊗ j n ) are controlled by H A 1 0 ( X ) ; in essence this was obser ved by G abber ( cf . [CTO89, Remark 1.1.3] ). Lemma 4 (See Lemma 4.7) . Su ppose k is a ﬁeld, and n is an inte ge r coprime to the char acteristic of k . If X is a smooth pr op er k - scheme , then ther e is a canonical bijection H i ur ( X, µ ⊗ j n ) ∼ − → Hom( H A 1 0 ( X ) , H i ´ et ( µ ⊗ j n )) , wher e the gr o up on the right hand side is computed in the cate gory of N isne vich sheav es of abel ian gr ou ps, and the she af H i ´ et ( µ ⊗ j n ) is deﬁned in Example 4.6 . Proposit ion 1 sho ws that if X is A 1 -conne cted, then H A 1 0 ( X ) is trivia l. Thus, for example, non-tr ivia lity of unramiﬁed ´ etale cohomolog y can be used to detect A 1 -disco nnectedne ss. More genera lly , via Lemma 3 .3, we will see that H A 1 0 ( X ) is a “uni ve rsal unramiﬁed in va riant” for smooth proper schemes, in an appropriat e sense; see L emma 4.2 for a precise statement. For example, H A 1 0 ( X ) contro ls unramiﬁed Milno r K-theory (see Example 4.9) and the un ramiﬁed W itt sheaf (see Example 4.12); this point of vie w is dev eloped in § 4. All of the theorie s describ ed in the pr evio us parag raph ha ve transfe rs of an a ppropriat e kind, and the ﬁrst two are e ven con trolled by Suslin homology . Howe ve r , the zeroth A 1 -homolo gy sheaf con- trols unramiﬁed in varian ts that do not possess transfe rs. Using this addition al informati on, if X is smooth and pr oper w e can s how that A 1 -conne ctedness is characteri zed by v anishing of unramiﬁed in v ariant s. More precisely , w e pro ve the follo wing result. Theor em 5 (See Theorem 4.15 ) . If k is a ﬁeld, and X is a smooth pr oper k -scheme, then X is A 1 -conne cted if and only if the canonica l morphism H A 1 0 ( X ) → Z is an isomorphism; the same statemen t hold s with r ational coefﬁ cients. The use of strictly A 1 -in v arian t shea ves that do not posses s transfers seems essential . Ind eed, Suslin homology (e ve n with integral coefﬁci ents) canno t detect A 1 -conne ctedness, in lar ge part becaus e of its inability to see the dif ference between rationa l points and 0 -cycles of degre e 1 . W e recall an exa mple of Parima la (see E xample 4.19), pointed out to us by Sasha Merkurj ev , showin g that ev en if X is a smooth projecti ve v ariety such that the degree morphism H S 0 ( X ) → Z is an isomorph ism, X need not be A 1 -conne cted. Section 2 is dev oted to brieﬂy revie wing aspec ts of A 1 -homoto py theory , V oe vods ky’ s theory of motiv es, and A 1 -homolo gy theory; in particular , we ﬁx our notatio n for the rest of the paper . Section 3 studies the birat ional properties of the zero th A 1 -homolo gy and Suslin homolo gy sheav es and relates these two objec ts to the zeroth A 1 -homoto py sheaf. Finally , Section 4 prov ides a ﬁeld theore tic poi nt of vie w useful for studying A 1 -homolo gy and Suslin homology shea ves together with the uni vers ality statemen t allu ded to above . 4 2 A 1 -homotopy , A 1 -homology and Suslin homology (sheav es) Relationship with other work This work is part of a sequence of papers including [AH11b, A H11a, A so11] studyin g the A 1 - homolog y sheaf, its relatio nship with ration al points, an d rationality questions. In [AH11b], we pro ve that if X i s smooth and prop er ov er a ﬁeld, then H A 1 0 ( X ) detects rati onal poin ts. More precis ely [AH 11b, Corolla ry 2.9] states that a s mooth proper v ariety X has a k - rational point if and only if the canon ical map H A 1 0 ( X ) → Z is an epimorphism. On the othe r hand π A 1 0 ( X ) controls the unramiﬁed cohomo logy of smooth va rieties X . In [Aso11], we produ ce rationall y connected , non-ra tional smooth pro per v arieties X where non-rationa lity is detected by a degree n unr amiﬁed cohomol ogy class, b ut canno t be detect ed by lo wer de gree in v ariant s; the c onnection with th is work is mention ed at the end of Section 4. General con v entions Through out the paper k will be a ﬁxed base ﬁ eld. Write S m k for the category of schemes tha t are smooth, separa ted and hav e ﬁnite type o ver k . When we use the word sheaf without modiﬁcation , we alwa ys mean N isne vich sheaf of sets on S m k . Acknowledgements W e were led to the study of cohomolo gical va nishing properties of A 1 -conne cted schemes by B. Bhatt’ s observ atio n that the Brauer group of a A 1 -conne cted smooth scheme is tri vial (this is un- publis hed, bu t see [Gil09, Theorem 4.3]), w hich suggeste d that higher cohomolo gical in varian ts could o bstruct A 1 -conne ctedness as well. This work also o wes an intellectual debt to Fabien More l, who has stressed the importanc e of A 1 -homolo gical in v ariants; we thank him for much encourage- ment an d many discussion s thro ugh the cou rse of this w ork. Finally , we thank Ch ristian H ¨ a semeyer for many discussions aro und this topic, Sasha Merkurje v for pointing out Example 4.19, and Jean- Louis Colliot-Th ´ el ` ene for helpful comments and correspond ence. 2 A 1 -homotopy , A 1 -homology and Suslin homology (shea ves) W e re vie w the con struction and basic prope rties of A 1 -deri ved categorie s and A 1 -homolo gy as ske tched in [Mor05b] and de velo ped in [Mor11]. Fo r completenes s, we also gi ve a detailed com- pariso n between A 1 -homolo gy and Suslin homology shea ves, which was alluded to in [Mor06] but has not been dev eloped in the literature (in deta il). For more discuss ion of the homological algebra underl ying the A 1 -deri ved cate gory , we refer the reader to [CD09, § 4]. The results stated in this sectio n are essen tial to the for mulation and proofs of results in subsequen t sections. Simplicial homotopy categ ories Let ∆ ◦ S hv N is ( S m k ) denote the cate gory of simplici al Nisnev ich shea ves on S m k ; we will refer to objec ts in this categor y as k -space s, or simply as space s if k is clear from conte xt. The Y oneda embeddi ng provides a fully-fait hful func tor S m k → ∆ ◦ S hv N is ( S m k ) . W e use this to identify S m k with a full subca tegory of ∆ ◦ S hv N is ( S m k ) , and syste matically abus e notation by denoti ng a smooth scheme and the corresp onding simplicial sheaf (the sheaf of n -simplices is the N isne vich 5 2 A 1 -homotopy , A 1 -homology and Suslin homology (sheav es) sheaf represente d by the scheme , and all face and de generac y morphisms are the ide ntity mor - phism) by the same roman letter . Generally , we use calligr aphic letters (e.g., X , Y ) for objec ts of ∆ ◦ S hv N is ( S m k ) . The cate gory ∆ ◦ S hv N is ( S m k ) admits a proper closed model structure w here the coﬁbrat ions are monomorph isms, the weak equiv alen ces are those morphis ms of simplic ial shea ves that stalk- wise induce weak equi vale nces of the corresp onding simplicial sets, and the ﬁ bration s are those morphisms ha ving the right lifting prop erty with respect to mor phisms that are simultaneou sly coﬁ- bratio ns and weak equi v alences (see, e.g., [MV99, § 2 The orem 1.4]). T he res ulting model structu re is called the injecti ve local model structure, or the Joyal-Jar dine model structu re. The simplicial homotop y cate gory , d enoted H s (( S m k ) N is ) , is the homotop y cate gory of this model struc ture. Through out, we write [ X , Y ] s for Hom H s (( S m k ) N is ) ( X , Y ) . Deriv ed categories of sheav es of R -modules For a commutati ve unital ring R , we let M od k ( R ) denote the cate gory of Nisnevi ch shea ves of R -modules. Similar ly , we let ∆ ◦ M od k ( R ) denote the catego ry of simplicial Nisnev ich shea ves of R -modules. Giv en an y obje ct X ∈ ∆ ◦ S hv N is ( S m k ) , write R ( X ) for the Nisne vich sh eaf of R -module s freely gen erated by the simplices of X ; R ( X ) an object of ∆ ◦ M od k ( R ) . This constr uction deﬁnes a functor R ( · ) : ∆ ◦ S hv N is ( S m k ) → ∆ ◦ M od k ( R ) that is left adjoint to the for getful functor ∆ ◦ M od k ( R ) → ∆ ◦ S hv N is ( S m k ) . Let C h ≥ 0 ( M od k ( R )) denote the catego ry of chain complex es (dif ferential of degr ee − 1 ) of Nisne vich shea v es of R -modules situat ed in degre es ≥ 0 . Ther e is a functor of normalized chain comple x N ( · ) : ∆ ◦ M od k ( R ) → C h ≥ 0 ( M od k ( R )) . The sheaf theore tic Dold -Kan correspon dence produ ces an adjo int equi v alence K ( · ) : C h ≥ 0 ( M od k ( R )) → ∆ ◦ M od k ( R ) . Let C h − ( M od k ( R )) denote the category of bounded below chain complex es of Nisne vich shea ves of R -modules; objects in this category will be referred to simply as comple xes . The categor y S m k is essentia lly small, so the cate gory C h − ( M od k ( R )) is the category of bounded belo w com- ple xes in a Grothendieck abelia n catego ry . T herefo re, results of B eke imply that C h − ( M od k ( R )) can be equipped with a model categ ory st ructure w here coﬁbratio ns are monomorph isms, w eak equi v alences are quasi- isomorphisms, and ﬁbrations are those morphisms ha ving the right lifting proper ty with respect to morphisms that ar e s imultaneous ly coﬁbrations and weak equiv alen ces (see [Bek00, Propo sition 3.13]). This model structure —the inject iv e local model structure—ha s homo- top y cate gory the b ounded belo w deri v ed cate gory D − ( M od k ( R )) . W e denote by (( − ) f , θ ) a ﬁx ed ﬁbrant resolutio n functor , i.e., ( − ) f is an endofunct or of C h − ( M od k ( R )) and θ : I d → ( − ) f is a natura l trans formation such that if A is comple x, the induced map A → A f is a quas i-isomorphis m and monomorph ism and A f is a ﬁbrant complex. The homotopy catego ry D − ( M od k ( R )) is a triang ulated cate gory with the usual shift functor . Notation 2.1. W e use homologica l con venti ons for complex es. More precisely , if C ∗ is a complex, then the shift funct or satisﬁes C ∗ [1] = C ∗ +1 so that H i ( C ∗ [1]) = H i − 1 ( C ∗ ) ; this con ven tion will be justi ﬁed in the nex t subsectio n. Any c omplex C ∗ can b e considere d as a cohomologic al comple x C ∗ with C i = C − i ; we use this con v ention when computing hypercoho mology . 6 2 A 1 -homotopy , A 1 -homology and Suslin homology (sheav es) Hurewic z th eory If X is a spa ce, set C ∗ ( X , R ) := N (( R ( X ))) . The assignment X → C ∗ ( X , R ) prov ides a functor ∆ ◦ S hv N is ( S m k ) − → C h ≥ 0 ( M od k ( R )) . This functor sen ds mono morphisms to monomorph isms, and sends weak equi v alences to quasi- isomorph isms. T hus, it descend s to a func tor H s (( S m k ) N is ) − → D − ( M od k ( R )) . There is a corres ponding version of this functo r in the setting of pointed spa ces as well . If X is a space, the structure morphism X → Sp ec k induces a morphism of complex es C ∗ ( X , R ) → C ∗ (Sp ec k , R ) ; we let ˜ C ∗ ( X , R ) denote the kernel of this morphism. If X is pointed, then C ∗ ( X , R ) → C ∗ (Sp ec k , R ) is split, and ˜ C ( X , R ) is a summand of C ∗ ( X , R ) . In the other directi on, the adjo int K ( · ) (coming from the D old-Kan corre spondenc e) composed with the inclus ion ∆ ◦ M od k ( R ) → ∆ ◦ S hv N is ( S m k ) produces a functor C h ≥ 0 ( M od k ( R )) − → ∆ ◦ S hv N is ( S m k ) . This composite functor sends quasi-isomorp hisms to w eak equiv alen ces, and using propert ies of adjunc tions can be sho wn to preserv e ﬁ bration s as well. In fact, there is an adjunc tion (2.1) [ X , K ( A, n )] s ∼ − → Hom D − ( M od k ( R )) ( R ( X ) , A [ n ]) , which we use freely in the sequel. W e set H i ( X , R ) := H i ( C ∗ ( X , R )) and if X is point ed ˜ H i ( X , R ) := H i ( ˜ C ∗ ( X , R )) . If S 1 s denote s the constant sheaf deﬁned by the simplicial circle, we let Σ 1 s X = Σ 1 s ∧ X . It is not hard to check that ˜ H i (Σ 1 s X , R ) = ˜ H i − 1 ( X , R ) . A 1 -homotopy catego ries The A 1 -homoto py category , construct ed in [MV99, § 2 Theorem 3.2], is obtaine d as a cate gorical localiz ation of ∆ ◦ S hv N is ( S m k ) . Recall that a space X is called A 1 -local if for any space Y the canon ical map [ Y , X ] s − → [ Y × A 1 , X ] s is a bijection. A morphi sm f : X → Y is an A 1 -weak equi valence if the induced map [ Y , Z ] s → [ X , Z ] s is a biject ion for all A 1 -local spaces Z . The categor y ∆ ◦ S hv N is ( S m k ) can be equipped with a model structure where weak equi v alences are A 1 -weak equi val ences, coﬁbr ations are monomor - phisms and ﬁbrations are those m orphis ms havi ng the right lifting property with respect to mor- phisms that are simultane ously coﬁbration s and A 1 -weak equi v alences . W e write H ( k ) for the re- sultin g h omotopy category , and [ X , Y ] A 1 for Hom H ( k ) ( X , Y ) . The full subcat egory of H s (( S m k ) N is ) spann ed by A 1 -local objects can be taken as a model for the A 1 -homoto py category . Thus, if X is A 1 -local, we ha ve [ Y , X ] s = [ Y , X ] A 1 ; we use this freely in the sequel . Deﬁnition 2.2. Suppose X is a space. The shea f of A 1 -conne cted compon ents of X , denoted π A 1 0 ( X ) is the Nisne vich sheaf associa ted with the presheaf U 7→ [ U, X ] A 1 . A space X is A 1 - conne cted if the canonical morphism π A 1 0 ( X ) → ∗ (where ∗ is the constant 1 point sheaf) is an isomorph ism. 7 2 A 1 -homotopy , A 1 -homology and Suslin homology (sheav es) A 1 -deriv ed categories There is an analogous abelianiz ed versio n of the A 1 -homoto py category; we recall the basic deﬁni- tions, which were origina lly introd uced by Morel. Deﬁnition 2.3. A complex A of Nisne vich she av es of abelia n grou ps is A 1 -local if for any smooth k -scheme U , and eve ry integer n H n N is ( U, A ) → H n N is ( U × A 1 , A ) is an isomor phism. If A is simply a sheaf of abelian groups, say A is stric tly A 1 -in va riant if it is A 1 -local vie wed as a complex of shea ve s. Remark 2.4 . A sheaf of sets S is A 1 -in va riant if for any smooth scheme U , the m ap S ( U ) → S ( U × A 1 ) is a bijection . A sheaf of group s G is str on gly A 1 -in va riant if for any smooth scheme U , and any inte ger i ∈ { 0 , 1 } , the maps H i ( U, G ) → H i ( U × A 1 , G ) are bijection s. Denote the localizin g subc ategory of D − ( M od k ( R )) genera ted by comple xes of the form R ( X × A 1 ) → R ( X ) for smoot h sch emes X by T ( A 1 , R ) . By the theory of localizing categori es, a quotient categor y D − ( M od k ( R )) /T ( A 1 , R ) exis ts; this cate gory , denoted D A 1 ( k , R ) , is called the A 1 -deri ved cate gory . Let D − ( M od k ( R )) A 1 − loc ⊂ D − ( M od k ( R )) denot e the full-subca tegory consis ting of A 1 -local comple xes. This inclu sion admits a left adjoint L A 1 : D − ( M od k ( R )) − → D − ( M od k ( R )) A 1 − loc that can be used to identify , up to equi va lence, D − ( M od k ( R )) /T ( A 1 , R ) with D − ( M od k ( R )) A 1 − loc . The fun ctor L A 1 is ca lled the A 1 -locali zation functor; for mor e details see, e.g., [CD09, Propositi on 4.3]. For simplicity , we will write D A 1 ( k ) for D A 1 ( k , Z ) . Observe that if A is an A 1 -local c omplex, then K ( A, n ) is an A 1 -local spac e by the ad junction of 2.1. Deﬁnition 2.5. The A 1 -singu lar chai n comple x of X with R -coef ﬁcients , denot ed C A 1 ∗ ( X , R ) , is deﬁned to be the A 1 -locali zation L A 1 ( C ∗ ( X , R )) . W e w rite C A 1 ∗ ( X ) for C A 1 ∗ ( X , Z ) . The A 1 - homolo gy shea ves of X with R -coef ﬁcients are deﬁne d by H A 1 i ( X , R ) := H i ( C A 1 ∗ ( X , R )) . If X is pointe d, the r educe d A 1 -homolo gy sheave s of X with R -coefﬁ cients are deﬁned by ˜ H A 1 i ( X , R ) := H i ( L A 1 ( ˜ C ∗ ( X , R ))) . Example 2.6 . Suppose k is a ﬁeld. The comple x C ∗ (Sp ec k ) is just the Nisn evi ch sheaf Z placed in deg ree 0 . T he Nisnevi ch sheaf Z is ev en Nisne vich ﬂ asque so all higher N isne vich cohomology of a smooth k -scheme with coef ﬁcients in Z v anishes . Sin ce the functor Z ( · ) is clear ly A 1 -in v arian t, it follows immediately that Z is strictly A 1 -in v arian t. As a conseque nce H A 1 0 (Sp ec k ) = Z and all the higher A 1 -homolo gy shea ve s of Sp ec k vani sh, as topologica l intuition suggest s. Remark 2.7 (Mayer -V iet oris) . If X is a smooth scheme, there is a Nisnevi ch Mayer-V ietoris se- quenc e allowin g computation g of H A 1 i ( X ) from piece s of an ope n cov er . This follo ws immediat ely from [MV99, § Remark 1.7] togeth er with the fac t that the A 1 -locali zation func tor is exact. 8 2 A 1 -homotopy , A 1 -homology and Suslin homology (sheav es) Sheav es with transfers and Suslin homology of motiv es Again, let R be a commutati ve unital ring. Writ e C or k ( X, Y ) for the free R -module generated by inte gral closed subschemes of X × Y that are ﬁnite and sur jecti ve ov er a compone nt of X (an element of this gro up is referred to as a ﬁnite corr esponde nce ). W e let R tr ( X ) denote the presheaf on S m k deﬁned by U 7→ C or k ( U, X ) . If X is a smooth s cheme, th e f unctor U 7→ R tr ( X )( U × ∆ • ) deﬁnes a simplicial abelian group for which we write C S ∗ ( R tr ( X )) . By deﬁnition C S ∗ ( R tr ( X )) is situate d in posit iv e (homological) degrees . Write C or k for th e cat egory whose objects are smooth schemes , and morphis ms are ﬁnite corre- spond ences from between smooth schemes (this category is described in detail in [MW06, Chapter 1]). There is a functor S m k → C or k that sends an element of Hom ( X, Y ) to the ﬁnite correspon - dence deﬁned by the graph. An addit iv e contra vari ant func tor from C or k to ab elian groups is call ed a preshe af with tr ansfers. Any preshea f with tran sfers can be viewed as a presheaf of abelian gro ups on S m k by restriction to S m k . A preshe af w ith transfers whose rest riction to S m k is a Nisne vich sheaf is called a Nisne vich sheaf with transfers. The preshea ves R tr ( X ) are all Nisne vich she av es with trans fers ([MW06, Lemma 6.2]). One can deﬁne a der iv ed categ ory of N isne vich sheav es with transfers: take the homoto py cat- ego ry of complex es of Nisne vich shea ves w ith transfers and localiz e at the qua si-isomorph isms. A comple x of Nisnevi ch sheav es with trans fers is A 1 -local if it is A 1 -local as a complex of Nisnev ich shea ves after forg etting the transfers . A strictly A 1 -in va riant sheaf with transf ers is a sheaf with transfe rs that is A 1 -local when viewed as a complex of Nisnevi ch sheav es with transfers. T ak ing the quo tient of the deri ved cate gory of Nisne vich shea ves with transfe rs by the localizing subcate - gory generated by A 1 -local complex es of Nisnev ich shea ves with transfe rs one ob tains V oe vod sky’ s (“big” ) deri ve d category of m oti ves DM ef f k , − . W e r efer the reader to [MW06, § 13] for a much more detaile d discus sion of this construction . The tec hniques giv en so far are suf ﬁcient to deﬁne Suslin homolog y for a smooth sch eme, bu t we w ill need to deﬁne Suslin homology of an arbitrary space for some later statement s. T o do this, recall the following result . Lemma 2.8 ([MV99, § 2 Lemma 1.16]) . Ther e is a pair (Φ r ep , θ ) consistin g of an endo functor Φ r ep : ∆ ◦ S hv N is ( S m k ) → ∆ ◦ S hv N is ( S m k ) and a natur al transfor mation θ : Φ r ep → I d such that for any s implicial shea f X , Φ r ep ( X ) n is a copr od uct of r epr esentable sheaves, an d Φ r ep ( X ) → X is a s implicial weak equivalen ce and stalkwise a ﬁbrati on of simplicial sets. W e refer to Φ r ep ( X ) as a resolution of X by represent ables. As abov e, let R be a com- mutati ve unital ring. Usin g Φ r ep , one can deﬁne a moti ve and Suslin homolo gy of any X ∈ ∆ ◦ S hv N is ( S m k ) . Deﬁnition 2.9. Suppose X is a k -space. The motive of X , deno ted M ( X ) , is the clas s of the normaliz ed complex R tr (Φ r ep ( X ) • ) in DM ef f k , − . The i -th Suslin homolog y sheaf of X , denote d H S i ( X ) , is deﬁned as H i ( L A 1 R tr (Φ r ep ( X ) • )) . Remark 2.10 . In fact, this construct ion can be exten ded to a functor H ( k ) → DM ef f k , − ; see, e.g., [W ei04] for more details regar ding this constructio n. While this is not the usual deﬁnition of S uslin homology , it coincide s with that one in case k is assumed perfect via the follo wing important foundatio nal result. 9 2 A 1 -homotopy , A 1 -homology and Suslin homology (sheav es) Theor em 2.11 ([MW06, Corollary 14.9] ) . If k is a perfect ﬁeld, and X is a smooth k -scheme, the comple x C S ∗ ( R tr ( X )) is A 1 -local . Remark 2.12 . S uppose k is a ﬁ eld havin g charac teristic p . If k is not perfect, it is not kn own whether C S ∗ ( R tr ( X )) is A 1 -local. On the other hand, unpublis hed work of Suslin establis hes that so long as p is in v ertible in R , then V oe vod sky’ s theorem that homotopy in v ariant preshea v es of R -modules with tra nsfers ha ve homotopy i n v ariant cohomolo gy [MW06, Theore m 13.8] still holds . Using Suslin ’ s result, if p is in v ertible in R , then one can sho w that C S ∗ ( R tr ( X )) is A 1 -local. Thus, in this situa tion, the deﬁnition of Suslin homol ogy gi ven abo ve agree s with the usua l deﬁniti on of Suslin homolog y . Comparing homology shea ves For any X ∈ ∆ ◦ S hv N is ( S m k ) , recall that S ing A 1 ∗ ( X ) is deﬁned to be the diagonal of the bisim- plicial sheaf ( i, j ) 7→ Hom (∆ i k , X j ) , where ∆ i k is the alge braic i -simplex and we write Hom for the inte rnal hom in the cat egory of Nisne vich sheav es of sets. By construct ion, there is a canon ical morphism X → S ing A 1 ∗ ( X ) that is an A 1 -weak equi v alence (see [MV99, p. 88 ]). In particular , for any smooth scheme X the morphism X → S ing A 1 ∗ ( X ) induces a morphism C ∗ ( X, R ) → C ∗ ( S ing A 1 ∗ ( X ) , R ) that becomes an isomorphism after A 1 -locali zation. For any smooth scheme Y , there is a canonica l monomorph ism of shea ves R ( Y ) → R tr ( Y ) (send a morphis m U → Y to the correspon dence deﬁned by its graph), and this constructio n induces a map N ( R ( S ing A 1 ∗ ( X ))) → C S ∗ ( R tr ( X )) . Combinin g this morphi sm with the discuss ion of the pre vious paragraph and applying the A 1 -locali zation func tor we get morphisms C A 1 ∗ ( X, R ) − → C A 1 ∗ ( S ing A 1 ∗ ( X ) , R ) − → L A 1 ( C S ∗ ( R tr ( X )) . W e th us obtai n a compa rison morphi sm from A 1 -homolo gy to Suslin homology , and we summarize this constr uction as follo ws. Cor ollary 2.13. F or any smooth sch eme X , ther e ar e comparison m aps H A 1 i ( X, R ) − → H S i ( X, R ) . induce d by the morphis m C A 1 ∗ ( X, R ) → L A 1 ( C S ∗ ( R tr ( X ))) . Remark 2.14 . W ith a l ittle more work, one ca n ext end this c omparison map to a natu ral transforma- tion of functor s on space s. The follo wing examples sho w this morphism is not an isomorphism in general. Example 2.15 . If ( X , x ) is a pointed smooth scheme , the zeroth Suslin homology sheaf splits H S 0 ( X ) ∼ → Z ⊕ ˜ H S 0 ( X ) . The morp hism of Corollary 2.13 is co mpatible with this spitt ing and there is an induced morphism ˜ H A 1 0 ( X ) → ˜ H S 0 ( X ) . In general, this morphism is not an isomor - phism, e.g., for X = G m . Indeed the sheaf of groups G m is strictly A 1 -in v arian t with transfers gi ven by the usual norm map on units. Lemma 3.3 sho ws that the identit y map G m → G m in- duces a homomorphi sm of strictly A 1 -in v arian t shea ves (with transf ers) ˜ H A 1 0 ( G m ) → G m (resp. ˜ H S 0 ( G m ) → G m ). Theorem 3.1 of [SV96] sho ws the map ˜ H S 0 ( G m ) → G m is an isomorphism. 10 2 A 1 -homotopy , A 1 -homology and Suslin homology (sheav es) More general ly , ˜ H S 0 ( G m ∧ n ) is closely related to Milnor K-theory . On the other hand, ˜ H A 1 0 ( G m ∧ n ) has b een compu ted by M orel (combine [Mor1 1 , The orem 2.37] and [Mor1 1 , Theor em 4.46]) : when n ≥ 1 , this sheaf is a mixture of M ilnor K -theory and W itt grou ps, which one refers to as Milnor- W itt K-theory . Similar ex amples can be constructed from any smooth proper curv e. Remark 2.16 . There is also a “stabili zed” vers ion H s A 1 0 ( X ) of H A 1 0 ( X ) where one “in verts G m . ” If S 0 s = Sp ec k + , then ˜ H s A 1 0 ( S 0 s ) = H s A 1 0 (Sp ec k ) essential ly by deﬁnition. One can also work directl y with the stable A 1 -homoto py cat egory to p rove H s A 1 0 (Sp ec k ) coincides with the 0 -th st able A 1 -homoto py she af of the m oti vic sphere spectrum (se e [Mo r05b, p. 7]). This sheaf has been identi ﬁed as K M W 0 by [Mor04, Coroll ary 6.4.1]. For an y ﬁnitely generated separable exte nsion L/k there is an isomorphism K M W 0 ( L ) ∼ → GW ( L ) (see [Mor05b, Remark 6.1.6b] or [Mor11, Lemmas 2.9-2.10]), where GW ( L ) denotes the Grothendiec k- W itt group of isomorphism classes of non-de generate symmetric bilinea r forms. On the other hand it follo ws from the deﬁnitions giv en above that H S 0 (Sp ec k ) = Z . These computa tions sugg est that D A 1 ( k ) , or perhap s its stabil ized vers ion, pro vides a versi on of V oe vo dsky’ s triangula ted categ ory of motiv es incorpo rating data from the theory of quadrati c forms. This point of vie w is further de velop ed in [AH11a]. Connectivi ty and the t -structure W e no w recall some basic facts re garding the structure of A 1 (or Suslin) homolog y she av es, all due to Morel. Recall that a c omplex of sheav es A ∗ is called ( − 1) -conn ected (or positive) if its homology shea ves H i ( A ∗ ) v anish for i < 0 . Each of the follo wing results was prove n in the context of stab le A 1 -homoto py theory by Morel in [Mor05b]. Howe ve r , the pro ofs he gi ves apply just as w ell (as he observ es) to the sett ing of deriv ed categ ories that we consid er . For this reason, we giv e references to the corres ponding state ments in sta ble A 1 -homoto py theory . Theor em 2.17 ([Mor05b, Theorem 6.1.8]) . If A ∗ is a ( − 1) -co nnected comple x of R -modules, then its A 1 -local ization L A 1 ( A ∗ ) is also ( − 1) -connect ed. Theor em 2.18 ([Mor05b, Theorem 6.2.7]) . If X is a k -space, then for eve ry inte g er i the sheaves H A 1 i ( X , R ) and H S i ( X , R ) ar e always stric tly A 1 -in va riant, and these sheaves ar e trivial if i < 0 . Pr oof. By const ruction, th e comple xes C A 1 ∗ ( X , R ) are A 1 -locali zations of the complex es C ∗ ( X , R ) . The latter comple xes are ( − 1) -connecte d by deﬁniti on. The resu lt follo ws immediatel y from The- orem 2.17. T he statemen t for Suslin homol ogy is prov en in an identical fashion. Before we state the nex t result, let us recall a vari ant of [B BD82, D ´ eﬁnition 1.3.1]. Deﬁnition 2.19 (Homologica l t -structure) . Let T be a triangulat ed catego ry . A homologi cal t - struct ur e on T consi sts of a pair of stric tly full subca tegorie s T ≤ 0 ⊂ T and T ≥ 0 ⊂ T suc h that, setting T ≤ n := T ≤ 0 [ n ] and T ≥ n := T ≥ 0 [ n ] , the follo wing properties hold: i) for any C ∈ T ≥ 0 and any D ∈ T ≤− 1 , one has Hom T ( C, D ) = 0 ; ii) there are inclusio ns T ≥ 1 ⊂ T ≥ 0 , and T ≤ 0 ⊂ T ≤ 1 ; 11 2 A 1 -homotopy , A 1 -homology and Suslin homology (sheav es) iii) for any object X ∈ T there is a distinguis hed triangle A − → X − → B − → A [1] with A ∈ T ≥ 0 and B ∈ T ≤ 1 . If ( T , T ≤ 0 , T ≥ 0 ) is a homological t -structure on T , then by [BB D82, Propositi on 1.3.3] there are tr uncation functors τ ≥ n : T → T ≥ n , an d τ ≤ n : T → T ≤ n adjoin t to the corr esponding inclusion functo rs. Using homologi cal con vention s as above , the proof of loc. cit. giv es the follo wing result. Pro position 2.20. Suppose ( T , T ≤ 0 , T ≥ 0 ) is a t -cate gory . If X is any object in T , ther e exist s a uniqu e morphi sm d ∈ Hom 1 ( τ ≤ 1 X, τ ≥ 0 X ) such that the tria ngle τ ≥ 0 X − → X − → τ ≤ 1 X d − → τ ≥ 0 X [1] is distin guished. A comple x A ∗ is called ne ga tive if H i ( A ∗ ) = 0 for i > 0 and positive if H i ( A ∗ ) = 0 for i < 0 . W e write D A 1 ( k ) ≤ 0 for the full subcateg ory of D A 1 ( k ) consisting of A 1 -local negati ve comple xes, and D A 1 ( k ) ≥ 0 for the full subc ategory consisting of A 1 -local positi ve comple xes. Pro position 2.21 ([Mor05b, Lemma 6.2.11]) . The trip le ( D A 1 ( k ) , D A 1 ( k ) ≤ 0 , D A 1 ( k ) ≥ 0 ) is a ho- molog ical t -struct ur e on D A 1 ( k ) . If w e w rite A b A 1 k for the cate gory of strictly A 1 -in v arian t shea v es. The categor y of strictly A 1 - in v ariant shea ves of grou ps can be identiﬁed as the hea rt of this t -structure. By [BBD82, Th ´ eor ` eme 1.3.6], we get the follo wing result. Cor ollary 2.22. The cat e gory A b A 1 k is abelian . If k is perfect, V oe vo dsky sho wed t hat DM ef f k , − admits a t -structure deﬁned in a manner identical to Proposition 2.21. T he heart of the resulting t -struc ture is precisely the cate gory of strictly A 1 - in v ariant Nisne vich shea ves with trans fers ; see [D ´ eg08 , § 4.3] for a discussion. W e write A b A 1 tr ,k for the abelian cate gory of strictly A 1 -in v arian t shea v es with transfers. W e can deﬁne H S 0 as a funct or from DM ef f k , − to A b A 1 tr ,k (see loc. cit. Formula 4.12a) . Gersten r esolutions Suppose A is an A 1 -local comple x. The axiomatic approach of [CTHK97] prov ides general ma- chiner y for producin g a Gersten resolu tion associate d with the N isne vich hyperc ohomology of A (see ibid. § 7). Let ( X, Z ) be a pair w here X is a smoot h scheme and Z ⊂ X is a closed s ubscheme. The functo r ( X, Z ) 7− → H ∗ Z ( X N is , A ) (i.e., Nisnev ich hyperco homology w ith supports on Z ) deﬁnes a cohomology theo ry with supports in the sen se of [CTHK97, Deﬁ nition 5.1.1]. Moreo ver , this theory satis ﬁes Nisne vich e xcision (Axiom COH1 of [CTHK 97, p. 55]) and A 1 -homoto py in v arian ce (Axiom COH3 of ibid p. 58). Let H n Z ar ( A ) denote the Zariski shea f associated with the preshe af U 7→ H n N is ( U, A ) ; this apparen t ab use of terminology w ill be just iﬁed momenta rily . 12 3 Birational geometry and strictly A 1 -in variant sheav es Pro position 2.23 ([CTHK97, Corollary 5.1.11]) . Suppose k is an inﬁnite ﬁeld. F or any smooth k -sche me X , and any A 1 -local comple x A , the complex H n Z ar ( A ) | X − → a x ∈ X (0) i x ∗ H n x ( X, A ) − → · · · − → a x ∈ X ( p ) i x ∗ H p + n x ( X, A ) − → · · · is a ﬂasque r esoluti on. Pr oof. W e just observ e that [CT HK97] Propositi on 5.3.2a, Axiom COH2 is impli ed by Axiom COH3 ). W e write H n N is ( A ) for the Nisnevic h sheaf associated with the presh eaf U 7→ H n N is ( U, A ) . W e use the follo wing fundamental comparison result. Theor em 2.24 ([CTHK97, Theorem 8.3.1]) . Suppose k is an inﬁnit e ﬁ eld. F or any smooth k -scheme X , and any A 1 -local comple x A , the canonical maps H i Z ar ( X, H n Z ar ( A )) − → H i N is ( X, H n N is ( A )) ar e isomorphi sms. 3 Birational geometry and strictly A 1 -in variant s hea ves In this section, we stud y the relation ship between the zer oth A 1 -homolo gy or Suslin homology sheaf (r ecall Deﬁnitions 2.5 and 2.9 ) and th e zeroth A 1 -homoto py sheaf (recall Deﬁnition 2.2 ). The princi pal results of this section imply Proposition 1 and T heorem 3 of the introd uction. W e prov e in L emma 3.3 that the zeroth A 1 -homolo gy (resp. Suslin homo logy) sheaf of a smooth scheme X is initial among strictly A 1 -in v arian t shea v es (with transfe rs) admit ting a morphism from X . In Proposit ion 3.5 we establis h that the zeroth A 1 -homolo gy (resp. Suslin homolo gy) sheaf of X is t he free strictly A 1 -in v arian t shea f (with transfers) generated by π A 1 0 ( X ) , and in Theore m 3.9 that it is a stable biratio nal in varian t for smooth proper schemes ov er inﬁnite ﬁelds. A facto rization lemma The functo r sending a Nisne vich sheaf F to the correspondi ng constant simplicial sheaf (all face and deg eneracy m aps are the identity) is ful ly-faith ful. The full-subca tegory of ∆ ◦ S hv N is ( S m k ) con- sisting of constant simplicial shea ve s will be referred to as the subcate gory of spac es of simplicial dimensio n 0 (see [MV 99, p. 47]). Spaces of simplicial dimens ion 0 are automatically simplicially ﬁbrant (see [MV99, § 2 Remark 1.14]); in particular , since smoot h schemes hav e simplici al dimen- sion 0 , the y are simplicially ﬁbrant. If X is any space, then the unstab le A 1 - 0 -con necti vity theorem [MV99, § 2 Corollary 3.22] giv es an epimorphism X → π A 1 0 ( X ) . W e begin by statin g a result that will be used repeat edly in the seque l. Lemma 3.1. Suppose X and Y ar e k -space s of simplicial dimension 0 , and Y is A 1 -local . The canon ical epimor phism X → π A 1 0 ( X ) induces a biject ion (3.1) Hom S hv N is ( S m k ) ( π A 1 0 ( X ) , Y ) − → Hom S hv N is ( S m k ) ( X , Y ) functo rial in both inpu ts. 13 3 Birational geometry and strictly A 1 -in variant sheav es Pr oof. Since X and Y ha ve simplicia l dimens ion 0 , the cano nical m ap Hom S hv N is ( S m k ) ( X , Y ) − → [ X , Y ] s is a bijec tion, as we observ ed just before the statement of the lemma. Since Y is A 1 -local, w e also ha ve identiﬁca tions [ X , Y ] A 1 = [ X , Y ] s . Since smooth schemes hav e simplicia l d imension 0 , for any smooth scheme U , we ha ve [ U, Y ] A 1 = Hom S hv N is ( S m k ) ( U, Y ) . Sheaﬁfying for the Nisne vich topo logy , we deduce that π A 1 0 ( Y ) = Y . T o ﬁnish, obse rve that any morphi sm X → π A 1 0 ( Y ) fac tors uniquely through π A 1 0 ( X ) by the deﬁnition of π A 1 0 ( · ) . Cor ollary 3.2. Suppo se X is an A 1 -conne cted smooth k -sch eme. If Y is an A 1 -local space of simplicia l dimens ion 0 , then the map Y ( k ) → Y ( X ) induced by the stru ctur e m ap is a bijection. In partic ular , if M is a strictly A 1 -in va riant sheaf , and X is an A 1 -conne cted smooth k -scheme , the canon ical map M ( k ) → M ( X ) is a bijection . Lemma 3.3. If M (r esp . M ′ ) is a strict ly A 1 -in va riant sheaf of R -modules (with trans fers), then for any space X ther e are b ijections H 0 N is ( X , M ) ∼ − → Hom A b A 1 k ( H A 1 0 ( X , R ) , M ) H 0 N is ( X , M ′ ) ∼ − → Hom A b A 1 k ( H S 0 ( X , R ) , M ′ ) functo rial in both X and M (r esp. M ′ ). Pr oof. As abov e, since M is A 1 -local we ha ve ident iﬁcations Hom S hv N is ( S m k ) ( X , M ) ∼ − → [ X , M ] A 1 = [ X , K ( M , 0)] A 1 . The adjun ction between the A 1 -homoto py and A 1 -deri ved cate gories allows one to identify the last abelia n grou p w ith Hom D A 1 ( k ) ( C A 1 ∗ ( X , R ) , M ) . No w , by Proposition 2.21, we kno w th at D A 1 ( k ) admits a homological t -s tructure. B y the stable A 1 -conne ctiv ity theorem, we kno w that C A 1 ∗ ( X , R ) ∈ D A 1 ( k ) ≥ 0 . The re sult follo ws from a genera l fact about homologi cal t -structures (see Deﬁnition 2.19). L et T be a triang ulated categor y with a homolog ical t -struc ture ( T ≥ 0 , T ≤ 0 ) and heart A . L et M ∈ A , and write M [0] for M vie wed as an object of T situated in degree 0 . For any object C ∈ T ≥ 0 , Proposi tion 2.20 and a shiftin g ar gument gi ve rise to the distin guished triang le τ ≥− 1 C − → C − → τ ≤ 0 C − → τ ≥− 1 C [1] . Applying the functo r Hom T ( · , M [0] ) to this distin guished triangle, we get a m ap Hom T ( τ ≤ 0 C, M [0]) − → Hom T ( C, M [0]) , and this map is an isomorphism directl y f rom Deﬁnition 2.19(i) and (ii), which sho w that Hom T ( τ ≥− 1 C, M [0]) and Hom T ( τ ≥− 1 C [1] , M [0]) v anish. Since τ ≥ 0 C = C , and H 0 ( C ) = τ ≤ 0 τ ≥ 0 C (see [BBD 82, Th ´ eor ` eme 1.3.6]) we get an isomorphis m Hom A ( H 0 ( C ) , M ) ∼ − → Hom T ( C, M [0]) . 14 3 Birational geometry and strictly A 1 -in variant sheav es The proof for S uslin homology shea v es is similar ; one uses [MW 06, Exercise 13.6 ] to identify Hom S hv N is ( S m k ) ( X , M ′ ) with Hom DM ef f k, − ( M ( X ) , M ) (since M ′ is A 1 -local) together with an identi cal trunc ation argumen t. Cor ollary 3.4. If X is a smooth k -scheme, and M is a strict ly A 1 -in va riant sheaf of R -modules with tra nsfers , then any morphism ϕ : H A 1 0 ( X, R ) → M factor s as a compos ite H A 1 0 ( X, R ) − → H S 0 ( X, R ) − → M , wher e the ﬁr st map is th e m orphis m of Corollary 2.13 . Pr oof. The morphi sm of Corollary 2.13 induces for any strictly A 1 -in v arian t sheaf of R -modules with trans fers a morphi sm Hom A b A 1 k ( H S 0 ( X, R ) , M ) − → Hom A b A 1 k ( H A 1 0 ( X, R ) , M ) . Lemma 3.3 implies that this morphi sm is a bije ction. Dependence on the sheaf of A 1 -connected components If M is a (suf ﬁciently nice) topologica l space, the M ayer -V ie toris sequen ce shows that the ordi- nary singul ar homology group H 0 ( M , R ) is the free R -module generat ed by the conne cted com- ponen ts of M . On the other hand, L emma 3.3 can be interpret ed as saying that H A 1 0 ( X , R ) (resp. H S 0 ( X , R ) ) is the fre e strictly A 1 -in v arian t shea f (with tr ansfers) on th e sheaf X . W e n ow show t hat H A 1 0 ( X , R ) (resp. H S 0 ( X , R ) ) is the free strictly A 1 -in v arian t sheaf (with transfers) on the sheaf of A 1 -conne cted compone nts of X . Pro position 3.5. F or any spac e X , and any commuta tive uni tal ring R , the maps H A 1 0 ( X , R ) − → H A 1 0 ( π A 1 0 ( X ) , R ) and H S 0 ( X , R ) − → H S 0 ( π A 1 0 ( X ) , R ) , induce d by the cano nical epimorphism X → π A 1 0 ( X ) ar e isomorp hisms. Pr oof. W e prov e the ﬁrst statement; the second statement is prov en in an essential ly identi cal man- ner . Assume ﬁ rst that X ha s simplicial dimension 0 . Let M be an arbitrary strictly A 1 -in v arian t sheaf (with transf ers for the seco nd state ment). W e hav e a commutati ve diagram Hom S hv N is ( S m k ) ( π A 1 0 ( X ) , M ) / /   Hom A b A 1 k ( H A 1 0 ( π A 1 0 ( X )) , M )   Hom S hv N is ( S m k ) ( X , M ) / / Hom A b A 1 k ( H A 1 0 ( X ) , M ) . By Lemma 3.3 the horizo ntal maps are isomorphisms, and by Lemma 3.1 the left vertical m ap is a bijecti on. Indeed, al l these bijections are functo rial in both variab les. It follo ws that the right 15 3 Birational geometry and strictly A 1 -in variant sheav es ver tical map is a biject ion functoriall y in both va riables as well. The result then follo ws from the Y oneda lemma. T o treat the ge neral case, it suf ﬁces to observ e that by [MV99, § 2 Propositio n 3.14] e very space X is A 1 -weakly equi v alent to a sp ace of simplicial dimension 0 . Remark 3.6 (A non-abelian v ariant) . One may also pro ve a non-abelia n v ersion of Proposition 3.5. Because one need s to k eep track of base points, this ver sion seems not as widely applica ble. Recall that if ( S , s ) is a pointe d sheaf of sets , we can con sider F A 1 ( S ) := π A 1 1 (Σ 1 s S ) . Results of [Mor11] sho w that this sheaf is strongly A 1 -in v arian t (see Remark 2.4). As abo ve, one can sho w that the canon ical map F A 1 ( S ) → F A 1 ( π A 1 0 ( S )) is an isomorphis m. This result is co mpatible with the pre vious results via the A 1 -Hure wicz theo rem (also prov en by Morel). The shea ve s F A 1 ( π A 1 0 ( S )) contai n “ non-abeli an” informatio n, e.g., related to ﬁnite cov ers with non-abelia n fundamen tal g roup. Lemma 3.7 ([Mor05b, Lemma 6.4.4]) . Sup pose M is a strictly A 1 -in va riant shea f of gr ou ps. If X is a smooth k -scheme, and U ⊂ X is an open subs cheme whose complement has codimension ≥ d in X , then the r estrictio n map H i N is ( X, M ) → H i N is ( U, M ) is a monomorphi sm if i ≤ d − 1 and a bijection if i ≤ d − 2 . Pro position 3 .8. Suppo se X is a smooth k -sch eme and U ⊂ X is an open s ubsche me of X . Assume the comple ment of U in X ha s codimens ion ≥ d , for some inte ger d > 0 . F or any commutat ive unital ring R , the canonic al maps H A 1 0 ( U, R ) − → H A 1 0 ( X, R ) , and H S 0 ( U, R ) − → H S 0 ( X, R ) ar e epimorphi sms if d = 1 , and isomorp hisms if d ≥ 2 . Pr oof. For the ﬁrst statement, if M is an arbitrary strictly A 1 -in v arian t shea f of R -modules, w e ha ve func torial bijections Hom A b A 1 k ( H A 1 0 ( X, R ) , M ) ∼ → M ( X ) by Lemma 3.3. Like wise, if M is an arbitrary strictly A 1 -in v arian t sheaf of R -modules with transfer s, we hav e functoria l bijectio ns Hom A b A 1 tr,k ( H S 0 ( X, R ) , M ) ∼ → M ( X ) by Lemma 3.3. Thu s, the result follo ws immediately from Lemma 3.7 and the Y oneda lemma. Stable birational equiva lence Recall t hat two smooth proper k -va rieties X and Y are stably k -biratio nally equiv alen t if X × P n is k -biration ally equi val ent to Y × P m for integ ers m, n ≥ 0 . In particu lar , if X is stably k -biration ally equi v alent to projecti ve space, then w e say tha t X is stably k -rational. Theor em 3.9. Supp ose k is an inﬁnite ﬁeld, an d R is a commutative unital ring. If X and X ′ ar e stably k -bir ationally equi valent smooth pr oper varieties then H A 1 0 ( X, R ) ∼ = H A 1 0 ( X ′ , R ) and H S 0 ( X, R ) ∼ = H S 0 ( X ′ , R ) . 16 3 Birational geometry and strictly A 1 -in variant sheav es Pr oof. Consider the compos ite map Y × A n ֒ → Y × P n − → Y . S ince H A 1 0 ( Y ) is A 1 -homoto py in v ariant , it follows th at the comp osite map H A 1 0 ( Y × A n ) − → H A 1 0 ( Y ) is an isomorph ism. On th e other hand, the map H A 1 0 ( Y × A n ) − → H A 1 0 ( Y × P n ) is an epimorphi sm by P roposit ion 3.8. A diagra m chase shows that projecti on map must then also be an isomorphism. The same ar gument works for Suslin homol ogy . If k has ch aracteristi c 0 , we may ﬁnish the proof by means of a stra ightforwa rd geometric ar gument using resolutio n of singula rities. Indeed, giv en any k -biration al morphism X → Y , there is a commutat iv e diag ram of k -birati onal morphisms of the form X ′ / /   Y ′   } } | | | | | | | | X / / Y where all the vertic al m aps are composites of a ﬁnite number of blow-ups with smooth centers. W e claim that it suf ﬁces to show that the morphism on zeroth A 1 -homolo gy sheav es induced by a blo w-up with smooth center is an iso morphism. If that is the case, since the composite map H A 1 0 ( X ′ ) → H A 1 0 ( Y ′ ) → H A 1 0 ( X ) is an is omorphism, we rea lize H A 1 0 ( X ) as a summan d of H A 1 0 ( Y ) and vice versa (by re ver sing the ro les of X and Y ). Let us check the result for f : X ′ → X , where f is a blo w-up at a codimension ≥ 2 smooth subsc heme Z ⊂ X . The induce d map X ′ \ f − 1 ( Z ) → X \ Z is an isomorph ism. The morphism X ′ \ f − 1 ( Z ) → X ′ is an open immersion w ith complement hav ing codimen sion 1 , and the morphism X \ Z → X is an open immersion with complement ha ving codimen sion ≥ 2 . By the pre vious prop osition, the map H A 1 0 ( X ′ \ f − 1 ( Z ) , R ) → H A 1 0 ( X ′ , R ) is an epi- morphism, the map H A 1 0 ( X ′ \ f − 1 ( Z ) , R ) → H A 1 0 ( X \ Z, R ) is an isomorphism, and the map H A 1 0 ( X \ Z, R ) → H A 1 0 ( X, R ) is an isomorphism. Composing the second and third of these iso- morphisms , w e see that the morphism H A 1 0 ( X ′ \ f − 1 ( Z ) , R ) → H A 1 0 ( X, R ) is an isomorph ism. Consequ ently , the morphism H A 1 0 ( X ′ , R ) → H A 1 0 ( X, R ) is an isomorphis m as well. The case of the zeroth Suslin homolog y shea f is ide ntical. If k is just inﬁnite, we ar gue as follo ws. If M is an arbitrary strictl y A 1 -in v arian t sheaf, then M is A 1 -local and so admits a Gersten resolution by Propositio n 2.23. By [CTHK97, Theorem 8 .5.1] it follo ws that M ( X ) is a birationa l in varian t of smooth pr oper v arieties (this e xplanat ion is exp anded slight ly in L emma 4.2). Since M was arbitrary , it follows from 3.3 that the same stateme nt holds for the zerot h A 1 -homolo gy sheaf. A n ana logous ar gument works for the zeroth S uslin homology sheaf. Remark 3.10 . As discuss ed in [AM11, § 2], w e kno w tha t A 1 -conne ctedness is a b irational in v ariant for ﬁelds ha ving charact eristic 0 . Ho wev er , we do not know whether the sheaf π A 1 0 ( X ) is itself a (stable ) birat ional in v arian t. The abo ve result sho ws that after abelian ization this is the case. Note also tha t the ﬁrst p roof abo ve implies t hat if X and X ′ are tw o schemes , not neces sarily proper , that can be linked by a chai n of blow-ups at smooth schemes, then H A 1 0 ( X ) ∼ = H A 1 0 ( X ′ ) . In fact, it is not at the moment kno wn whether π A 1 0 ( X ) is uncha nged by blo w-ups along smooth schemes! Remark 3.11 . In E xample 4.10 we will see that if k is a perfect ﬁeld and X is a smooth proper k - v ariety , then for any sepa rable ﬁ nitely gene rated extensio n L/k , one can iden tify H S 0 ( X )( L ) with C H 0 ( X L ) , functoriall y in L . Howe ver , biratio nal in v ariance for the C ho w group of 0 -cycles is a 17 4 An unramiﬁed characterization of A 1 -connectedness much older result. Indeed, if k has charact eristic 0 then [CTC 79, Proposition 6.3] establish es k - biratio nal in v ariance of C H 0 ( X ) , and Fulton [Ful98, Example 16.1.11 ] generalize s this t o arbitra ry charac teristic. 4 An unramiﬁed characterization of A 1 -connectedness In this section, we reca ll aspects of a “ﬁeld theoretic” or “unramiﬁed” approach to strictly A 1 - in v ariant shea ves pioneered by Morel [Mor05a, Mor05b, M or11] follo wing foundatio nal work of Rost [Ros96 ]. If M is a strictly A 1 -in v arian t sh eaf (see Deﬁnition 2.3), we now explai n how to identi fy section s of M over a smo oth scheme X in terms of the function ﬁeld k ( X ) of X a nd codimen sion 1 geometry of X , i.e., geometric discret e v aluations on k ( X ) . W e then provid e a number of examples of strictly A 1 -in v arian t shea ves. Combini ng thi s result with the discuss ion of § 3 (speciﬁcally Lemma 3.3) Lemma 4.2 explain s th e sense in which H A 1 0 ( X ) is a “uni versa l unramiﬁed in v ariant” as mentioned in the introduc tion. Theorem 4.15 and the subsequ ent corollar y gi ve the the unramiﬁed characteriza tion of A 1 -conne ctedness st ated in the introductio n. By means of an exampl e, we sho w that S uslin homolo gy is not suf ﬁciently reﬁned to detect A 1 -conne ctedness, or more loosely tha t A 1 -conne ctedness cannot be characterize d solely by m eans of “unramiﬁed in v ariant s with transfers ;” see Propositi on 4.17, Example 4.18 and Example 4.19 for more deta ils. Unramiﬁed elements Fix a ﬁeld k , and suppose M is a strictly A 1 -in v arian t sheaf (on S m k ). Suppose S is an essentia lly smooth k -scheme, i.e., a ﬁltering in verse limit of smooth schemes with smooth afﬁne transition morphisms . If we write S = lim X α , we can deﬁne M ( S ) = colim M ( X α ) . One can check that this colimit is inde pendent of the choice of ﬁltering in verse system deﬁning S . Thus, we can ext end M uniquely to a functo r on the category of ess entially smooth k -schemes. If F k denote s the cate gory of ﬁnitely gener ated exte nsion ﬁelds (morphisms are inclu sions of ﬁelds), then M gi ves rise to a (cov arian t) functor on F k . Abusing notati on, we will denot e all th ese functors by M . By Lemma 3.7, for an open immersion of smooth schemes U ֒ → X , the restriction map M ( X ) → M ( U ) is injecti v e. If L/k is a ﬁnitely generated exte nsion of k , ν is a geomet ric dis- crete va luation of L with v aluation ring O ν , and κ ν is the associated residu e ﬁeld then we hav e a morphism M ( O ν ) → M ( L ) ; this morphism is in jecti ve by what we’ ve just said. W e no w use these observ atio ns to deﬁne unramiﬁed groups associat ed with any stric tly A 1 -in v arian t sheaf. Deﬁnition 4.1. Suppose X is an irreduc ible smooth k -scheme. Gi ven x ∈ X (1) , write ν x for the corres ponding discre te valuat ion. For any x ∈ X (1) , the map M ( X ) → M ( O ν x ) is injecti ve. Set M ur ( X ) := \ x ∈ X (1) M ( O ν x ) , where the intersec tion is tak en in M ( k ( X )) . Lemma 4.2. The induced map M ( X ) → M ur ( X ) is an isomorphism. Thus , if X is a smooth variet y , the func tor M 7→ M ur ( X ) (fr om the cate gor y of strictly A 1 -in va riant shea ves of gr ou ps to the cate g ory of ab elian gr oups) is re pr esentabl e on A b A 1 k by the sheaf H A 1 0 ( X ) . 18 4 An unramiﬁed characterization of A 1 -connectedness Pr oof. A class α ∈ M ur ( X ) comes from a class M ( k ( X )) lying in the image of M ( O ν x ) as x ranges over the codimension 1 points of X . If α is in the image of O ν , then by deﬁnition there is an open subscheme U ν ⊂ X on w hich α is deﬁned. Thus, we can ﬁnd a collect ion of open subsch emes U i such tha t α ext ends to a class on U i for ea ch i . Using the sheaf prop erty and induct ion, these classes glue to giv e a class on the union U of the U i . By assumptio n, this unio n contai ns all co dimension 1 points of X . By Lemma 3.7, we kno w that if U ⊂ X is an open subsch eme whose closed complement has codimensi on ≥ 2 , the restriction map M ( X ) → M ( U ) is an isomorphi sm. T he secon d statemen t follo ws immediately from the ﬁrst one via Lemma 3.3. Cor ollary 4.3. Given M , M ′ ∈ A b A 1 k , then f : M → M ′ is an isomorphism if an d only if fo r every separ able , ﬁnitely gener ated extens ion L/k the morphism M ( L ) → M ′ ( L ) is an isomorphism. Pr oof. Since A b A 1 k is abelian, it suf ﬁces to prov e that the strictly A 1 -in v arian t shea ves ker( f ) and cok er( f ) are tri vial. Howe ver , it follows immediately from Lemma 4.2 that a strict ly A 1 -in v arian t sheaf A is trivi al if and only if A ( L ) is triv ial. Remark 4.4 . If M admits transfer s, the n the point of vie w on strictly A 1 -in v arian t shea ves (with transfe rs) discussed abo ve is closely relat ed to Rost’ s the ory of cycle module s [Ros96]. In fact, all of the examples of stric tly A 1 -in v arian t shea ves used belo w can be constructe d using either Rost’ s theory or a modiﬁcation dev eloped by Morel. The relations hip between strictl y A 1 -in v arian t shea v es with transfers and Rost’ s theory of cycl e modules has been dev eloped by D ´ eglise [D ´ eg08, D ´ eg10] (the former categor y is a localizati on of the latter). The counterpa rt of L emma 4.2 in the setting of cycle modules is gi ven by a result of Merkurje v [Mer08, Theorem 2.10]. In fact, the strictly A 1 -in v arian t sheaf with transfe rs associ ated with Merkurj ev’ s cy cle module by D ´ eglise’ s theory (or rather its deg ree 0 part) is precisely the 0 -th Suslin homology she af as we e xplain bel ow in Example 4.10. Remark 4.5 . There is a quotie nt m ap O ν → κ ν , and this induces a morphi sm M ( O ν ) → M ( κ ν ) . By choosi ng local paramet ers, one can deﬁne appropr iate notions of residu e maps for strictly A 1 - in v ariant she av es, though if M does n ot admit transfers, these residues depen d on th e cho ices made. This point of vie w is dev eloped in [Mor11, § 1], but we will not use this theo ry below . Unramiﬁed ´ etale cohomology and other examples Recall by Corollary 3.2 , if M is a strictly A 1 -in v arian t sheaf, and X is an A 1 -conne cted smooth scheme, the pullback map M ( Sp ec k ) → M ( X ) is a bijection . In this section, we gi ve a number of exampl es of unramiﬁed shea ves to show what kind of “v anish ing” statemen ts A 1 -conne ctedness entails . Example 4.6 . Suppos e k is a ﬁ eld, and n is an integ er that is not di visible by the characte ristic of k . Let H p ´ et ( µ ⊗ q n ) denote the (Nisne vich) she af (on S m k ) associate d with the presheaf U 7→ H p ´ et ( U, µ ⊗ q n ) . The sheaf H p ´ et ( µ ⊗ q n ) is strictly A 1 -in v arian t. T here are many ways to see this; for exa mple, it follows with a bit of work from homotop y in v ariance for ´ etale cohomology ([SGA73, Expose XV Lemme 4.2]). 19 4 An unramiﬁed characterization of A 1 -connectedness Lemma 4.7. Suppo se n is an inte ge r that is coprime to the char acteristic of k . If X ∈ S m k , then we have Hom A b A 1 k ( H A 1 0 ( X ) , H p ´ et ( µ ⊗ q n )) = H 0 N is ( X, H p ´ et ( µ ⊗ q n )) . If furth ermor e X is pr ope r , then the latt er gr oup is pr ecisely the gr oup H p ur ( k ( X ) /k , µ ⊗ q n ) . Pr oof. Since H p ´ et ( µ ⊗ q n ) is strictly A 1 -in v arian t, the equality in the state ment follo ws imm ediatel y from Lemma 3.3. By Lemma 4.2, if X is an irreduci ble smooth sche me, H p ´ et ( µ ⊗ q n )( X ) coincides with the subgr oup of H p ´ et ( k ( X ) , µ ⊗ q n ) consisting of unramiﬁed elements. Remark 4.8 . For a dev elopment of u nramiﬁed ´ etal e coh omology see, e.g., [CT95, § 4 ]. These groups admit an alternate descripti on. If L/k is a ﬁ nitely generated ﬁeld extens ion an d ν is a discrete v aluation of L/k with residue ﬁeld κ , there are residue maps ∂ ν : H p ´ et ( L, µ ⊗ q n ) − → H p − 1 ´ et ( κ, µ ⊗ q − 1 n ) . The subgro up of H p ´ et ( L, µ ⊗ q n ) can be identiﬁed with the intersectio n of the kernels of the residue maps ∂ ν as ν ranges over the dis crete valuat ions of L/k . Example 4.9 . Let X be a smooth k -var iety , w ith funct ion ﬁeld k ( X ) . Giv en a codimension 1 point, we write ∂ x for the res idue map associate d w ith the valuat ion ring d eﬁned b y x (see [Mil70 , L emma 2.1] for the const ruction of these resi due maps). W e deﬁne K M n ( X ) := k er ( K M n ( k ( X )) L x ∈ X (1) ∂ x − → M x ∈ X (1) K M n − 1 ( κ ν )) . W e recal l one functorial ity prope rty of these resid ue maps immediately subse quent to this examp le. If ϕ : A → B is a ring homomorphi sm, the induced map K M n ( A ) → K M n ( B ) is usually denot ed by either ϕ ∗ or Res A/B . One can sho w that K M n is a strictl y A 1 -in v arian t sheaf (see [Mor05a, § 2.2] for more details), and in fact K M n is a strictly A 1 -in v arian t sh eaf with transfe rs. For any integer m , multiplicatio n by m exte nds to a morphism of shea ves K M n × m − → K M n . The cate gory of strictly A 1 -in v arian t shea ves is abelian (see Corollary 2.22), an d the coke rnel of this morphis m o f shea ves, which is necessar ily str ictly A 1 -in v arian t, is denoted K M n /m . By construct ion, we hav e identiﬁcatio ns K M n ( L ) = K M n ( L ) and K M n /m ( L ) = K M n ( L ) /m for any ﬁni tely generated extensi on L/k . Example 4 .10 . Let k be a perf ect ﬁeld and let X be a s mooth proper k -va riety . Consider the functor assign ing to a ﬁnitely generated sep arable exte nsion L/k the grou p C H 0 ( X L ) . By mean s of duality in V oe vod sky’ s deri ved cate gory of motiv es, one can sho w (see [HK06, T heorem 2.2] or [D ´ eg10, § 3.4]) that for L as abo ve, there is a canoni cal ident iﬁcation H S 0 ( X )( L ) ∼ − → C H 0 ( X L ) . If one r eplaces C H 0 ( X L ) by its r ationaliza tion, a similar statement is true for S uslin homology with Q -coef ﬁcients. The sec tions of H S 0 ( X ) o ver a smoot h scheme U w ith function ﬁeld L can thus be descri bed either in terms of unramiﬁed elements, or as follo ws. If u is a codimension 1 point of U , there are specializati on maps C H 0 ( X L ) → C H 0 ( X κ ν ) [Ful98, § 20.3], and H S 0 ( X )( U ) can be realize d as the inters ection of the k ernels of these specializa tion maps. 20 4 An unramiﬁed characterization of A 1 -connectedness Remark 4.11 . B y the equi val ence of categorie s between an approp riate cate gory of strictly A 1 - in v ariant shea ves with transfers and Rost’ s cate gory of cycle modules [D ´ eg08, T h ´ eor ` eme 3.3], H S 0 ( X ) gi ves rise to Merkurje v’ s uni v ersal cycle module from [Mer08, § 2.3]. Lemma 3.3 com- bined with this observ ation can be use d to giv e an altern ate proof of [Mer08, Theorem 2.11] under the hypoth esis that k is perfect. Example 4.12 . Let k be a ﬁeld havi ng characterist ic unequal to 2 . For an y smooth k -scheme X , let W ( X ) be the associat ed W itt group. U sing the purity results of [OP99], one can study the Nisne vich sheaﬁﬁcatio n of this presheaf. Indeed, the Nisne vich sheaﬁﬁcatio n of the functor X 7→ W ( X ) deﬁnes a sheaf W , which we refer to as the unramiﬁed W itt sheaf. O ne can identi fy the group of sectio ns W ( X ) as the subgro up of W ( k ( X )) with triv ial (second) residues at points of codimen sion 1 of X . S ee [CTO89, Append ice], [Mor05a, § 2.1] and the referen ces therein fo r more details . While W itt groups do not admit transfer s in the same sense as Milnor K-theory or unramiﬁed ´ etale cohomol ogy , there is a notion of transfer for W itt groups. Example 4.13 . Let W be the un ramiﬁed W itt sh eaf a s ju st de ﬁned. Let I ( k ) denote the fun damental ideal in th e W itt ring of k , i.e., the ideal of ev en dimension al forms, an d let I n denote the n -th power of the fundamenta l ideal (which is kno wn to be additi vely generated by Pﬁ ster forms. W e then set I n ( X ) = I n ( k ( X )) ∩ W ( X ) . The presh eaf U 7→ I n ( U ) is a strictly A 1 -in v arian t sheaf by , e.g., [Mor05a, Theorem 2.3]. There is a monomorphis m of strictly A 1 -in v arian t sh eav es I n +1 ֒ → I n (coming fr om the corre sponding injecti ve maps o n sectio ns over ﬁelds), and it follows th at I n / I n +1 is also strictly A 1 -in v arian t. Example 4.14 . If k is a ﬁeld havi ng characte ristic expon ent p , let G m ′ denote the ´ etale sheaf G m ⊗ Z Z [ 1 p ] . Let H 2 ´ et ( G m ′ ) denote the Nisne vich sheaf associated with the presh eaf U 7→ H 2 ´ et ( U, G m ′ ) . One can sho w that H 2 ´ et ( U, G m ′ ) is strictl y A 1 -in v arian t. Using Lemm a 3.3 on e deduces that Hom A b A 1 k ( H A 1 0 ( X ) , H 2 ´ et ( G m ′ )) = H 0 N is ( X, H 2 ´ et ( G m ′ )) . Furthermor e, one can sho w using purity that if X is smooth a nd proper H 0 N is ( X, H 2 ´ et ( G m ′ )) is pre- cisely the coho mological Brauer group H 2 ´ et ( X, G m ′ ) . It follo ws from Propositi on 3.5 and Lemma 3.3 that if X is an A 1 -conne cted smooth scheme ov er an algebrai cally clo sed ﬁeld, then H 0 N is ( X, H 2 ´ et ( G m ′ )) is triv ial. That H 2 ´ et ( X, G m ′ ) is tri vial if X is A 1 -conne cted was ﬁrst observ ed by B. Bha tt; this result is stated (w ith proof) in [Gil09, Theorem 4.3]. For yet another proof of this statement, see [AM11, Proposit ion 4.2]. Giv en any object in the stable A 1 -homoto py cat egory (i.e., a P 1 -spect rum), Morel’ s connec tivit y results (recalled here as T heorem 2.17) sho w that the assoc iated stable A 1 -homoto py sheav es are strictl y A 1 -in v arian t. Thus, gi ven any cohomol ogy theory representabl e in the stable A 1 -homoto py cate gory , one can get correspon ding strictly A 1 -in v arian t shea ves; this applies notably to moti vic cohomol ogy , algebraic K-t heory , Hermitian K-theory , etc. Another notable examp le comes from the stable A 1 -homoto py group s of moti vic spheres; the known computation s are related to the Milnor - W itt K-theory shea ves mentione d in Example 2.15. Detecting A 1 -connectedness with A 1 -homology and birational shea ves Finally , we prov ide th e “unramiﬁed” character ization of A 1 -conne ctedness state d i n the introductio n as Theorem 5. This result can be viewed as an extensio n of [AH11b, Theorem 1], and t he techniques 21 4 An unramiﬁed characterization of A 1 -connectedness are similar . Theor em 4.15. If k is a ﬁeld, R is a commutative unital ring (e.g ., Z or Q ) and X is a smooth pr op er k -scheme , then X is A 1 -conne cted if and only if the canonical map H A 1 0 ( X, R ) → R is an isomorph ism. Pr oof. W e prove only the statement w ith Z -coef ﬁcients; th e correspo nding stat ement with R -coef ﬁcents follo ws by re peating the ar gument word for word w ith Z replaced by R . If X is A 1 -conne cted, then the canonical map in question is an isomorphis m by Proposition 3.5 (note: this does not require proper ness). In the oth er direct ion, suppose X is not A 1 -conne cted. It sufﬁce s to provide a strictly A 1 -in v arian t sheaf M such that the map M ( k ) → M ( X ) is not an isomorph ism. B y Lemma 3.3 this is equi v alent to proving that the map Hom A b A 1 k ( Z , M ) − → Hom A b A 1 k ( H A 1 0 ( X ) , M ) is not a bijec tion. Recall that a presheaf of sets F on S m k is called birational if for an y open dense immersion U ֒ → X the map F ( X ) → F ( U ) is a bijection . In [AM11, Theorem 6.2.1], we sho wed that if X is a smooth prop er k -scheme, then there are a biration al and A 1 -in v arian t sheaf π b A 1 0 ( X ) and a morphism X → π b A 1 0 ( X ) (funct orial in morp hisms o f smoot h proper schemes) c haracterize d by the proper ty that if L is any ﬁ nitely generat ed separable extens ion of k , then π b A 1 0 ( X )( L ) = X ( L ) /R . Here, the set X ( L ) /R is the set of R -equi vale nce classes of points in X ( L ) . Now , either X ( k ) is empty or not. Case 1 . Suppose X is A 1 -disco nnected, but X ( k ) is empty . In [AH11b, Lemma 2.4], we prov ed that the fr ee she af of abelian g roups on π b A 1 0 ( X ) , deno ted Z ( π b A 1 0 ( X )) , is birationa l and s trictly A 1 - in v ariant . Homomorphisms Z → Z ( π b A 1 0 ( X )) correspon d precisely to elements of π b A 1 0 ( X )( k ) . In [AH11b, Corollar y 2.9], we sho wed that X ( k ) is non- empty if and only if the map H A 1 0 ( X ) → Z is an epimorph ism. Case 2 . Assume that X is A 1 -disco nnected, b ut X ( k ) is non-empty . A ny rational point in X ( k ) induces a splitting Z → H A 1 0 ( X ) , and a corres ponding splitting Z → Z ( π b A 1 0 ( X )) . Since the catego ry of strictly A 1 -in v arian t shea ves of groups is abelian (see Corollary 2.22), we ha ve direct sum deco mpositions H A 1 0 ( X ) ∼ = Z ⊕ ˜ H A 1 0 ( X ) and Z ( π b A 1 0 ( X )) ∼ = Z ⊕ ^ Z ( π b A 1 0 ( X )) , and these split tings are compatibl e in the sense that the morphis m H A 1 0 ( X ) → Z ( π b A 1 0 ( X )) induces a morphism Z ⊕ ˜ H A 1 0 ( X ) → Z ⊕ ^ Z ( π b A 1 0 ( X )) that is the identity morphism on the ﬁrst summand. By [AM11, Corollary 2.4.4], X is A 1 -conne cted if and only if for e very ﬁnite ly generated separa ble ex tension L/k the set π b A 1 0 ( X )( L ) is reduced to a point. Thus, by assumption, there exi sts a separable e xtension K /k such that π b A 1 0 ( X )( K ) consis ts of (strictly) more than 1 element. Write X K for the base ex tension of X to S p ec K . Pullback giv es an ident iﬁcation H A 1 0 ( X )( K ) = H A 1 0 ( X K )( K ) by [Mor05 b , § 5.1 ], see in particul ar E xample 5.1.3. Thus, withou t loss o f genera lity , we can assume k = K and that π b A 1 0 ( X )( k ) consists of strictly more than 1 element. Each element of π b A 1 0 ( X )( k ) determine s a homomorph ism Z → H A 1 0 ( X ) that is non-tri vial, since the composi te m orphism Z → H A 1 0 ( X ) → Z ( π b A 1 0 ( X )) is non-tri vial. T aking the sum of these homomorphi sms giv es rise to a non-tri vial homomorp hism Z ( π b A 1 0 ( X ))( k ) → H A 1 0 ( X )( k ) . It follo ws immediately that ˜ H A 1 0 ( X )( k ) is non-tri vial. 22 4 An unramiﬁed characterization of A 1 -connectedness Combining Lemma 4.2 with Theorem 4.15 we deduce the follo wing result. Cor ollary 4.16. I f k is a ﬁeld and X is a smooth pr oper k -scheme, then X is A 1 -conne cted if and only if for eve ry strict ly A 1 -in va riant sheaf M , the canon ical m ap M ( k ) → M ur ( X ) is a bi jection. Detecting A 1 -connectedness with Suslin homology As it turns out, the key po int in the proof of Theore m 4.15 is th e use of strictly A 1 -in v arian t shea ves that do not necessarily possess transf ers. In gener al, e.g., if X is a smooth curv e of genus g ≥ 1 , the shea ves Z ( π b A 1 0 ( X )) do not possess transfer s; for a discussio n of this point, see [Lev10, § 2]. If X is a smooth proper k -v ariety , neither the zeroth Suslin homolog y sheaf of X with inte gral nor the v ariant with rationa l coef ﬁcients can detect A 1 -conne ctedness. W ith rational coef ﬁcients, this ﬁ ts into a general statemen t about S uslin homology of separ ably rationally connected varieti es (see, e.g., [Kol9 6 , Chapter 4 Deﬁnition 3.2]) and appeal to Example 4.18. If k is a perfect ﬁeld, [K ol96, Chap ter 4 Theorem 3.9.4] shows that sep arably rationa lly connected v arieties X /k hav e the proper ty that for ev ery separa bly close d extens ion L/k , ev ery two L -points can be connected by a P 1 . Pro position 4.17. If k is a perfect ﬁeld, and X is a smooth pr op er k -scheme such that for ever y separ ably closed ﬁeld L/k w e hav e X ( L ) /R = ∗ , then the cano nical morphism H S 0 ( X, Q ) → Q is an isomorph ism. Pr oof. Since the sheaf H S 0 ( X, Q ) is strictly A 1 -in v arian t, it suf ﬁces by C orollar y 4.3 to prov e that the map in qu estion is a n iso morphism on s ections ov er e very ﬁnitely ge nerated separa ble extensio n L/k . B y m eans of the identiﬁcation H S 0 ( X, Q )( L ) = C H 0 ( X L ) Q from E xample 4.10 it therefore suf ﬁces to prov e C H 0 ( X L ) Q is isomorp hic to Q under the stated hypothes es. If ¯ L is an algeb raic clos ure of L , then we hav e a restriction map for Chow gro ups C H 0 ( X L ) → C H 0 ( X ¯ L ) . The kernel of this m ap is a tors ion subg roup. Indeed, if Z is a cycle in C H 0 ( X L ) that goes to zero in C H 0 ( X ¯ L ) , then Z necessar ily goes to zero in a ﬁ nite exten sion L ′ /L . On the other hand p ullback follo wed by pushforw ard is multipli cation by [ L ′ : L ] , an d thus [ L ′ : L ] Z = 0 . Under the assumption on X , we know that C H 0 ( X ¯ L ) = Z . Upon tensoring with Q , restrictio n becomes an isomorph ism: it is surjecti ve since X L has a 0 -cycle of ﬁnite degre e coming from a point ov er some ﬁnite exten sion L ′ /L and injecti ve since the kernel of restriction is torsion and therefo re becomes tri vial after tensoring with Q . Example 4.18 . The classic examples of [AM72] provid e unirational (hence separably rationa lly co n- nected ) smooth proper v arietie s X ov er C that are no n-rationa l, bu t for which B r ( X ) is non-t rivi al. In parti cular , these varie ties hav e H S 0 ( X, Q ) = Q , but are not A 1 -conne cted, e.g., by Example 4.14 and Corollary 3.2. O ther e xamples along these lines are provided in [CTO89, Pe y93] and [Aso11]. Finally , we observe that the zeroth inte gral Suslin homology shea f of a smooth projecti ve v ari- ety cann ot detect A 1 -conne ctedness. Sin ce w e kno w that the zero th A 1 -homolo gy detect s rational points , and the zero th Suslin homology is relat ed to 0 -cy cles, a natural p lace to look for a countere x- ample i s among t he smooth p rojecti ve k -v arieties that po ssess a 0 - cycle of degree 1 b ut that h av e no k -rationa l point; we t hank Sash a Merkurje v fo r pointi ng out the follo wing e xample due to Pa rimala. 23 REFERENCES Example 4.19 . If X i s a smooth projecti ve varie ty such that the morphism H S 0 ( X ) → Z is an isomorph ism, it need not be the case that X is A 1 -conne cted. In [Par05, Theorem 3], Parimala gi ves a ﬁeld k (ha ving charac teristic 0 ) and a projec tiv e homoge neous space X under a connected reduct iv e linear algebra ic group over k such that i) X has a point o ver a degr ee 2 and deg ree p ( p odd) extensio n of k , b ut ii) has no k -rationa l point. P oint (ii) gu arantees that X is no t A 1 -conne cted. Again combining 4.10 and Corollary 4.3, to prov e that H S 0 ( X ) → Z is an isomorphism, it suf ﬁces to prove that C H 0 ( X L ) → Z is an isomorphi sm for ev ery ﬁnitely generated extensi on, separa ble exten sions L/k . I f K is a ﬁ eld, and X is a proje ctiv e homogene ous space under a con- nected reducti ve linear algebra ic group such that X ( K ) is non- empty , then choice of x ∈ X ( K ) determin es an isomor phism G/P ∼ → X , where P is a K -parabolic subg roup of G . 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