Toric varieties, monoid schemes and $cdh$ descent

TORIC V ARIETIES, MONOID SCHEMES AND cdh DESCENT G. COR TI ˜ NAS, C. HAESEMEYER, MARK E. W ALKER, AND C. WEIBEL Abstract. W e giv e conditions for the May er-Vietoris prop ert y to hold f or the algebraic K -theory of blow-up squares of toric v ar ieties and sche mes, using the theory of monoid sc hemes. These conditions are used to relate algebraic K -theory to topological cyclic homology in characte ristic p . T o ac hieve our goals, we dev elop man y notions for monoid sche mes based on classical algebraic geometry , suc h as separated and prop er maps and resolution of singularities. The goal o f this pap er is to prove Haesemeyer’s The o rem [18, 3.12] fo r toric schemes in any characteristic. It is prov en below as Co rollary 14.4. Theorem 0.1. Assume k is a c ommutative r e gular no etherian ring c ont aining an inﬁnite ﬁeld and let G b e a pr eshe af of sp e ctr a deﬁne d on the c ate gory of schemes of ﬁnite typ e over k . If G satisﬁes the Mayer-Vietoris pr op erty for Zariski c overs, ﬁnite abstr act blow-up squar es, and blow-ups along r e gularly emb e dde d close d subschemes, then G satisﬁes the Mayer-Vietoris pr op erty for al l abstr act blow-up squar es of t oric k -schemes obtaine d fr om su b dividing a fan. The application we hav e in mind is to understand the rela tionship b e t w een the algebraic K -theor y K ∗ ( X ) = π ∗ K ( X ) and top ological cyclic homolo gy T C ∗ ( X ) = { π ∗ T C ν ( X, p ) } of a to ric scheme ov er a r egular ring of ch ara cteristic p (and in particular o f tor ic v arie ties over a ﬁeld of c hara cteristic p ). Th us we cons ide r the presheaf of homotopy ﬁb ers {F ν ( X ) } of the ma p of pro- s pectra from K ( X ) to { T C ν ( X, p ) } . W or k of Geisser-Hess elholt [11, Thm. B], [1 2] shows that this ho- motopy ﬁber (r egarded as a pro-pr esheaf of sp ectra) satisﬁes the h yp otheses of Theorem 0.1 a nd hence a slight mo diﬁcation of the pro of of our theorem implies that it satis ﬁe s the May er-Vietoris prop ert y for all a bstract blow-up squares of tor ic schemes. W e will g iv e a rigoro us pro of of this in Cor o llary 14 .8 b elow. One ma jor to ol in o ur pro of will b e a theorem o f Biersto ne-Milman [1] which says that the s ingularities of a toric v ar iet y (or scheme) can b e resolved b y a s e quence o f blow-ups X C → X along a cent er C that is a smo oth, eq uiv ariant closed subscheme of X along whic h X is nor mally ﬂat. If o ne o nly had to consider toric schemes, this would allow one to use Haesemeyer’s or iginal argument to prov e Theo rem 0.1, Date : Octob er 22, 2018. 2000 Mathematics Subje ct Classiﬁc ation. 19D55, 14M25, 19D25, 14L32. Key wor ds and phr ases. abeli an monoids, monoid schemes toric v arieties, algebraic K - theo ry . Corti ˜ nas’ research was supp orted by Conicet and partially supp orted b y gran ts UBA CyT W386, PIP 112-200801-00900, and MTM2007-64704 (F eder funds). Haesemey er’s research was partially suppor ted by NSF grant DMS- 0966 821. W al ker’s research was partially supported by NSF grant DMS-0601666. W eib el’s research was supp orted by NSA and NSF grant s. 1 2 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL since toric schemes over a r egular ring ar e nor mal and Cohen-Mac a ula y . How ever, examples s ho w that the blow-up of a toric scheme along a smo oth center (ev en a po in t) can b e non-no rmal. Thus, even star ting with a toric scheme, the tow er o f blow-ups constr ucted by Bierstone- Milman will o ften inv olve non-nor mal schemes with a torus a ction. The pro of of o ur theorem requires us to work with a la rger class o f schemes, one co n taining all the schemes in this tow er. Beyond this, we need a class of schemes which is closed under pas sage to (po ssibly non-reduced) equiv ar ian t closed subschemes, pullba c ks and blow-ups. It turns out tha t all thes e op erations may be lifted to the categor y of monoid schemes of ﬁnite t yp e, and tha t the rea lizations of monoid schemes ov er a com- m utative regular ring k containing a ﬁeld fo rm a cla ss of schemes with the a b ov e- men tioned prop erties. The k -realiza tion of an a ﬃne monoid scheme is a scheme of the form Spec k [ A ], with A an a b elian mo noid; the k -r ealization of a monoid scheme (Deﬁnition 5.3) is a scheme over k which is cov ered b y aﬃne o pen sub- schemes of this form, with homo morphisms of the underlying monoids inducing the gluing maps b et ween these op en subschemes. T o achiev e o ur go als, it is e asier to work dir e c tly with the catego r y of mo noid schemes, a nd Sections 1 – 3 of this pap e r are devoted to a in tro duction to mono id schemes. T oric monoid schemes are introduced in Section 4 and the relatio n to toric v arie ties is carefully describ ed. In Sec tio ns 5 a nd 6, w e prove that the k -realization functor pr eserves limits and show that many monoid scheme-theoretic pr oper ties translate well into algebra ic geo metry . Pro jectiv e mo noid schemes, blow-ups a nd prop er maps a r e in tro duced in Sections 7 a nd 8. After int ro ducing the technical notion of p ctf monoid schemes in Section 9 , bir ational maps and res o lution of singularities are g iv en in Sections 10 and 11. The last pa rt of this pap er (Sections 1 2 – 14) is devoted to the notion o f coho - mological descent (Deﬁnition 1 2.11), the pr o of of our Ma in Theorem 0.1 and its application to alg ebraic K -theory and top ologica l cyclic homolog y . As far a s the authors are aware, this pap er pr e s en ts the ﬁrst attempt at a sys tem- atic study of geometric pro perties of monoid schemes within the category of monoid schemes, and the rela tionship of these with the geometric pro perties o f their real- izations. The idea of a monoid scheme itself go es ba c k at lea st to Kato [22], and general deﬁnitio ns were given by Deitmar in [8] and (under the na me M 0 -schemes) by Co nnes, Consani and Marcolli in [4]. Deitmar studies no tio ns of ﬂatness and ´ etaleness fo r monoid schemes, and intro duces discrete v aluation mo noids. New in this pap er is our systematic inv e s tigation o f separ atedness, prop erness, general v aluatio n monoids and the v aluative criteria, pro jectivity and blowing up, and the introduction of a cla ss o f mono id s chemes (the ab ov e mentioned p ctf monoid schemes) with b e tter formal prope rties than o nly those g iven b y fans, yet av oiding the worst pa thologies of non-ca nce llativ e mono ids. 1. Monoids Since we know of no suitable r eference for the facts we need concer ning mono ids and their pr ime s pectra, we b e g in with a sho r t e x pos´ e of this bas ic material. Unless o ther wise stated, a monoid in this pap er is a p ointed ab elian monoid; i.e., an ab elian monoid ob ject in the symmetric monoida l catego ry of pointed sets with smash pr oduct as monoidal pr oduct. More explicitly , a monoid is a p oin ted set A with basep oint 0 , equipp ed with a pa iring µ : A ∧ A → A (wr itten µ ( a, b ) = ab ) TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 3 that is ass ocia tive and co mmutative and has a n ide ntit y ele ment 1. The basep oint is unique b ecause it is character ized by the pro perty that 0 a = 0 for a ll a ∈ A . F or example, if R is a co mm utative ring, then forg etting addition gives a monoid ( R , × ) of this type. Sometimes + notation is used for µ , for example in applications to toric v arieties; in these ca ses we wr ite 0 for the identit y element, and ∞ for the basep oint . W e can conv ert any unp oin ted abelia n monoid B into a p oin ted abelia n monoid B ∗ by adjoining a basep oint. Neither the zero monoid { 0 } no r the monoid { 0 , 1 , t } with t 2 = 0 are of this form. A morphis m of monoids is a ma p of p oin ted sets preserving the m ultiplicative ident ity a nd multiplication. The initial mo noid is S 0 = { 0 , 1 } with 1 · 1 = 1, and the initial map ι A : S 0 → A is such that the identit y on A eq uals the comp osition A ∼ = / / S 0 ∧ A ι A ∧ id / / A ∧ A µ / / A. L o c alization. A multiplic atively close d subset S ⊂ A is a subset cont aning 1 a nd closed under mu ltiplication. Giv en a multiplicativ ely closed subset S of A , the lo c alization S − 1 A consists of equiv ale nce clas s es of fractions of the form a s with a ∈ A and s ∈ S . As usual, a s = a ′ s ′ if and only if as ′ s ′′ = a ′ ss ′′ for some s ′′ ∈ S , and the op eration in S − 1 A is g iv en b y multiplication of fractions. There is a canonica l monoid ho momorphism A → S − 1 A s ending a to a 1 , and a, b ∈ A are mapp ed to the same element o f S − 1 A if and o nly if as = bs for some s ∈ S . An ide al I in a monoid A is a p oint ed subs e t such that AI ⊆ I . If I ⊂ A is an ideal, A/I is the monoid obtaine d by co lla psing I to 0 — i.e., it is ca nonically isomorphic to ( A \ I ) ∪ { 0 } with the unique mu ltiplication rule that ma k es the canonical surjection A ։ A/I into a mo rphism of monoids. More g enerally , any surjective homomor phism of monoids A → B is the quotient by a c ongruenc e , i.e. an equiv a le nc e relation compa tible with the monoid o pera tion. Every no n-zero monoid A has a unique maximal idea l (written m A ), namely the complement of the submono id o f units U ( A ) := { a ∈ A | a b = 1 for so me b } . W e say that a monoid morphism g : A → B is lo c al if g ( m A ) ⊆ m B or, equiv alent ly , if g − 1 ( U ( B )) ⊆ U ( A ). A prime ide al is a prop er idea l p ( p 6 = A ) whose complement S = A \ p is clo sed under mu ltiplication; in this case we write A p for the lo caliza tion S − 1 A . The dimension of A is the supr e mum of the lengths of a ll ch ains o f pr ime ideals, and the height of p is the dimens io n of A p . Since the intersection of a n arbitrary chain of primes is pr ime, every prime ideal co n tains a minimal prime ideal (by Zor n’s Lemma). Lemma 1.1. F or every multiplic atively close d subset S of A with 0 6∈ S , ther e is a prime ide al p of A such that S − 1 A = A p . Pr o of. Since S − 1 A is a no n-zero mo noid it has a maxima l (pro per) ideal m ; the inv er se image o f m in A is a prime idea l p . Let T denote A \ p ; then S ⊂ T and any t ∈ T is a unit in S − 1 A . Hence there ar e homo morphisms S − 1 A → T − 1 A = A p and T − 1 A → S − 1 A covering the identit y of A . Hence b oth comp osites S − 1 A → S − 1 A and A p → A p are identit y maps, by the universal prop erty of lo calizatio n.  W e let MSp ec( A ) denote the set of pr ime ideals of A ; it is a top ologic a l space when eq uipped with its Zar is ki top ology , in which closed s ubsets ar e those of the 4 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL form V ( I ) = { p | I ⊂ p } for a n ideal I of A . The pr incipal op e n subsets D ( s ) = { p ∈ MSp e c( A ) | s / ∈ p } = MSp ec( A [1 /s ]) form a basis for the Zarisk i top ology . The space MSp ec( A ) is quasi-co mpact, since any op en D ( s ) cont aining the unique maximal ideal m A m ust hav e D ( s ) = MSpec( A ). There is a shea f of monoids A on MSp ec( A ) whose stalk at p is A p ; if U is op en then A ( U ) is the subset of Q p ∈ U A p consisting of elements which lo cally co me from some S − 1 A . Explicitly , A ( U ) = { a ∈ Y p ∈ U A p : ( ∀ p ∈ U )( ∃ s / ∈ p , x ∈ A )( ∀ q ∈ U ) s / ∈ q ⇒ a q = a s } . In pa rticular, A = A m A , and A ( D ( s )) = A [1 /s ]. More gener a lly any ide a l I of A determines a sheaf I on MSp ec A by I ( U ) = { a ∈ A ( U ) : ( ∀ p ∈ U ) a p ∈ A p · I } Example 1.2. The free (a belian) p oin ted monoid on the set { t 1 , . . . , t n } is the mul- tiplicative mono id F n consisting of a ll monomials in the p olynomial ring Z [ t 1 , ..., t n ] (together with 0). Ea c h of the 2 n subsets of { t 1 , ..., t n } generates a prime ideal p , and every prime ideal of F n has this for m. W e write A n for MSp ec( F n ). If A → B is a mor phism of monoids, then the inv erse ima ge o f a prime ideal is a prime ideal, and w e have a contin uous map MSpec ( B ) → MSp ec( A ). If I is an ideal of A then MSp ec( A/I ) → MSp ec( A ) is a closed injection onto V ( I ). If S is m ultiplicatively clo s ed in A then either S − 1 A = 0 (in which ca se MSp ec S − 1 A = ∅ ) or S − 1 A = A p for some p (Lemma 1.1); in either cas e ι : MSpec ( S − 1 A ) → MSpec( A ) is an injection o nto the set o f primes that ar e disjoin t from S . The restriction ι − 1 ( A ) to this subset is the sheaf of monoids on MSp ec( A p ). Recall that a po in t x 1 of a top ological spa ce X is c a lled gener alization o f a p oin t x 0 (and x 0 is ca lle d a sp e cialization of x 1 ) if x 0 is in the closure of x 1 . F or example, if p , q ∈ MSp ec A , then p genera liz e s q if and only if p ⊂ q . Lemma 1. 3. L et p b e a prime ide al in a monoid A . Then MSpec( A p ) → MSpec ( A ) is an inje ction, close d under gener alization, and the fol lowing ar e e qu ival ent: (i) MSp ec( A p ) is op en in MSp ec( A ) . (ii) MSp e c ( A p ) = D ( s ) for some s ∈ A . (iii) Ther e is an s ∈ A such that A p = A [1 / s ] . Pr o of. The ﬁrst a ssertion w as o bserved ab ov e. Since D ( s ) = MSp ec( A [1 /s ]), (iii) is equiv alent to (ii), a sp ecial case of (i). Co n versely , supp ose that U = MSpec( A p ) is the co mplemen t of V ( I ) for so me ideal I of A . Then U = ∪ s ∈ I D ( s ). In par ticular, there is a n s in I such that p ∈ D ( s ). But then U ⊆ D ( s ) and hence U = D ( s ).  Example 1.4. Let A b e the free p oin ted abe lian monoid generated b y the inﬁnite set { t 1 , t 2 , . . . } . If p is the prime idea l generated by some ﬁnite subset of the t i ’s then MSpec( A p ) ca nnot b e op en in MSp ec( A ). Indeed, if it were op en then by Lemma 1.3 it would hav e the fo rm D ( s ) for so me element s ∈ A . But any s inv olves only a ﬁnite nu mber o f v ariables, so the prime ideal t j A be longs to D ( s ) for inﬁnitely many t j 6∈ p . In par ticular, D ( s ) cannot b e contained in MSp ec A p . TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 5 Lemma 1.5. If A is ﬁnitely gener ate d as a monoid, then MSp ec( A ) is a ﬁnite p artial ly or der e d set. If S is a mu ltiplic ative subset of A , t hen S − 1 A is also ﬁnitely gener ate d, and MSp ec( S − 1 A ) is op en in MSp ec( A ) . Pr o of. Suppose A is g e nerated by x 1 , . . . , x m . Then for any prime ideal p , the m ultiplicative subset S = A \ p is ge nerated by { x i | x i / ∈ p } . Indeed, if s ∈ S , then s = Q i x e i i with e i = 0 whenever x i ∈ p . Thus A has at most 2 m prime ideals. By Lemma 1.1, we may assume S = A \ p for some pr ime p . If s is the pro duct of the gener ators of S , then A p = A [1 /s ]. By L emma 1.3, MSpe c ( A p ) is o pen.  W e say A is c anc el lative if for a, b, c ∈ A the conditions ab = ac and a 6 = 0 together imply that b = c . In this case, the unp ointed monoid A \ { 0 } injects into its gr oup co mpletio n and { 0 } is the unique minimal prime idea l o f A . W e deﬁne the p ointe d gr oup c ompletion of A to be the p ointed monoid A + obtained b y adjoining a basep oint to the usual g r oup co mpletion of the unp ointed monoid A \ { 0 } . No te that A is a p oin ted submonoid o f A + , and that A + is the lo caliza tion A { 0 } of A a t the minimal prime ideal. W e say A is torsionfr e e if whenever a n = b n for a, b ∈ A and some n ≥ 1, we hav e a = b . The monoid { 0 , ± 1 } is cancellative but not tor sionfree. If A is cancella tiv e and A + \ { 0 } is a tor sionfree abelia n group, then A is torsionfree . An element is nilp otent if a n = 0 for some n , a nd the nilr adic al o f A is the set nil( A ) of nilpo ten t elements. It is easy to prove (using Zorn’s lemma as in ring theory), that nil( A ) is the intersection of the minimal prime idea ls of A . W e say that A is re duc e d if nil( A ) = 0, and set A red = A/ nil( A ). An y clo sed subset Z of X = MSp ec( A ) deﬁnes a larg e s t ideal I s uc h tha t Z = V ( I ), and A/I is a r e duce d monoid. Indeed, if Z = V ( I 0 ) then A/I = ( A/I 0 ) red ; I is the intersection of the prime ideals co n taining I 0 . An ticipating Lemma 2.9, we write ¯ Z eq for MSpec ( A/I ) and call it the e qu ivari ant closur e of Z in X . F o r example, ¯ X eq is MSp ec( A red ). Another impo rtant sp ecial cas e is when Z = { p 1 , . . . , p l } is a set of prime idea ls of A ; in this case ¯ Z eq = MSp ec( A/ ∩ p i ). Deﬁnition 1. 6 . The normalization of a ca ncellative monoid A is deﬁned to be the submonoid A nor = { α ∈ A + | α n ∈ A for some n ≥ 1 } of A + . W e say that A is normal if it is cancellative a nd A = A nor . The normalizatio n of S − 1 A is S − 1 A nor . If A is torsio nfr ee then s o is A nor . R emark 1.6.1 . If A is cancellative then MSp e c( A nor ) → MSp ec( A ) is a top ological homeomorphism. Indeed, if p is a prime ideal of A then p nor := { b ∈ A nor | ( ∃ n ) b n ∈ p } is a pr ime ideal of A nor and p = p nor ∩ A . It is ea sily s een that every prime idea l of A nor has the for m p nor for some p . R emark 1.6.2 . If A is normal a nd p is a prime ideal, then A/ p is also nor ma l. Indeed if x, y ∈ A a nd s ∈ A \ p ar e such that x n and s n y ar e mapp ed to the same element of A/ p , then either x n = s n y in A or x, y ∈ p . Since A is assumed normal, it follows that either x ∈ p or there is a z ∈ A such that x = sz in A . More gener a lly , let f : A → B b e a morphism of monoids. W e say that f is inte gr al if for every b ∈ B ther e is an integer n ≥ 1 such tha t b n lies in the image of A , and w e say that f is ﬁnite if there ex is t b 1 , . . . , b n ∈ B ( n ≥ 1) such that B = S i Ab i . The normalizatio n A → A nor is integral but not always ﬁnite. 6 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL Lemma 1.7. L et A f − → B b e a monoid m orphism with B ﬁn itely gener ate d over A . i) If f is inte gr al, then f is ﬁ nite. ii) If f is ﬁnite and B is c anc el lative, then f is int e gr al. Pr o of. Choo se a surjection A [ t 1 , . . . , t n ] ։ B , with the t i mapping ont o generators b i of B over A . If f is integral, then ther e is an m such tha t b m i is in the image of A for a ll i ; thus every element of B ca n be written as a pro duct f ( a ) c j , where a ∈ A and c j is a mono mial on the b i with exp onents ≤ m . This proves i). Next assume that f is ﬁnite a nd that B is cancella tiv e. Let b 1 , . . . , b n ∈ B b e such that B = S i Ab i . F or ea c h i , we choos e an index π ( i ) and a i ∈ A such tha t b 2 i = a i b π ( i ) ; then π is a map fro m the ﬁnite set { 1 , . . . , n } to itself. F or each ﬁxed i , the iterates π r ( i ) ca nnot all b e dis tinct, so ther e exist s ≥ 1 and r ≥ 1 such that j = π r ( i ) s a tisﬁes π s ( j ) = j . Hence ther e is a n a ∈ A and m ≥ 1 such that b m j = ab j . Because B is cancella tive, this implies that b m − 1 j = f ( a ). Th us b j and hence b i is integral ov e r A , as requir ed.  R emark 1 .7 .1 . The hypothesis that B b e cancellative in par t ii) of Lemma 1.7 is necessary . F or ex ample, the monoid B g enerated by x, y sub ject to y 2 = xy contains the free monoid A genera ted by x ; the extensio n A ⊂ B is ﬁnite but not integral. F or a p oin ted se t X a nd commutativ e r ing k , k [ X ] deno tes the free k -mo dule on X , mo dulo the summand indexed by the base p oint o f X . If A is a p ointed monoid, k [ A ] is a ring in the usual wa y , with multiplication g iv en b y the pr o duct rule for A . If B is an unp ointed monoid, k [ B ∗ ] coincides with the usual monoid ring for B with k co eﬃcients. If I is an idea l of the monoid A then k [ I ] is an ideal of the ring k [ A ], and k [ A/I ] = k [ A ] /k [ I ]. If I is prime, k [ I ] need no t be a prime ideal. The category of p oint ed monoids has all small co limits. F or ex a mple, the co- pro duct of A 1 and A 2 is the smash pro duct A 1 ∧ A 2 ; the maps from A 1 and A 2 to A 1 ∧ A 2 send a 1 to a 1 ∧ 1 and a 2 to 1 ∧ a 2 . The functor A 7→ k [ A ] pres erv es co limits since it has a right adjo int, se nding an alg ebra R to ( R , × ), the underlying mult i- plicative monoid of R ; in particular, the natural map k [ A 1 ] ⊗ k k [ A 2 ] → k [ A 1 ∧ A 2 ] is an isomor phism. More g enerally , the pusho ut A 1 ∧ C A 2 of a diagra m (1.8) C f / / g   A 2   A 1 / / A 1 ∧ C A 2 is the quotient of A 1 ∧ A 2 by the congruence generated by ( a 1 f ( c ) , a 2 ) ∼ ( a 1 , g ( c ) a 2 ). Note that k [ A 1 ∧ C A 2 ] ∼ = k [ A 1 ] ⊗ k [ C ] k [ A 2 ]. Lemma 1.9. Every prime ide al p of A 1 ∧ A 2 has the form p 1 ∧ A 2 ∪ A 1 ∧ p 2 for unique prime ide als p 1 and p 2 . Explicitly, p i is the inverse image of p under the c anonic al inclusion A i → A 1 ∧ A 2 . Pr o of. Giv en a pr ime ideal p of A 1 ∧ A 2 , set p 1 = p ∩ A 1 , p 2 = p ∩ A 2 and q = p 1 ∧ A 2 ∪ A 1 ∧ p 2 . Then q is prime b ecause its complement is ( A 1 \ p 1 ) × ( A 2 \ p 2 ), which is multiplicativ ely closed. Clearly q ⊆ p ; to see that q = p , co nsider a n element a 1 ∧ a 2 of p . As p is prime, either a 1 ∧ 1 or 1 ∧ a 2 is in p . In the ﬁrst ca se, a 1 ∈ p 1 so a 1 ∧ a 2 is in p 1 ∧ A 2 ⊆ q ; in the second ca se, a 2 ∈ p 2 so a 1 ∧ a 2 is in A 1 ∧ p 2 ⊆ q .  TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 7 Example 1.10. If T is the free monoid o n one elemen t t , then A ∧ T is the analogue of a p olynomial ring ov er A , and k [ A ∧ T ] = k [ A ][ t ]. F or any prime idea l p of A there a re exactly tw o primes of A ∧ T o ver p : the extended prime p ∧ T and the prime gener ated by p and t (i.e., p ∧ T ∪ A ∧ { t n : n ≥ 1 } ). The map MSpec( A ∧ T ) → MSp ec( A ) induced by the canonical inclusion A → A ∧ T is b oth op en a nd clo sed, b ecause the imag e of D ( at n ) is D ( a ) and the ima ge of V ( I ) is V ( I ∩ A ). Prop osition 1. 1 1. Given a pushout diagr am (1 .8 ) , every prime ide al of A 1 ∧ C A 2 has the form p 1 ∧ A 2 ∪ A 1 ∧ p 2 for unique prime ide als p 1 in A 1 , p 2 in A 2 . Mor e over, the ide al p 1 ∧ A 2 ∪ A 1 ∧ p 2 of A 1 ∧ C A 2 is prime if and only if p 1 and p 2 have a c ommon inverse image in C . Pr o of. If p is a pr ime in A 1 ∧ C A 2 , its inv erse image in A 1 ∧ A 2 is prime; by L e mma 1.9 it has the form p 1 ∧ A 2 ∪ A 1 ∧ p 2 , where p i ⊂ A i are the inv erse ima ges of p . Since A 1 ∧ C A 2 is a quotient, this pr o ves the ﬁrst as sertion; b ecause (1.8) commutes, p 1 and p 2 hav e a common inv erse image in C . Conv ersely , supp ose that p 1 and p 2 hav e a co mmon inverse image q in C , and set S 1 = A 1 \ p 1 , S 2 = A 2 \ p 2 and I = p 1 ∧ A 2 ∪ A 1 ∧ p 2 ⊂ A 1 ∧ C A 2 . T o see that the ideal I is prime, it suﬃces to show tha t the image o f S 1 × S 2 in A 1 ∧ C A 2 is disjoint from I . Since p 1 and p 2 are pr ime, a 1 f ( c ) ∈ S 1 if and only if a 1 ∈ S 1 and c 6∈ q , while g ( c ) a 2 ∈ S 2 if and o nly if a 2 ∈ S 2 and c 6∈ q . It follows that ( a 1 f ( c ) , a 2 ) is in S 1 × S 2 if and o nly if ( a 1 , g ( c ) a 2 ) is. T hus S 1 × S 2 is clos ed under the e quiv alence relation deﬁning A 1 ∧ C A 2 , and its image in A 1 ∧ C A 2 is disjoint from I .  2. Monoid schemes W e will need to consider monoid schemes , so metimes k no wn as “ sc hemes ov er the ﬁeld with one ele men t”. These a re the ob jects which result by gluing together sp ectra of p ointed monoids alo ng op en s ubsets, and will be related to classica l schemes in Section 5. The theory of monoid schemes was developed by Kato [22], Deitmar [8 ], Connes-Co nsani-Marcolli [6], [4], [5], etc. The survey [23] by L´ opez Pe˜ na and Lo rscheid gives a nice ov erview of this notion a nd related idea s (but s ee Remark 4.4.1 b elow). A monoid sp ac e is a pair ( X , A X ) consis ting of a top ological space X a nd a shea f A X of p ointed ab elian monoids on X . A morphism of monoid sp ac es fro m ( X , A X ) to ( Y , A Y ) is given by a c o n tinuous map f : X → Y to g ether with a mor phism of sheav es f # : f − 1 A Y → A X on X (or, equiv alently , a morphism f # : A Y → f ∗ A X of sheav es o n Y ) that is lo c al in the sense that the maps on stalks A Y ,f ( x ) → A X,x are loca l mo rphisms o f monoids, for all x ∈ X . By abuse of no tation, w e will often simply write X for the monoid spa ce ( X, A X ). The a sso ciation A 7→ MSp ec( A ) ex tends to a fully faithful contra v ar ian t functor from mo noids to monoid spaces, which we will call MSp ec by abuse o f notatio n. An aﬃne monoid scheme is a monoid spa ce isomor phic to MSp ec( A ) for s ome monoid A . A monoid scheme is a mono id space ( X, A ) such that every p oint has an op en neighbo r hoo d U such that ( U, A| U ) is isomo r phic to an aﬃne monoid scheme. If ( U, A| U ) ∼ = MSpec A we shall often abuse notation and write U = MSpec A . A mor phis m o f monoid schemes is just a mo rphism o f the under lying monoid spaces. The dimension of a mo noid scheme is the largest dimension of its aﬃne op en neighborho o ds. 8 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL Lemma 2. 1. L et ( X , A ) b e a monoid s cheme. F or any op en U ⊆ X , the monoid sp ac e ( U, A| U ) is a monoid scheme. The scheme ( U , A| U ) is c al le d the op en subscheme of X asso ciate d to U . Pr o of. If x ∈ U and V = MSp ec( A ) is an aﬃne o p en neighbor hoo d of x in X , U ∩ V is also op en. Since U ∩ V is the union of basic op en subschemes D ( s ) o f V , x ha s a neighborho o d of the form D ( s ), and D ( s ) = MSpec ( A [1 /s ]) is aﬃne.  W e say that a monoid scheme is cancellative (r e s p., reduced, no rmal, ...) if its stalks are ca ncellativ e mo noids (r esp., reduced, normal, ... monoids), or equiv a- lent ly , if its mono ids of sections ar e cancella tiv e (resp., ...). Example 2. 2. The pro jective line P 1 is obtained by gluing MSp ec( { t n , n ≥ 0 } ∗ ) and MSp ec( { t n , n ≤ 0 } ∗ ) alo ng MSp ec( { t n , n ∈ Z } ∗ ). This monoid scheme is connected, torsio nfr ee and normal. Partial or der, maximal and minimal p oints. Recall that the points of an y topo logical space may be partially o rdered b y the relation that x ≤ y if and only if y is in the closure of { x } . In this wa y we can sp eak o f maximal and minimal po in ts. The maximal p oints are the clo sed p oin ts; minimal p oint s ar e also ca lled generic p oints. F or the to polog ical space MSp ec( A ) of a mo noid A , we have p ≤ q if and only if p ⊆ q . Minimal p oints ex ist in a n y monoid scheme b ecause, as noted b efore 1.1, every prime idea l contains a minimal prime idea l. Lemma 2. 3. Each c anc el lative monoid scheme X de c omp oses as t he disjoint union of (close d and op en) monoid su bschemes X η , e ach the closur e of a un ique minimal p oint η of X . In p articular, if X is c onne ct e d then it has a unique minimal p oint. Pr o of. F or each minimal p oin t η ∈ X , let X η denote the closur e of η in X . Giv en x ∈ X , cho o se an a ﬃne neig h b o rho od U x = MSpec ( A ) of x . If y is the p oint of X corresp onding to m A , then A = A y . Since A is ca ncellativ e, U x has a unique minimal p oint η , so U x ⊆ X η . It follows that X η = ∪ U x is open (and closed) in X , and that X is the disjoint union of the X η .  Lemma 2.4. L et X b e a monoid scheme and U ⊆ X an op en subscheme. The n the fol lowing ar e e quivalent. (i) U is an aﬃne monoid scheme. (ii) U has a unique maximal p oint. If X = MSpec ( A ) , every aﬃne op en subscheme is MSpec ( A p ) for some p . Pr o of. Since mono ids hav e unique maximal ideals, (i) implies (ii). Conv ersely , suppo se that U has a unique maximal p oin t x . Note that U = { y | y ≤ x } by deﬁnition of the or der relation. If MSpec( A ) is an a ﬃne o p en neighbor hoo d of x , then U ⊆ MSp ec( A ), so w e may assume that X = MSpe c ( A ). In this case U = MSp ec( A x ) by Le mma 1.3.  Deﬁnition 2.5. Let f : Y → X b e a map o f mono id s c hemes. W e say that f is a close d immersion if it induces a homeomorphism of Y on to its image (equipp ed with the subspa c e top ology), and for every aﬃne op en subscheme U = MSp ec( A ) of X (i) the op en s ubs c heme V = U × X Y of Y is aﬃne (p ossibly empty) and (ii) the map A X ( U ) → A Y ( V ) is surjective. A close d su bscheme of a mo noid scheme TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 9 X is an iso morphism class o f closed immer sions into X . E a c h closed subscheme is represented b y a monoid scheme ( Z, A Z ) s uc h that f is a subspace inclusio n Z ⊂ X . A clo s ed immersion f : Y → X is called e quivariant if in a ddition each such A X ( U ) → A Y ( V ) is the quotient by an ideal. The terminology “equiv a riant c lo sed immers ion” co mes from the theory of toric v arie ties : the equiv ariant clo sed subschemes of a to r ic v a riet y ar e pr ecisely those closed s ubsc hemes that are equiv a r ian t for the ac tio n of the under ly ing torus. W e will see in Sec tio n 4 tha t a tor ic v a r iet y has an asso ciated toric mo noid scheme, and that the equiv ar ia n t closed subschemes of the monoid scheme determine equiv ariant closed subschemes o f the toric v ariety . Example 2.6. Given a closed subset Z of a monoid scheme X , there is an equi- v aria n t reduced closed subscheme Z red asso ciated to Z , deﬁned by patching; if X = MSp ec( A ) a nd Z = V ( I ) then Z red = MSp ec( A/I ) red . Lemma 2.7. Any surje ct ion of monoids A π − → B determines a close d immersion MSpec( B ) ⊆ MSp ec( A ) . If B = A/I then it is an e quivariant close d subscheme. Pr o of. Set Y = MSp ec( B ) and X = MSpec ( A ). The map π ∗ : Y → X of underlying spaces is injective, since if q 1 6 = q 2 then π − 1 ( q 1 ) 6 = π − 1 ( q 2 ). If a ∈ A , the ima g e of the basic o p en D ( π ( a )) ⊆ Y is D ( a ) ∩ π ∗ ( Y ). Thus Y is homeomorphic to π ∗ ( Y ). Let U ⊆ MSp ec( A ) b e a n aﬃne op en subscheme. By Lemma 2.4 there is a prime p o f A such that U = MSp ec( A p ); b y Lemma 1 .3, U = D ( s ) for some s . Hence U ∩ Y = D ( π ( s )) = MSpec ( B [1 /s ]), which is a ﬃne or empty . Since A [1 /s ] → B [1 /s ] is onto, Y → X is a close d immer sion.  R emark 2.7.1 . A clo sed subscheme Y ⊂ X need not determine a closed subset of the underlying top ological space. F or example, the diagonal embedding A 1 → A 2 is a clo s ed immersio n by Lemma 2.7, but it is not top o logically closed, b ecause it takes the generic p oint of A 1 to the g eneric p oint of A 2 and the maximal p oint to the maximal p oint; the int ermediate p oint s ar e not in the ima ge. Deﬁnition 2.8 . If ( X , A ) is a mono id scheme, a shea f of ideals I is said to be quasi-c oher ent if its res triction to any aﬃne op en subscheme U of X is the sheaf asso ciated to the ideal I ( U ) of the mono id A ( U ). Giv en any clos e d immers ion i : Y → X , the inv e r se image I of 0 under A X → i ∗ A Y is quas i-coherent. Lemma 2.7 shows that co n versely a n y quasi-co heren t sheaf I deﬁnes an equiv ar ian t closed immersion. Lemma 2.9. F or any monoid scheme X and any subset Z of the u nderlying p oset, ther e is an e quivariant close d sub cheme ¯ Z e q of X t hat c ontains Z and is c ontaine d in every other e quivariant close d s u bscheme of X c ontaining Z . We c al l ¯ Z e q the equiv ar ian t closure of Z in X . If U is an op en subscheme of X then ¯ Z e q ∩ U is Z ∩ U e q . Pr o of. W e saw in Section 1 that if Z is a ny subset of MSpe c( A ), there is a n equi- v aria n t clo sed subscheme ¯ Z eq = MSpec( A/I ) which contains Z (and its closure), and which is minimal with this prop erty . Indeed, if the closure of Z is V ( I 0 ), then A/I = ( A/I 0 ) red . Since S − 1 ( A/I ) = ( S − 1 A/S − 1 I 0 ) red , this construction patches to give a genera l constr uc tio n.  R emark 2.9.1 . If every po in t in Z has height at least i in X then every p oin t in ¯ Z eq has height at least i in X . This follows fro m the lo cal description of ¯ Z eq . 10 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL Finite typ e. W e say that a monoid scheme ha s ﬁnite t yp e if it admits a ﬁnite o pen cov er by a ﬃne monoid sch emes asso ciated to ﬁnitely gener ated monoids. These monoid s c hemes a re the a nalogues of No etherian schemes, just as ﬁnitely genera ted monoids a r e the analogues o f commutativ e No etherian rings: if A is a ﬁnitely generated monoid then every ideal is ﬁnitely genera ted, and A has the a scending chain conditio n o n ide a ls. (The usual pro of of the Hilb ert Basis Theore m works.) By Le mma 1.5, if ( X , A ) is a monoid scheme of ﬁnite type, then X is a ﬁnite po set, with the pos et top ology . The sheaf of mono ids A of a monoid scheme X determines a (contra v a riant ) functor A fro m the pos e t X to monoids , called the stalk functor of ( X , A ), sending x to A x . It is useful to introduce the notio n of a monoid p oset as a context for thinking ab out a stalk functor A . A monoid p oset is a pair ( Y , B ) co nsisting of a p oset Y and a contra v ar ian t functor B from Y to mo noids. There is a categ o ry of monoid p osets; a mo rphism f : ( X , A ) → ( Y , B ) of monoid pose ts is a p oset map g : X → Y A to the down ward- closed subset W ( x ) = { y ∈ X | y ≤ x } . T he r e is a morphis m o f mono id p osets (2.10) ι x : ( W ( x ) , A | W ( x ) ) → F (MSpec A ( x )) whose p oset map sends a p oint y o f W ( x ) to the inv e r se image p y ∈ MSp ec A ( x ) of the maximal idea l of A ( y ) under A ( x ) → A ( y ); the maps A ( x ) p y → A ( y ) de- termine the natural trans formation A ◦ ι x ⇒ A | W ( x ) . If the mor phism (2 .10) is a n isomorphism for all x ∈ X , we will say that the monoid p oset ( X , A ) is scheme-like and (by abuse of notation) we will call A a stalk functor . W e say that a monoid p oset ( X , A ) is of ﬁnite typ e if X is a ﬁnite p oset and ea ch A ( x ) is a ﬁnitely generated monoid. If X is a mono id scheme of ﬁnite t yp e, then F ( X ) is a monoid p oset of ﬁnite type. The following pro positio n shows that the stalk functor is alwa ys eno ugh to determine a monoid scheme of ﬁnite type. Prop osition 2. 1 1. The functor F ( X , A ) = ( X , A ) induc es an e qu ival enc e b etwe en the ful l s ub c ate gory of monoid schemes of ﬁnite typ e and the ful l s ub c ate gory of scheme-like monoid p osets ( X , A ) of ﬁnite typ e. Pr o of. If ( X , A ) is a monoid p oset, we may equip X with the p oset top ology , and deﬁne the sheaf A on X by the formula A ( U ) = lim ← − x ∈ U A ( x ) . Thu s G ( X, A ) = ( X, A ) is a monoid space. It is clear from the formula for A ( U ) that a mo rphism ( Y , B ) → ( X , A ) of mo noid po s ets induces a mo r phism G ( Y , B ) → G ( X, A ) of mono id spaces. Th us G is a functor . Because each W ( x ) has x as its maximal p oint , A ( W ( x )) = A ( x ). Thus F ( G ( X , A ) is isomor phic to ( X , A ). If ( X, A ) is sc heme-like o f ﬁnite type, then G ( X , A ) is a monoid scheme of ﬁnite t yp e. Conv ersely if X is a mo noid scheme of ﬁnite type a nd U is an aﬃne op en in X , we know by Lemma 2.4 that there is a unique x ∈ X such that U = MSpec ( A ( x )) and hence A ( x ) = A ( U ). Given a n o pen U in X , any p oint y in U lies in an aﬃne op en V ⊂ U , and V = MSp ec( A ( x )) for so me x ∈ U with y ≤ x by Lemma 2.4. It follows that GF ( X ) ∼ = X .  A mo no id scheme ( X , A ) of ﬁnite type will often b e sp eciﬁed b y its monoid po set, viz., ( X , A ). T o avoid confusion, we sha ll use ro man letters for stalk functors and script letters for s hea ves. TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 11 R emark 2 .12 . The pro of o f Prop osition 2.11 shows that an y scheme-like mo noid po set ( X , A ) ca n b e recov ered fro m the monoid space G ( X, A ) b ecause F G ( X , A ) ∼ = ( X, A ). If ( X , A ) is an ar bitrary monoid scheme with s ta lk functor A , then the top ology o f X may b e coar ser than the p o set top ology . How ever the ar gumen t of the pr oo f of the pro position shows that w e ca n rec over A from A a nd the top ological space underlying X , using the formula A ( U ) = lim ← − x ∈ U A ( x ). 3. Basechange an d sep ara ted mo rphisms It is use ful to simplify constr uctions using ba se-change. F o r this, we need pull- back squar es in the catego ry o f mo no id schemes. There is a ca nonical mor phism ν : X → MSpec ( A ( X )) which is universal for maps from X to aﬃne mo noid schemes. It sends a point x to the preimage ν x of the maximal ideal of A x . The s heaf homomo r phism ν # is that induced by the canonica l maps A ( X )[1 / s ] → A ( ν − 1 ( D ( s )). The univ ersa l pr operty shows that the (contra v a riant ) functor X 7→ A X ( X ) from mono id schemes to monoids is left adjoint to the functor MSpec , i.e., that a ﬃne mo noid schemes ar e a reﬂective sub c ategory of all monoid schemes. It follows that MSp ec conv erts pushouts o f diagrams of monoids to pullba c ks o f diagrams in the categ ory of all monoid schemes. In particular, for any pushout diagr am of monoids (1.8), the induced diagr am is cartesian: MSpec ( A 1 ∧ C A 2 )   / / MSpec A 2   MSpec A 1 / / MSpec C. Prop osition 3.1. The pul lb ack X × S Y of a diagr am of monoid schemes X × S Y / /   X   Y / / S exists in the c ate gory of al l monoid schemes. Its un derlyi ng top olo gic al sp ac e is the pul lb ack X × S Y in t he c ate gory of top olo gic al sp ac es. Pr o of. Existence o f the pullback X × S Y is derived from the existence of pullbacks of aﬃne mono id s chemes, just as for usual schemes ([16, Thm. 3.3 ]). T o prov e the assertio n ab out underlying top ological spaces, it suﬃces to co nsider the a ﬃne case. Using the notation o f (1.8), wr ite P for the pullba c k of MSp ec( A 1 ) and MSp ec( A 2 ) over MSp ec( C ) in T op. The canonical map f : MSp ec( A 1 ∧ C A 2 ) → P is a co n tin uous bijection b y Prop osition 1.11. T o sho w that f is a homeomorphism, it suﬃces to show that it ta kes any basic op en set D ( s ) to a n op en set of P . W rite s = s 1 ∧ s 2 ; then s 6∈ p if and only if s 1 ∧ 1 , 1 ∧ s 2 6∈ p . W e saw in Prop osition 1.11 that if p maps to ( p 1 , p 2 ) then p = p 1 ∧ A 2 ∪ A 1 ∧ p 2 , and that s 1 ∧ 1 6∈ p (resp., 1 ∧ s 2 6∈ p ) is equiv a len t to s 1 6∈ p 1 (resp., s 2 6∈ p 2 ). This shows that f takes D ( s ) to the op en set ( D ( s 1 ) × D ( s 2 )) ∩ P , a s required.  Example 3.1.1. The pr oduct X × Y is just the pullback when S is the terminal monoid scheme MSp ec( S 0 ). 12 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL R emark 3 .1.2 . Let X and Y b e monoid schemes of ﬁnite type, ov er a common S . Then the pullbac k X × S Y ha s ﬁnite type. Indeed, it has a ﬁnite cov er by aﬃne op ens of the form MSp ec( A 1 ∧ C A 2 ), a nd in ea c h case A 1 ∧ C A 2 is ﬁnitely genera ted bec ause A 1 and A 2 are. Example 3.2. Pro positio n 3.1 shows that given tw o close d subschemes Z 1 , Z 2 of X , the pullback Z 1 × X Z 2 is a subscheme whose underlying top ologica l space is the intersection of the tw o subspaces of X . More genera lly , given any family of closed immersions Z i ֒ → X , we can form the inv erse limit lim Z i ֒ → X by patching the inverse limits on each aﬃne op en MSpec( A ), b ecause the colimit of a family o f surjections A ։ B i exists and is a sur jection. Sep ar ate d morphisms. An imp ortant hypothesis in many theor ems ab out monoid schemes, o ften overlooked in the literature , is that they be separ ated. Deﬁnition 3.3. A mor phism f : X → S of monoid schemes is sep ar ate d if the diagonal map ∆ : X → X × S X is a clos ed immersion. W e say that X is separa ted if it is sepa rated ov er MSp ec( S 0 ) where we recall S 0 = { 0 , 1 } . Being s e pa rated is lo cal o n the base: if S has an op en cover { U } then f is separated if and o nly if ea c h f − 1 ( U ) → U is separ ated. Lemma 3.4 . If A → B is a morphism of monoids then MSp ec( B ) → MSp ec( A ) is a sep ar ate d morphism of m onoid schemes. In p articular, close d immersions ar e sep ar ate d. Pr o of. By Pro positio n 3.1, the diagona l ma p ∆ corresp onds to the multiplication map B ∧ A B → B , w hich is s ur jectiv e. By Lemma 2.7, ∆ is a closed immersio n.  R emark 3 .4.1 . E x ample 1.1 0 shows that X × A 1 → X is separated and universally closed for every monoid s c heme X . This s ho ws that “separ ated a nd universally closed” do es not provide a g oo d no tion of pro p er mor phism of monoid schemes; we will discuss a n a ppropriate deﬁnition in Sectio n 8 . Example 3.5. Here is a n e x ample of a monoid scheme which is non-sepa r ated. Let A and B each b e the free a belian monoid with tw o gener ators, F 2 (see Ex ample 1.2). Let U be the ope n s ubs et o f ea c h of MSp ec( A ) and MSpec( B ) g iv en by removing the unique clo s ed point (asso ciated to the maxima l ideal in ea c h monoid); explicitly U = {h t 1 i , h t 2 i , { 0 }} . Then we may glue MSp ec( A ) and MSp ec( B ) along U to form a monoid s cheme X of ﬁnite type. As a p oset, X has ﬁve elements, tw o of which are maximal — the tw o copies of h t 1 , t 2 i — and the rest ar e in U . The k -realizatio n of X (deﬁned in 5 .3 b elow) is the non-sepa r ated scheme given by the a ﬃne pla ne with the o rigin do ubled. Lemma 3.6. A map f : ( X , A ) → ( S, B ) of m onoid s chemes is sep ar ate d if and only if for every x 1 , x 2 in X such that f ( x 1 ) = f ( x 2 ) and su ch that MSp ec( A x 1 ) and MSpec( A x 2 ) ar e op en, either ther e is n o lower b ound for { x 1 , x 2 } in the p oset X or else t her e is a unique maximal lower b ou n d x 0 = x 1 ∩ x 2 , and A x 1 ∧ B f ( x 1 ) A x 2 → A x 0 is onto. Pr o of. By Prop osition 3.1 and Lemma 2.4, a n a ﬃne op en subset o f X × S X has the form U = ( U 1 × U 2 ) ∩ ( X × S X ), where the maximal p oint ( x 1 , x 2 ) of U determines the a ﬃne open subsets U i = MSp ec( A x i ) of X . Since ∆ − 1 ( U ) = U 1 ∩ U 2 , TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 13 Prop osition 3.1 implies that X → ∆( X ) is a homeomor phism a nd that the p oset underlying U 1 ∩ U 2 is the subset { z ∈ X | z ≤ x 1 , z ≤ x 2 } of low er bo unds fo r { x 1 , x 2 } . If U 1 ∩ U 2 = ∅ , { x 1 , x 2 } has no lower b ound. By Lemma 2 .4, U 1 ∩ U 2 is nonempty a ﬃne if and only if it has a unique maximal element. Th us ∆ is a clo sed immers ion if a nd only if, in the ab o ve situatio n, whenever U 1 ∩ U 2 is nonempty it is aﬃne (and hence has a unique max ima l low er bo und x 0 ), and A x 1 ∧ C A x 2 → A x 0 is onto, where s = f ( x 1 ) = f ( x 2 ) and C = B s .  Corollary 3.7. If X is a monoid scheme of ﬁn ite t yp e with stalk functor A , then X is sep ar ate d if and only if whenever t wo p oints x 1 , x 2 of X have a lower b ound they have a gr e atest lower b ound x 1 ∩ x 2 , and A ( x 1 ) ∧ A ( x 2 ) → A ( x 1 ∩ x 2 ) is onto. Pr o of. Comb ine Lemma 3.6 and P r opo sition 2.11.  Corollary 3.8. The interse ct ion of t wo aﬃne op en subschemes of a sep ar ate d monoid scheme is aﬃne. Pr o of. Suppose X is a s e parated mono id scheme, with U 1 , U 2 aﬃne and o pen in X . Let x 1 , x 2 be the uniq ue closed points of U 1 , U 2 . If x 1 and x 2 do no t have a common lower bound in X , then U 1 ∩ U 2 = ∅ . Otherwise, by Lemma 3.6, they have a greatest low er b ound, whic h is the unique max imal p oint of U 0 = U 1 ∩ U 2 . By Lemma 2.4, U 0 is aﬃne.  4. Toric monoid schemes As obser v ed by Ka to [22] and Deitmar [8], the fan asso ciated to a toric v ariety pro duces a mono id scheme. In this section we clarify this cor resp o ndence, using the following deﬁnition. Deﬁnition 4.1. A toric monoid scheme is a separ ated, connected, tor sionfree, normal monoid scheme of ﬁnite type. Recall that a fan co nsists of a free ab elian group N of ﬁnite rank (wr itten ad- ditively) together with a ﬁnite co llection ∆ of stro ngly conv ex ratio na l polyhedr al cones σ in N R (hereafter referr e d to as just c ones ), satisfying the conditions that (1) every face o f a member of ∆ is als o a member of ∆ and (2) the intersection of any tw o member s of ∆ is a fa c e of each. Here a st ro ngly c onvex r ational p olyhe dr al c one is a cone with a pex at the origin, ge nerated by ﬁnitely many elements of N , that contains no lines thro ugh the origin. Note that ∆ is a ﬁnite p oset under containmen t; we now c o nstruct a monoid po set (∆ , A ) a nd us e Prop osition 2.11 to deﬁne the asso ciated monoid s c heme. Construction 4.2 . Given a fan ( N , ∆), set M = Hom Z ( N , Z ) and M R = M ⊗ R . W e deﬁne a co n trav ariant functor A from ∆ to monoids (written additively) by A ( σ ) = ( σ ∨ ∩ M ) ∗ , σ ∨ = { m ∈ M R | m ( σ ) ≥ 0 } . Each such mono id is tor sion-free, normal and ﬁnitely gener ated (Gordon’s Lemma). If τ is a face o f σ , then there is an m ∈ A ( σ ) such that A ( τ ) = A ( σ )[ − m ]. Hence by Lemma 1.1 ther e is a pr ime ideal P σ ( τ ) of A ( σ ) such that A ( τ ) = A ( σ ) P ( τ ) . By Prop osition 2.11 and Corollar y 3.7, A is the stalk functor of a to ric mo noid scheme X ( N , ∆), which by abus e o f notation we wr ite as X (∆) = (∆ , A ) . 14 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL Thu s any fan ∆ determines a toric mono id s c heme in the s ense of Deﬁnition 4.1. A morphism of fans, from ( N , ∆) to ( N ′ , ∆ ′ ), is given b y a g roup ho momorphism φ : N → N ′ such that the image of each cone in ∆ under the induced map N R → N ′ R is co n tained in a cone in ∆ ′ . Such a map o f fans induces a p oset ma p ∆ → ∆ ′ , sending σ to the smallest cone σ ′ in ∆ ′ that contains φ ( σ ), and precomp osition with φ y ie lds a natura l transformatio n (( σ ′ ) ∨ ∩ M ′ ) ∗ → ( σ ∨ ∩ M ) ∗ of stalk functors. According to P rop o sition 2.11, this data determines a morphism of monoid sc hemes: X ( φ ) : X (∆) → X (∆ ′ ) . If φ 1 6 = φ 2 then X ( φ 1 ) 6 = X ( φ 2 ), as φ ∗ 1 6 = φ ∗ 2 on some A ( σ ). Thus we hav e a faithful functor X from fans to toric mono id s c hemes. Example 4.3. F o r the cone σ in the plane spanned by (0 , 1) and (1 , − 2), A ( σ ) = σ ∨ ∩ M is the submono id of Z 2 spanned by { (1 , 0) , (1 , 1 ) , (1 , 2) } . If ∆ is the fan spanned by σ and its faces, then (∆ , A ) = MSp ec A ( σ ). If ( X , A ) is a toric monoid scheme a nd x ∈ X , we will write M x for the g roup completion of the unpo in ted monoid A ( x ) \ { 0 } . Each M x is a torsionfr ee ab elian group of ﬁnite rank. The gr oups M x are a ll isomorphic, b ecause X has a unique minimal p oint η by Lemma 2.3, and M x → M η = A ( η ) \ { 0 } is an isomorphism for all x . Theorem 4. 4. The faithful functor ∆ 7→ X (∆) fr om fans to toric monoid schemes, deﬁne d by Construction 4.2 has t he fol lowing pr op erties. (1) Every toric monoid scheme ( X, A ) is isomorphic to X ( N , ∆) , wher e: a) The latt ic e N is the Z -line ar dual of M = M η , wher e η is t he unique minimal p oint of X . b) The p oset ∆ of c ones in N R is isomorphic to the p oset underlying X . F or e ach x ∈ X , the c one σ x in N R is t he dual c one of the c onvex hul l of A ( x ) \ { 0 } in M R . (2) F or fans ( N , ∆) and ( N ′ , ∆ ′ ) , a morphism f : X (∆) → X (∆ ′ ) of monoid schemes is given by a (ne c essarily unique) morphism of fans if and only if f maps the generic (i.e., minimal) p oint η of X (∆) to the generic p oint η ′ of X (∆ ′ ) . In this c ase, the map of fans ( N , ∆) → ( N ′ , ∆ ′ ) is given by the Z -line ar dual of t he gr oup homomorphism f # η : M ′ = ( A ′ ( η ′ ) \ { 0 } ) → ( A ( η ) \ { 0 } ) = M . Pr o of. Throughout this pr oo f, for a canc e lla tiv e monoid A , we write A o for the unpo in ted monoid A \ { 0 } , wr itten a dditiv ely , and we identif y ea ch A o ( x ) with a submonoid of M . Let ( X , A ) be a toric monoid scheme. W e ﬁrst show that ( N , ∆) as deﬁned in the statement is a fan. F or x ∈ X , let σ ∨ x ⊂ M R denote the co n vex h ull of A o ( x ) in M R . Note that this deﬁnes a cone σ x = ( σ ∨ x ) ∨ in N via the identiﬁcation N = N ∗∗ . The cone σ ∨ x is a ra tional p olyhedra l co ne beca use it is spa nned by a ﬁnite se t { a i } of genera tors of A o ( x ); the cone σ x is thus also a rationa l p olyhedral cone, and it is strong ly convex since A o ( x ) + = M . T o see that A ( x ) = ( σ ∨ x ∩ M ) ∗ , let b = P q i a i be an element of σ ∨ x ∩ M , wr itten as a po sitiv e Q -linear combination of the a i . Clearing denominator s, nb is a pos itiv e Z -linear combination of the a i for some p ositive integer n and hence is in A o ( x ). Because A ( x ) is normal, b is in A o ( x ), as r equired. TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 15 If τ is a face of σ x , it is deﬁned by the v anishing o f some m ∈ σ ∨ x . Cle a ring denominators and using a gain that A ( x ) is nor mal, we may assume m ∈ A o ( x ). By deﬁnition, τ is the set of linea r functionals on M R that are no n-negative o n A o ( x )[ − m ]. By Lemma 1.1, A ( x )[ − m ] coincides with A ( y ) for some y ≤ x , a nd th us the fac e τ is the ele ment σ y of ∆. If x, y ∈ X , we claim that the intersection σ x ∩ σ y is a cone in of ∆. Since X is separated, x and y have a unique grea test common lower bo und, wr itten x ∩ y , and the map A ( x ) o × A ( y ) o → A o ( x ∩ y ) is surjective, b y the additive version o f Corollar y 3.7; mo reov er b ecause X is cancellative, it is an isomorphism. A linear functional on M R is non-neg ativ e o n A o ( x ) × A o ( y ) if and only if it is no n- negative on A o ( x ) and A o ( y ), and thus w e have the r e quired iden tity: σ x ∩ σ y = σ x ∩ y . Moreov er, σ x ∩ y is a face o f bo th σ x and σ y , b ecause by Lemmas 1.3 and 1.5 there are m 1 , m 2 such that A o ( σ x ∩ y ) = A o ( σ x )[ − m 1 ] = A o ( σ y )[ − m 2 ], This prov es that ∆ is a fan. By Construction 4.2, the fan ( N , ∆) determines a monoid s c heme (∆ , B ). The bijection σ : X → ∆ ( x 7→ σ x ) is order pre serving, b ecause if x < y in X , then A o ( y ) ⊂ A o ( x ) ⊆ M . By cons tr uction, w e have a natural is omorphism A ( x ) = ( σ ∨ x ∩ M ) ∗ = B ( σ x ). This proves that σ deter mines an isomorphism of monoid schemes, co mpleting the pro of of prop erty 1). Construction 4.2 shows that the co ndition in prop erty 2 ) is necess ary , since a morphism of fans sends the z ero c o ne to the zer o cone. Co n versely , if f ( η ) = η ′ , then f # η induces a mono id map A ′ ( η ′ ) = M ′ ∗ → M ∗ = A ( η ); since a n y such map sends units to units, it induces a gro up homomo rphism M ′ → M . Let φ : N → N ′ be the Z -linear dual of this map. Since for e ac h x ∈ X , the map f # x is the restrictio n of f # η , it follows that f = X ( φ ), as desire d.  R emark 4 .4 .1 . There ar e diﬀering ass e rtions in the litera ture rela ted to Theo rem 4.4. Using a diﬀeren t deﬁnition of ‘toric v ariety’ it is cla imed in [8, Thm. 4.1] that any co nnected cancella tiv e monoid scheme of ﬁnite type yields a toric v ar iet y , but not every such “to r ic v ariety” is asso ciated to a fan. F or example, MSpec of the cusp monoid C = { t 2 , t 3 , ... } ∗ yields the cusp. In [23, 2.1 ], the ﬂaw ed [8, Thm. 4.1] is used to claim that the functor of Theor em 4.4 is an equiv alence, under the weaker hypothesis that A has no tor sion; the cusp mo noid is also a counterexample to the assertion in lo c. cit. W e conclude this section with a description of s eparated normal monoid s chemes. If X is connected and cancellative, with minimal prime η , then M η is a ﬁnitely generated ab elian gro up. T he r efore there is a non-canonical isomor phism M η ∼ = M × T , wher e M is a free ab elian group and T is a ﬁnite to rsion group. Prop osition 4.5 . Any s ep ar ate d, c onne cte d, normal m onoid scheme of ﬁ n ite typ e de c omp oses as a c artesian pr o duct of monoid schemes X ∼ = ( X, A ) × MSpe c( T ∗ ) , wher e ( X , A ) is a toric monoid scheme and T is a ﬁnite ab elian gr oup. Pr o of. If MSpec( A ) is an a ﬃne open of X then A is a submonoid of A η = ( M × T ) ∗ ; since A is normal, T ∗ is a submonoid o f A . Every elemen t o f A η \{ 0 } can b e wr itten uniquely as a pro duct mt with m ∈ M a nd t ∈ T ; since t ∈ A , if mt ∈ A then 16 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL m ∈ A ∩ M . Thus if we set B = A ∩ M ∗ there is a decomp osition A ∼ = B ∧ T ∗ . In other words, MSp ec( A ) ∼ = MSpec( B ) × MSp ec( T ∗ ) Since every lo calization of A ha s the form A p = B ′ ∧ T ∗ , the aﬃne open subse ts of MSp ec( A ) are ar e all of the form MSp ec( B ′ ) × MSp ec( T ∗ ). Gluing thes e together gives the dec o mpositio n o f X .  Note that the factoriza tion in P rop o sition 4.5 is not unique; it dep ends up on the choice of is omorphism A η ∼ = ( M × T ) ∗ . Corollary 4 .6. If f : X → X ′ is a morphism b etwe en sep ar ate d and c onne cte d normal monoid schemes of ﬁnite typ e, inducing an isomorphi sm f ∗ : A ′ η ′ → A η of gr oup c ompletions, then f is isomorphic to t he pr o duct of a morphism X (∆) → X (∆ ′ ) of toric m onoid schemes and an isomorphism MSpec ( T ∗ ) → MSpec ( T ′ ∗ ) . Pr o of. By a ssumption, f maps the ge ne r ic p oin t η of X to the generic po in t η ′ of X ′ . Choosing a decomp osition A η ∼ = ( M × T ) ∗ , w e have an implicitly deﬁned decomp osition A ′ η ′ ∼ = ( M × T ) ∗ . Then for ea c h x ∈ X the decomp ositions A x ∼ = B x ∧ T ∗ , A ′ f ( x ) ∼ = B ′ f ( x ) ∧ T ∗ of Prop osition 4.5 sa tisfy f ∗ ( B ′ f ( x ) ) ⊆ B x ⊆ M ∗ . Therefore the map A ′ f ( x ) → A x factors as a pro duct of f ∗ ( B ′ f ( x ) ) ⊆ B x and T ∗ ∼ = T ∗ , for each x . The result follows.  R emark 4 .6.1 . Not every morphism ( X , A ) × MSpec ( T ∗ ) → ( X ′ , A ′ ) × MSpec( T ′ ∗ ) betw een connected normal mo noid schemes o f ﬁnite type will facto r as a cartesia n pro duct of maps ( X, A ) → ( X ′ , A ′ ) and MSp ec( T ∗ ) → MSp ec( T ′ ∗ ). F or example, this fails for the cano nical MSpec (( Z /n ) ∗ ) → MSp ec( Z ∗ ). How ever, such a map de- termines b oth a toric map ( X , A ) → ( X ′ , A ′ ) and a map MSpec( T ∗ ) → MSp ec( T ′ ∗ ). 5. Realiza tion s of mono id schemes In this se ction we ﬁx a co mm utative ring k . If A is a mo no id, the ring k [ A ] gives rise to a scheme Sp ec( k [ A ]), which is called the k -r e alization of MSp ec( A ). The aﬃne spac e s A n k = Sp ec( k [ t 1 , ..., t n ]) o f 1.2 a re use ful examples. The k -realizatio n is a faithful functor from monoids to aﬃne k -schemes; a mono id morphism A → B naturally gives rise to a morphis m Sp ec( k [ B ]) → Sp ec( k [ A ]). If X is a n a ﬃne monoid scheme, w e write X k for its r ealization: MSpec ( A ) k = Sp ec( k [ A ]) . W e saw in (1.8) tha t the k -re a lization functor commut es with pullback for aﬃne monoid schemes, beca use it has a left adjoint (deﬁned o n the catego ry of aﬃne k -schemes) sending Sp ec( R ) to MSp ec( R, × ), where ( R, × ) is the multiplicativ e monoid whos e underly ing p oin ted set is R . Thu s if X = MSp ec( A ) is an a ﬃne monoid, the adjunction Hom(Spec ( R ) , X k ) ∼ = Hom MSch (MSpec( R , × ) , X ) means that X k represents the functor sending Sp e c R to Hom MSch (MSpec ( R, × ) , X ). Deﬁnition 5. 1. Let X b e a monoid scheme and k a ring. Deﬁne a contra v ar ia n t functor F X from the categ ory of aﬃne k -schemes to sets to b e the Zar iski shea ﬁﬁ- cation of the pr esheaf Spec R 7→ Hom MSch (MSpec ( R, × ) , X ) . If X is aﬃne, the preshea f is a lready a sheaf since it is repre sen ted by X k . TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 17 Recall from [9, VI-14] that a co n trav ariant functor F from a ﬃne k -schemes to sets is represe n ted b y a unique k -scheme X if and only if F is a Zariski sheaf and F admits a cov ering by o pen subfunctors F α , each of which is repre sen ted by a n aﬃne scheme U α . If so, the represe nting scheme X is obtained by gluing the U α together. Here, a s ubfunctor F α ⊆ F is op en if for e very k -alg ebra R and every morphism Hom( − , Sp ec R ) → F , i.e., for every elemen t of F (Sp ec R ), the pullba c k functor F α × F Hom( − , Sp ec R ) is r epresent ed by an op en subscheme of Sp ec R . A collection of subfunctors { F α } of F c overs F if fo r every k -algebra L which is a ﬁeld, we have F (Spec L ) = S α F α (Spec L ). Theorem 5.2. The functor F X is r epr esente d by a scheme X k . Pr o of. Suppose that U = MSp ec( A ) is any aﬃne monoid subscheme o f X . Since sheaﬁﬁcation pres erves monomorphisms such a s Hom( − , U ) ⊆ Hom( − , X ), F U is a subfunctor of F X . If Λ is a lo cal k -algebra and L = Sp ec(Λ) then (5.2a) F X ( L ) = Hom MSch (MSpec (Λ , × ) , X ) . Since MSpe c (Λ , × ) has a unique p oint, each map MSp ec(Λ , × ) → X factors throug h an aﬃne op e n submonoid U . Therefore F X is co vered b y the collection of subfunc- tors F U , a s U ranges over all aﬃne op en mono id s ubsc hemes o f X . W e will show that the F U are open subfunctor s of F X ; we hav e s een that each F U is represented by the a ﬃne scheme U k . By [9, VI-1 4], this will pr o ve that F X is repr esen table by the k -scheme which is obtained by gluing the a ﬃne s chemes U k . Fix an aﬃne o pen monoid subscheme U = MSp ec( A ). T o prov e that F U is op en, ﬁx a k -algebra R and consider a morphism Hom( − , Sp ec R ) → F X and its co rresp onding element φ ∈ F X (Spec R ). W e have to show that the pullback G = F U × F X Hom( − , Sp ec R ) is represe n ted b y an op en subscheme V of Sp ec( R ). Since F X is a shea f, Sp ec( R ) ha s an aﬃne o pen cov ering { Spec R [1 /s ] | s ∈ S } such that the r estriction of φ to F X (Spec R [1 /s ]) is represented by a morphism φ s : MSp ec( R [1 /s ] , × ) → X of monoid schemes. By Observ a tion 5.2.1 b elow, there are contin uo us maps Spec ( R [1 /s ]) ֒ → MSpec ( R [1 /s ] , × ) φ s − → X. Let V ′ s denote the in verse image o f U under φ s and let V s denote the o p en subspace V ′ s ∩ Sp ec( R [1 /s ]); w e regard V s as an op en subscheme of Spec( R [1 /s ]) and hence of Sp ec( R ). W e claim that G is r epresented b y the op en s ubs cheme V = ∪ V s of Spec R . T o prov e our claim, it suﬃces to consider a lo cal k -scheme L = Spec(Λ) a nd prove that G ( L ) = Hom( L, V ) as subsets of Hom( L, Sp ec R ). Since L is lo cal, w e have F U ( L ) = Hom( A, (Λ , × )), and (5.2a) ho lds for X . Thus G ( L ) is the s et of all f : L → Sp ec R such that MSpec(Λ , × ) f × − → MSp ec( R, × ) φ − → X maps the clos e d po in t m of L into U . If the ima ge of f lies in V , m lands in some V s and hence f × maps the close d p oint ( m , × ) of MSpec (Λ , × ) into V ′ s . It follows that φf × ( m , × ) ∈ U , i.e., f ∈ G ( L ). Thus Hom( L, V ) ⊆ G ( L ). Conv ersely , if f : L → Spec( R ) is in G ( L ) then f fa ctors throug h some f s : L → Spec ( R [1 /s ]) and φ s f × s maps the closed p oin t ( m , × ) o f MSp ec(Λ , × ) to a po int in the subset U of X , s o f s ( m ) ∈ V s . But s ince L is lo cal, this implies tha t f s ( L ) ⊆ V s . The desired equality G ( L ) = Hom( L, V ) follows.  18 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL Observation 5 .2.1 . Let R b e an y commutativ e r ing, and ( R , × ) its underlying mul- tiplicative monoid. If p is a prime ideal of the ring R , then ( p , × ) is a pr ime ideal o f the monoid ( R, × ). The resulting inclusion Sp ec( R ) ֒ → MSp ec( R, × ) is cont inuous bec ause if s ∈ R the op en subspa ce D ( s ) of MSp ec( R ) int ersec ts Sp ec( R ) in the op en subspac e { p ⊂ R | s 6∈ p } . If R is lo cal, the maximal ideal m of R maps to the maximal prime ( m , × ) of MSp ec( R, × ). Deﬁnition 5.3. Given a commutativ e ring k and a scheme ( X , A ), we deﬁne its k -r e alization X k to b e the s cheme representing F X . R emark 5.3.1 . O bserve that X k = X Z × Sp ec Z Spec k fo r any monoid scheme X and commutativ e r ing k . Those preferring the notio n o f a ﬁeld with one ele ment ( F 1 ) might prefer wr iting X k as X × Sp ec F 1 Spec k or just X × F 1 k . Corollary 5.4. The k -r e alization fu n ctor X 7→ X k pr eserves arbitr ary limits (when they exist). In p articular, it pr eserves pul lb acks. Pr o of. Suppose that { X i , i ∈ I } is a diagr am of mono id s chemes and that its limit X exists in the c a tegory of monoid schemes. It suﬃces to prove the cano nical map F X → F = lim ← − F X i is an isomorphism of sheav es on the c a tegory o f a ﬃne k -s c hemes. Recall that the limit of a diagram of sheav es exists and co incides with the limit as pre shea ves. That is, we hav e F (Spec R ) = lim ← − F X i (Spec R ). When R is lo cal, we hav e F X (Spec R ) = Hom(MSpec( R , × ) , X ) a nd also F (Sp ec R ) = lim ← − Hom(MSpec ( R, × ) , X i ) ∼ = Hom(MSpec ( R, × ) , X ) , where the second isomorphism ho lds since X = lim ← − i X i . Since the shea f map F X → F is a n isomorphism on all lo cal rings, it is an isomor phism o f sheaves.  In Pr o positio n 5.7 b elow w e shall g iv e an explicit cons tr uction of X k for separ ated X . W e need some preliminar ie s . Lemma 5.5. If S is mu lt ipli c atively close d in A , S − 1 k [ A ] ∼ = k [ S − 1 A ] . Pr o of. The monoid map A → S − 1 A is initial among mono id maps A → B tha t take S to units. Similar ly , the map k [ A ] → S − 1 k [ A ] is initial a mo ng k -alg ebra homomorphisms k [ A ] → C that take S to units. Being a left adjoint, the functor k [ − ] preserves initial ob jects.  R emark 5 .6 . Let A be a monoid. An y aﬃne open monoid subscheme of MSp ec( A ) has the form MSpec ( A p ) for so me prime ideal p of A , by Lemma 2.4, and A p = A [1 /s ] by Le mma 1.3. Hence Sp ec( k [ A p ]) → Spec ( k [ A ]) is an o pen immersion, by Lemma 5.5. F or the next Prop osition, let us say tha t a p oint x in a monoid scheme X is nic e if the canonica l ma p U = MSpe c( A x ) → X is an op en immersio n. Every closed po in t is nice by Lemma 2.4, but the p oints of E xample 1.4 ar e not nice. If X is of ﬁnite type, then every p oint is nice by Lemma 1.5. The nice p oint s x ∈ X are a coﬁna l subset of the pos et underlying X by Lemmas 1.3 and 2.4, beca use the closed p oints in any op en subscheme are nice. If x < y are tw o nice p oin ts then Spec ( k [ A x ]) → Sp ec( k [ A y ]) is a n op en immersion by Lemma 5.5. The criterion for separatedness in Lemma 3.6 uses nic e po in ts. TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 19 Prop osition 5.7. Le t k b e a c ommutative ring and ( X , A ) a sep ar ate d monoid scheme. Then the k -r e alization of X is X k = lim − → x ∈ X Spec ( k [ A x ]) . Pr o of. Put U x = MSp ec( A x ). Because nice points are coﬁnal in the p oset underly - ing X , the limit can b e taken ov er the nice po in ts. If x is nice, then U x ⊂ X is an op en immersion; set V x = ( U x ) k . If y is als o nice, then U x ∩ U y is an aﬃne op en, b e- cause the in tersection of tw o aﬃne op e n subschemes of a separated monoid scheme is aﬃne ope n by Corolla ry 3 .8. By Co rollary 5.4 we hav e ( U x ∩ U y ) k = V x × X k V y . Let V x,y be the image of the pro jection π x : V x × X k V y → V x . Then V x,y is op en in V x and we have a n isomo rphism ψ x,y = π y ( π x ) − 1 : V x,y → V y ,x . Hence the family of schemes V x indexed by the nice p oin ts of X tog ether with the op en subschemes V x,y ⊂ V x and the isomorphis ms ψ x,y satisfy the hypothesis of [EGA 0 I , (4.1.7)] (or [16, Ex. I I.2.12]). Therefore the limit of the prop osition exists, and is the s c heme obtained by g luing the realizatio ns of the op en aﬃne subschemes of X . Since this is also the deﬁnition of X k , the prop osition follows.  The k -rea liz ation functor from monoid schemes to k -schemes is faithful, beca use it is so lo cally: MSp ec( A ) k = Sp ec( k [ A ]). (This is clear if X is separa ted, a nd follows from Theorem 5.2 if it is not sepa rated.) It is not full beca use k -schemes such as A 1 k hav e many mor e endomor phisms than their monoida l c o un terparts. The realization functor loses information, b ecause distinct monoid schemes can hav e is omorphic realizations . This is a w ell known phenomenon even for to r ic v arie ties , where the additional data o f a (faithful) tor us action is needed to recov er the fan. Example 5.8. F or a fan ∆ and a n y ﬁeld k , the v ariety X (∆) k is the usual toric k -v ariety a sso ciated to ∆. This is c le a r fro m Construction 4.2. Example 5. 9. Let T be a ﬁnite ab elian gro up. The k -realiza tion of MSp ec( T ∗ ) is the co group scheme Sp ec( k [ T ]). If | T | is a unit (o r nonzero divisor) in k then k [ T ] is reduced, but this fails if k is a ﬁeld o f characteristic p > 0 and T has p -torsion. Lemma 5.10. L et k b e an inte gr al domain and A a c anc el lative monoid. Set X = MSp ec( A ) and U = MSp ec( A + ) . (1) If A + is torsionfr e e then k [ A ] is a domain (i.e., X k is inte gr al). (2) S u pp ose that k is a normal domain c ontaining a ﬁeld; if char( k ) = p > 0 assume also t hat A + has no p -t orsion. Then k [ A + ] is normal and its sub algebr a k [ A ] is r e duc e d. That is, U k is normal and X k is r e duc e d. (3) S u pp ose t hat char( k ) = p > 0 and A + has p -t orsion. Then k [ A ] is not r e duc e d; k [ A ] r e d = k [ B ] , wher e the monoid B is the quotient of A by the c ongruenc e r elation that a 1 ∼ a 2 if and only if a p e 1 = a p e 2 for some e ≥ 0 . Pr o of. Since A is the union of its ﬁnitely generated submonoids A i , and k [ A ] = ∪ k [ A i ], we may assume that A is ﬁnitely genera ted. As no ted b efore 4.5, we can write A + = ( M × T ) ∗ where M is a free ab elian g roup and T is a ﬁnite to rsion group. Since A is a submonoid of A + , k [ A ] is a subalgebra o f k [ A + ]. If T is trivial, k [ A ] is a subring of k [ M ], which is manifestly a domain. If k ⊃ Q or if char( k ) = p and p ∤ | T | then k → k [ T ] is a ﬁnite ´ etale extension and k [ A ] is a subring of 20 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL k [ A + ] = k [ T ][ M ], which is manifestly no rmal if k is normal. Hence k [ A + ] and its subalgebra k [ A ] are r e duced in this ca se. Finally , s uppose that c har( k ) = p and tha t the p -torsio n s ubg roup T p of T is non-trivial. Since k [ T p ] red = k and k [ A + /T p ] is reduced b y (2 ), we hav e k [ A + ] red = k [ A + /T p ]. If B is the imag e o f A → A + /T p then k [ A ] red is the image k [ B ] of k [ A ] → k [ A + /T p ]. Two elemen ts a 1 , a 2 ∈ A go to the same elemen t of A + /T p if and only if their quo tien t is p -tor sion, i.e., if and only if they ar e congruent under the rela tio n ∼ of the lemma. It fo llows tha t B = A/ ∼ ; this co nc ludes the pro of.  R emark 5 .1 0.1 . If ( X , A ) is a cancella tive mo noid scheme of ﬁnite type, and k is of characteristic p > 0, Lemma 5 .1 0 (3) implies that ( X k ) red is the k -realiz a tion of ( X, B ), where B = A/ ∼ is the quotient stalk functor of A deﬁned a s in 5.10(3). Prop osition 5.1 1. If ( Y , B ) f − → ( X, A ) is a close d immersion of monoid schemes then f k : Y k → X k is a close d immersion of schemes for al l rings k . Pr o of. If V ⊆ X is a n aﬃne op en subscheme, then by Lemma 2.4 ther e exists x ∈ X such that V = MSp ec( A x ). W e shall abuse notation and wr ite V ∩ Y for V × X Y . If V ∩ Y = ∅ then V k ∩ Y k = ( V ∩ Y ) k = ∅ . Otherwise V ∩ Y = MSpec ( B y ) for some y , a nd A x → B y is onto, by Deﬁnition 2 .5. Since k -realizatio n pr eserves pullbacks by Co rollary 5.4, we hav e f − 1 k ( V k ) = f − 1 ( V ) k = Sp ec( k [ B y ]) and the restr ic tion f − 1 k ( V k ) → V k = Sp ec k [ A x ] of f is induced by the surjection k [ A x ] → k [ B y ]. This prov es that the restrictio n Y k ∩ V k → V k of f k is a clo s ed immer sion. Since V is an arbitrary a ﬃne op en subs cheme o f X , this proves that Y k → X k is a clo sed immersion.  A partial co n verse of this prop osition is true. Lemma 5. 1 2. Supp ose i : Y → X is a morphism of monoid schemes such that the underlying map of t op olo gic al sp ac es induc es a home omorphism onto its image. F or any ring k , if i k : Y k → X k is a close d immersion, then i is a close d immersion of monoid schemes. Pr o of. It suﬃces to prov e that if X = MSp ec( A ) is aﬃne, then Y is also a ﬃne and the asso ciated map of mo noids is surjective. Let B b e the sheaf of monoids for the scheme Y and se t B = Γ( Y , B ). The map Y → X fac tors as Y → MSp ec B → MSpec A. Upo n taking k -rea lizations w e have Y k = Sp ec( R ) and the map induced b y Y k → X k is a surjection: k [ A ] ։ R . Since this surjection factors through the map k [ A ] → k [ B ], which is induced b y a map of monoids A → B , we see that k [ B ] ։ R is surjection as well. Let Y = ∪ j W j be a cov ering by op en aﬃne subschemes, with W j = MSp ec B j . Then the ma p B → Q j B j is injectiv e and hence so is the map k [ B ] → Q j k [ B j ]. Since the la tter map facto r s as k [ B ] → R → Q J k [ B j ], it follows that k [ B ] ∼ = − → R is an isomor phis m. That is, the k -rea lization o f Y → MSp ec( B ) is an isomor phism. Mor eo ver, since k [ A ] ։ k [ B ] is o n to, so is the ma p A ։ B , and hence MSpe c ( B ) → X is a closed immersio n. In particular , the ma p o f underlying top ological spaces is a ho meomorphism onto its imag e. It follows from this (and TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 21 our a ssumption) that the map of top ological spaces underlying Y → MSpec ( B ) is a homeomor phism onto its image. W e ma y thus assume that the k -re alization Y k → X k = Sp ec( k [ A ]) is an iso- morphism. W e next claim that Y → X is a surjection on p oints, and hence (b y our as s umption that Y is homeo mo rphic to its imag e) a homeomo rphism on un- derlying top ologica l space s. T o see this, ﬁx a po in t p ∈ X and co nsider the mono id map i p : A → S 0 = { 0 , 1 } sending p to 0 and A \ p to 1. Let Y ′ denote the pullback o f Y → X along the map MSp ec S 0 i p − → X . By Coro llary 5.4, the map Y ′ k →  MSpec S 0  k = Sp ec k is an is omorphism, so in particula r Y ′ is non-empty . By Prop osition 3.1, it follows that Y → X is onto. Since X ha s a unique maximal p oin t, so do es Y . By Lemma 2.4, Y is aﬃne. Since Y k ∼ = Spec ( k [ A ]) we co nclude that Y ∼ = X .  Prop osition 5.1 3. F or any ring k and morphism of monoid schemes f : Y → X , the map f is a sep ar ate d morphism of monoid schemes if and only if its k -r e alizatio n f k : Y k → X k is a sep ar ate d morphism of s chemes. Pr o of. One direction is immediate from Co rollary 5.4 and Pro position 5.11. Assume f k is sepa rated. Since the underlying topo logical space of Y × X Y is given by the pullback in the categ ory of top ologica l spaces , it follows that Y ∆ − → Y × X Y is a homeomorphism onto its image. (Observe that Y → ∆( Y ) and ∆( Y ) π 1 − → Y are contin uo us, and b oth comp ositions are the identit y , where ∆( Y ) ⊂ Y × X Y is given the s ubs pace top ology .) Since ∆ k is a clo sed immersio n, Lemma 5.12 applies to ﬁnish the pr oo f.  6. Normal and smooth mono id schemes Throughout this section, k denotes an int egra lly closed domain co n taining a ﬁeld. The normalizatio n A nor of a canc e lla tiv e monoid A is deﬁned in Deﬁnition 1.6; since ( A p ) nor = ( A nor ) p nor , it makes sense to talk ab out the no rmalization of any cancellative monoid scheme. The k -realizatio n of X canno t b e normal unless X k is r educed. Lemma 5.10 shows that k [ A ] is reduced unles s p > 0 and A + has p -torsion, in which cas e k [ A ] red is k [ B ], wher e B is a particular quotient of A , describ ed there. Prop osition 6. 1. L et X = ( X , A ) b e a c anc el lative monoid scheme of ﬁnite t yp e such t hat its k -r e alization X k is a r e duc e d s cheme. Then (1) t he normalization of X k is the k -r e alization of ( X, A nor ) . (2) if X is normal, c onne cte d and sep ar ate d, ther e is a de c omp osition X k = X ′ k × k Spec k [ T ] wher e X ′ k is a toric k -variety and k [ T ] is ﬁ nite ´ etale over k . As in Remark 4 .6.1, the decomp osition in Pr o positio n 6.1(2) is not natura l in X . Pr o of. Part (2) is immediate from P ropo sition 4 .5 and Coro llary 5.4. Since the norma lization of a reduce d scheme is the scheme constructed by patch- ing together the normaliza tions of an a ﬃne cov er, we may ass ume that X is aﬃne, i.e, X = MSp ec( A ). Since k [ A nor ] is integral over k [ A ], we may ass ume that A = A nor . In this situation, where A is a nor mal monoid of ﬁnite t yp e, Pro positio n 4.5 states that A ∼ = A ′ ∧ T ∗ where A ′ is torsionfre e and T is a ﬁnite ab elian gro up. 22 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL Since X k is reduced, we know from Lemma 5 .10(3) and Exa mple 5.9 that T has no p -torsio n and k [ T ] is ﬁnite ´ etale over k . Since k [ A ] = k [ T ][ A ′ ], we are reduced to the case in which A is nor mal a nd tors io nfree, i.e ., X = MSp ec( A ) is an aﬃne toric monoid scheme. By Theor em 4.4, X is asso ciated to a fan ∆; by Example 5.8, X k is the toric v ariety asso ciated to ∆, and in particular X k is nor mal.  R emark 6 .1.1 . It is p ossible to give an elementary pro of of this result using that if A is a tor sionfree normal monoid then k [ A ] is integrally closed; see [14, 12.6]. Finite morphisms. W e will need to know that the no rmalization of a mo noid scheme is a ﬁnite mor phism, at least when X is of ﬁnite type . W e say tha t a morphism of monoid schemes f : Y → X is aﬃne if X can be cov ered by a ﬃne op en subschemes U i = MSpec( A i ) such that f − 1 ( U i ) is aﬃne. Equiv a le n tly , f is aﬃne if f − 1 ( U ) is aﬃne for every aﬃne op en subscheme U ⊂ X . Deﬁnition 6. 2 . Let f : Y → X b e a morphism of monoid schemes. W e say tha t f is ﬁn ite if it is aﬃne and A X ( U ) → A Y ( f − 1 ( U )) is ﬁnite for every aﬃne s ubs c heme U ⊂ X . W e say that f is inte gr al if it is aﬃne a nd A X ( U ) → A Y ( f − 1 ( U )) is int egra l for every aﬃne subscheme U ⊂ X . If X is cancella tiv e, its nor malization X nor → X is a n integral morphism. T o see this, we may assume X = MSp ec( A ) is aﬃne so tha t X nor → X is g iv en by A ֒ → A nor , where the normalization A nor is integral by Deﬁnition 1.6. W e now show that if X is als o of ﬁnite type, then X nor → X is ﬁnite. Prop osition 6.3. If X is a c anc el lative monoid scheme of ﬁnite typ e, the normal- ization X nor → X is a ﬁn ite morphism. Pr o of. It suﬃces to s ho w that if A is a cancellative monoid of ﬁnite type then A → A nor is ﬁnite. Since A nor is integral ov er A it suﬃces by Lemma 1.7(i) to show that A nor is of ﬁnite type. Be cause the group completion A + is ﬁnitely genera ted, it has the form ( M × T ) ∗ where T is a ﬁnite ab elian group and M is free ab elian. Since A [ T ] = S At is ﬁnite ov er A , we may replace A b y A [ T ] to a ssume that T ⊂ A . As in the pro of of Prop osition 4 .5, this implies that A = B ∧ T ∗ where B = A ∩ M ∗ is a ﬁnitely gener ated s ubmonoid o f M . If β is the rationa l conv ex p olyhedra l cone of M R spanned by the generator s of B , B nor is ( β ∩ M ) ∗ . By Gordo n’s Lemma [1 0], B nor is ﬁnitely generated. A fortiori, A nor = B nor ∧ T ∗ is ﬁnitely generated.  Smo othness. Deﬁnition 6.4. Let p be a prime. A separa ted monoid scheme o f ﬁnite type is p - smo oth if e a c h stalk (equiv alently , each maxima l stalk) is the smash pro duct S ∧ T ∗ , where S = G ∗ ∧ F is the smash pro duct of a fre e ab elian group with a p oin t adjoined and a free ab elian monoid, and T is a ﬁnite a belian gr oup having no p -torsion. A separated mono id scheme is 0 - smo oth if ea c h stalk ha s the form S ∧ T ∗ with T a n arbitrar y ﬁnite ab elian g r oup. W e will say that X is sm o oth if it is p -smo oth for all p , i.e., if each stalk is the pro duct of a free gr oup of ﬁnite rank a nd a free mono id o f ﬁnite rank. TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 23 A cone in a fan ( N , ∆) is said to b e n onsingu lar if it is s panned by par t o f a Z -basis for the la ttice N , in which case each monoid σ ∨ ∩ M is the pro duct of a free abelia n group and a free abe lia n monoid. A fan is sa id to b e nonsingular if all its cones are nonsingular. Prop osition 6.5. L et X = ( X, A ) b e a sep ar ate d c anc el lative monoid scheme of ﬁnite typ e. Its k -r e alization X k is smo oth over a ﬁeld k of char acteristic p ≥ 0 if and only if X is p -smo oth. If X is c onne ct e d and p -smo oth then, un der the de c omp osition X = ( X , A ′ ) × MSpec ( T ) of Pr op osition 6.1, t he fan u nderlying ( X , A ′ ) is nonsingular. Pr o of. Recall from [1 0, 2.1] tha t the toric v ariety asso ciated to a fan is smo oth if and only if each of its co nes is no nsingular. Therefore the pro positio n is a n immediate corolla r y of Pro position 6.1 a nd Lemma 5.10.  Example 6.5. 1. The hypothesis in 6 .5 that X b e cancella tiv e is necessa ry . F or example, consider the mono id A = h t, e | e = e 2 = te i , which has k [ A ] ∼ = k [ x ] × k . Thu s X = MSp ec( A ) is not p -smo oth but X k is smo oth for every k . 7. MPro j and Blow-ups An N -gra ding of a monoid A is a p oint ed s et decomp osition A = ∞ _ i =0 A i such that A i · A j ⊆ A i + j ; Z -grading s are deﬁned similarly . F or each nonzero a in A , let | a | denote the unique i such that a ∈ A i . F or every multiplicativ e set S , the lo calization S − 1 A is Z -graded b y | a/s | = | a | − | s | . F or exa mple, if s ∈ A i is non-zero we have A [ 1 s ] 0 = n a s n | | a | = | s n | = ni, n ≥ 0 o ∪ { 0 } . Let A ≥ 1 denote the ideal W i ≥ 1 A i = { a | | a | > 0 } ∪ { 0 } , so that A/ A ≥ 1 ∼ = A 0 ; the image o f the corre s ponding map MSp ec( A 0 ) → MSp ec( A ) consists o f the prime ideals of A co n taining A ≥ 1 . Deﬁnition 7. 1. If A is a n N -gr aded monoid, we deﬁne MPro j( A ) = ( X , B ) to b e the following monoid scheme. The under lying top ological spac e is X = MSp ec( A ) \ MSpec( A 0 ) — i.e., the op en subspace of those prime idea ls of A that do no t co n- tain A ≥ 1 . The stalks o f B on X a re deﬁned by sending p ∈ MSpec( A ) \ MSpec( A 0 ) to B p = ( A p ) 0 , the degr e e zero part of A p . If MSp ec( A p ) ⊂ X is op en, that is , if A p = A [1 / s ] for some s ∈ A ≥ 1 , then the map MSp ec( A p ) → MSp ec( A p ) 0 is a home- omorphism. Indeed this follows from the fact that a prime ideal q of A [1 /s ] contains an element a/s n if and only if q ∩ ( A [1 /s ]) 0 contains a n | s | /s n | a | . Thus MPr o j( A ) is cov ered by the aﬃne op en subschemes D + ( s ) = MSp ec( A [ 1 s ] 0 ) where s ∈ A ≥ 1 , and moreov er, every aﬃne op en subscheme is of this form. Hence MP r o j( A ) is a monoid scheme of ﬁnite t yp e whenever A is a ﬁnitely ge nerated monoid. The maps A 0 → ( A p ) 0 induce a struc tur e morphism MPro j( A ) → MSpe c ( A 0 ). R emark 7 .1 .1 . The k -realization of A is the gra ded ring k [ A ], and k [ A [ 1 s ] 0 ] is the degree 0 part of the ring k [ A ] [ 1 s ] , so the k -realiza tio n of MPr o j( A ) is Pro j( k [ A ]). 24 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL Observation 7.1.2 . T he co nstruction is natural in A for maps A → A ′ of graded monoids such that A ′ = A · A ′ 0 . F or such maps there is a canonical morphism MPro j( A ′ ) → MPro j( A ) induced by the restriction o f MSpec ( A ′ ) → MSp ec( A ). If s ∈ A ≥ 1 , the a ﬃne op en MSp ec( A ′ [ 1 s ] 0 ) maps to the aﬃne op en MSp ec( A [ 1 s ] 0 ). If S ⊂ A 0 is multiplicatively closed, S − 1 A is graded a nd MPro j( S − 1 A ) is MPro j( A ) × MSp ec( A 0 ) MSpec( S − 1 A 0 ). I t follows that this c o nstruction may b e sheaﬁﬁed: for any mono id scheme ( X , A 0 ) and any shea f A of gr aded monoids on X with ( A x ) 0 = ( A 0 ) x for a ll x ∈ X , there is a monoid scheme MPro j( A ) ov er X whose s ta lk at each x is MP ro j( A x ). Moreov er, if f : ( X ′ , A ′ 0 ) → ( X , A 0 ) is a mor - phism of mo noid schemes, eq uipped with s he aves A ′ and A of gr aded mono ids as ab o ve, any graded extension f − 1 A → A ′ of f − 1 A 0 → A ′ 0 such that A ′ = f − 1 A · A ′ 0 induces a canonica l mo rphism MPro j( A ′ ) → MPro j( A ) ov er f . Lemma 7.2. If f : A → B is a surje ctive homomorphism of gr ade d monoids, t hen the induc e d map MP ro j( B ) → MP ro j( A ) is a close d immersion. Pr o of. As noted a b ov e, any aﬃne op en subscheme U ⊂ MPr o j( A ) is of the for m U = MSp ec( A [ 1 s ] 0 ) for some s ∈ A ≥ 1 . But U ∩ MPr o j( B ) = MSpec( B [ 1 f ( s ) ] 0 ) is aﬃne, so we are in the case of Lemma 2.7.  Pr oje ctive monoid schemes. F or a mo no id A and indeterminates T 0 , . . . , T n , let A [ T 0 , . . . , T n ] denote the monoid freely genera ted by A and the T i . It is a grade d monoid, where each element of A has deg ree 0 a nd each T i has degree 1, a nd w e deﬁne P n A to b e MPro j( A [ T 0 , . . . , T n ]). More gener a lly , for any mo no id scheme X = ( X , A ), deﬁne P n X to be MPr o j( B ) wher e B is the sheaf of grade d monoids on X deﬁned by sending an op en subse t U to A ( U )[ T 0 , . . . , T n ]. In o ther words, P n X is deﬁned by patching together the monoid schemes of the form P n A as MSp ec( A ) ranges ov er aﬃne op en subschemes o f X . If X has ﬁnite type, so do es P n X . A mor phism of monoid s c hemes Y → X is pr oje ctive if, lo cally on X , it factors as a clos e d immer s ion Y → P n X for some n follow ed b y the pro jection P n X → X . Lemma 7.3. Pr oje ctive morphisms ar e sep ar ate d. Although this follows from Prop osition 5.13, we give an elementary pro of here. Pr o of. Since closed immersions are separa ted by Lemma 3 .4, it suﬃces to show that the morphisms P n X → X a re separa ted. W e may assume that X = MSp ec( A ), so that P n X = MPro j( A [ T 0 , . . . , T n ]). B y Deﬁnition 7 .1, p oin ts of P n X corres p ond to pr ime ideals in A [ T 0 , . . . , T n ] not containing { T 0 , . . . , T n } . By Lemma 1.9 and Example 1.2, every such prime ideal has the for m P S, p = A ∧ h S i ∪ p [ T 0 , . . . , T n ] where p is a prime ideal of A a nd h S i is the pr ime ideal gener ated by a prop er subset S o f { T 0 , . . . , T n } ; moreover p and S are unique a nd the pr o jection to MSp ec( A ) sends the p oint P S, p to p . Accor ding to Lemma 3 .6, it suﬃces to observe that for every P S, p and P S ′ , p the prime P S ∩ S ′ , p is a unique low er b ound. (The sur jectivit y condition of Lemma 3.6 is eas y , and left to the re a der.)  Example 7.3. 1 . If B is a ﬁnitely generated graded monoid, then MPro j( B ) → MSpec( B 0 ) is pro jective and hence se parated by Lemma 7.3. Indeed, this is a particular case o f Lemma 7.2, since B is a quotient of some B 0 [ T 0 , . . . , T n ]. TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 25 Blow-ups. Given a monoid A a nd an ideal I , we c o nsider the gra ded monoid A ∨ I ∨ I 2 ∨ · · · , where I n has degree n . It is useful to int ro duce a v ariable t , and rewrite this as A [ I t ] = _ n ≥ 0 I n t n ⊆ A ∧ F 1 . If S is multiplicativ ely clo sed in A , then S − 1 ( A [ I t ]) ∼ = ( S − 1 A )[ S − 1 I t ]. It follows that if I is a quasi-co heren t sheaf of ideals in a monoid scheme ( X , A ) then there is a monoid scheme MPr o j( A [ I t ]) over ( X , A ) obtained b y patching the MPro j( A [ I t ]) in the evident manner. Deﬁnition 7.4. If X = ( X , A ) is a monoid scheme and Z ⊆ X is a eq uiv ariant closed subscheme, g iv en by a quasi-coher en t sheaf of ideals I , w e de ﬁne the blow-up of X along Z to b e the mono id s cheme X Z = MP ro j( A [ I t ]). R emark 7.4.1 . If X = MSpec( A ) is aﬃne and Z = MSpec( A/I ) then X Z = MPro j( A [ I t ]), together with the structure mor phis m MP r o j( A [ I t ]) → MSp ec( A ). Since MPro j( A [ t ]) ∼ = MSpec( A ), it follows that for U = X \ Z w e hav e X Z × X U ∼ = U . The blow-up construction is natural in the pair ( A, I ) in the following sense. If A → B is a morphism o f monoids, I is an idea l of A and J = I B , ther e is a canonical graded mor phism A [ I t ] → B [ J t ] sa tisfying the hypotheses of 7.1 .2. Hence there is a morphism MPro j( B [ J t ]) → MP ro j( A [ I t ]) of the blow-ups ov er MSp ec( B ) → MSpec( A ). Mor e genera lly , if f : X ′ → X is a morphism o f monoid schemes, I is a q ua si-coherent s he a f of ideals on X and J = f − 1 I · A ′ , then the morphism f − 1 A [ I t ] → A ′ [ J t ] induces a canonical morphism MP ro j( A ′ [ J t ]) → MPro j( A [ I t ]) ov er f , descr ibed in 7.1.2. R emark 7.4.2 . The blow-up of X alo ng a quasi-coher en t sheaf of ideals I is pro- jective provided I is given lo cally on X by ﬁnitely ge nerated idea ls, by 7.3.1. F o r example, if X has ﬁnite type then the blowup of X a long any qua si-coherent s heaf of ideals is pro jective. Example 7. 5. Sup p ose N is a free a belian group with basis { v 1 , . . . , v n } , and { x 1 , . . . , x n } is the dual basis of M . Let σ be the cone in N R generated b y { v 1 , . . . , v d } , the cor respo nding aﬃne mono id scheme is X ( σ ) = MSpec ( A ), wher e A is generated by x 1 , . . . , x n and x − 1 d +1 , . . . , x − 1 n sub ject to x i x − 1 i = 1 for d < i ≤ n . The blow-up o f X ( σ ) alo ng the ideal gener ated by x 1 , . . . , x d is the to r ic monoid scheme X (∆), wher e ∆ is the sub division o f the fan { σ } given by insertion of the ray spanned b y v 0 = v 1 + ... + v d . T o see this, it suﬃces to copy the corresp onding argument for tor ic v arieties g iven in [10, p. 41]. Example 7 . 5.1. If Z is an equiv ariant clo s ed subscheme of X , deﬁned by a quasi- coherent s heaf of idea ls I , and f : X ′ → X is a morphism, then by naturality of the blow-up construction, discus s ed ab ov e, ther e is a canonica l mo rphism over f , from the blo w-up X ′ Z ′ of X ′ along the pullback Z ′ = Z × X X ′ to the blow-up X Z . Lemma 7.6. L et f : X ′ → X b e a ﬁnite morphism of monoid schemes (6.2 ). L et Z b e an e quivariant close d subscheme of X , X Z the blow-up along Z , and X ′ Z ′ the blow-up of X ′ along the pul lb ack Z ′ = Z × X X ′ . Then ˜ f : X ′ Z ′ → X Z is a ﬁn ite morphism. 26 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL Pr o of. W e may assume that X , and hence X ′ , is aﬃne. Then f is induced b y a map A → A ′ , Z is deﬁned by a n ideal I ⊂ A and Z ′ is deﬁned by J = I A ′ . Mo reov e r bec ause f is a ssumed ﬁnite, there are elemen ts c 1 , . . . , c r ∈ B such that B = S i Ac i . If a 0 , . . . , a n generate I , and b 0 , . . . , b n are their imag es in B , then ˜ f restricts to maps D + ( b i ) → D + ( a i ) induced by the monoid maps A i = A [ a 0 /a i , . . . , a n /a i ] → B i = B [ b 0 /b i , . . . , b n /b i ]. By insp ection, B i = ∪ n j =1 A i c j .  Prop osition 7. 7. L et Z b e an e quivariant close d subscheme of a monoid scheme X of ﬁnite typ e. Then for any c ommutative ring k the blow-up of X k along Z k is c anonic al ly isomorphic to the k -r e alization of the blow-up of X along Z . Pr o of. It suﬃces to consider the case X = MSp ec( A ), Z = MSpec ( A/I ). In this case S = k [ A [ I t ]] is the usual Rees ring k [ A ][ J t ], J = k [ I ]. Since the blowing- up of X k = Spec( k [ A ]) along Z k = Spec( k [ A/I ]) is Pro j( S ), we hav e the desired ident iﬁcation P r o j( S ) = Pro j( k [ A [ I t ]]) = MPr o j( A [ I t ]) k .  W e conclude this sec tio n b y observing that blow-ups o f monoid schemes satisfy a universal pr oper t y analo gous to that for blow-ups of us ua l schemes. T o s ta te it, we need some no tation. W e deﬁne a princip al invertible idea l of A to b e an ideal I such that ther e is a n x ∈ I s uc h that the ma p A x − → I ( a 7→ ax ) is a bijection. If I is a pr incipal inv ertible idea l of A then the ca nonical map MP ro j( A [ I t ]) → MSpec ( A ) is an isomor phism. A quasi- coherent sheaf of ideals of a monoid scheme X is s aid to b e invertible if X can b e cov ered by aﬃne o pen subschemes U such that I ( U ) is a principal inv er tible ideal of A X ( U ). If ( X, A ) is a monoid s cheme and I ⊂ A is a quasi- coherent sheaf of ideals (see Deﬁnition 2.8), we say that a morphis m f : Y → X inverts I if f − 1 I · B is an inv ertible sheaf on Y . Prop osition 7. 8. L et X b e a monoid scheme of ﬁnite typ e, Z an e quivariant close d subscheme deﬁne d by a quasi-c oher ent she af of ide als I , and π : e X → X the blow-up of X along Z . Then π inverts I and is universal with this pr op erty in the sense that if Y is of ﬁnite typ e and f : Y → X inverts I , then t he dotte d arr ow in t he diagr am b elow ex ist s and is u nique. Y / / f   ❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ e X π   X Pr o of. W e may ass ume that X = MSp ec( A ) for so me ﬁnitely genera ted monoid A , that I corr espo nds to an ideal I of A , and tha t e X = MPr o j( A [ I t ]). The map π inv erts I beca use the r estriction of π − 1 I to D + ( s ) is g enerated b y s for each s ∈ I . Let B be the structure sheaf of Y , and write J for the sheaf of ideals f − 1 I · B . By Example 7.5 .1, there is a unique mor phism fr om the blow-up e Y = MPro j( B [ J t ]) to e X over f . By as sumption, J is an inv ertible shea f. Hence e Y → Y is an isomorphism, b ecause lo cally J is a principal inv ertible idea l J of B and MPro j( B [ J t ]) ∼ = MSpec ( B ).  TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 27 8. Pr oper morphisms A monoid V is calle d a valuation monoid if V is ca ncellative and for every non- zero element α ∈ V + , at leas t o ne o f α or 1 α belo ngs to V . F o r example, if R is a v a luation ring, then the underlying multiplicativ e monoid ( R , × ) is a v aluation monoid. Also, the free p ointed monoid on one gener a tor is a v aluation monoid. Given a v aluation monoid V , the monoid V ∧ M ∗ is also a v a luation monoid for any ab elian group M . F or example, the monoid  y ± 1 1 , . . . , y ± 1 n , x  is a v aluation monoid. Given a v aluation monoid V , the units U ( V ) ar e a subgroup of V + \ 0 a nd the quotient group ( V + \ 0) /U ( V ) is a totally ordered ab e lia n gro up with the total ordering deﬁned by x ≥ y if and only if x y belo ngs to the image of V \ 0. T o conform to usual custom, we conv ert the group law for ( V + \ 0) /U ( V ) into +. W e also adjoin a base p oint, w r itten ∞ , to o btain the totally ordered p ointed (additive) monoid Γ :=  ( V + \ 0) /U ( V )  ∗ . W e extend the to tal or dering to Γ by declaring that γ ≤ ∞ for all γ ∈ Γ. W e ca ll Γ the value monoid o f the v a luation monoid V . The canonic a l surjectio n (8.1) ord : V + ։ Γ is called the valuation map o f V . T he monoid V is then identiﬁed with the set of x ∈ V + such that ord( x ) ≥ 0 (where, reca ll 0 is the identit y of Γ), and the maximal ideal m of V is { x | o r d( x ) > 0 } (since or d( x ) = 0 just in ca se x is a unit of V ). Note that (8.1) sa tisﬁes o r d( x ) ≤ ∞ , ord( xy ) = ord( x ) + or d( y ) and ord( x ) = ∞ if and only if x = 0. Conv ersely , given an ab elian group M and a sur jective morphism ord : M ∗ → Γ onto a tota lly or dered mono id (Γ , + , 0 , ∞ ) that satisﬁes these conditions, the set C = { a ∈ M | o rd( a ) ≥ 0 } is a v aluation monoid whos e po in ted group completio n is M ∗ and whose ass ocia ted v aluation map is ord. Lemma 8.2. A valuation monoid V has no ﬁnite extens ions c ont aine d in V + . Pr o of. Suppose that V ⊆ B ⊆ V + with B ﬁnite ov er V . By Lemma 1.7(ii), B is int egra l ov er V . F or every nonzero b ∈ B there is a n n ≥ 1 so that b n ∈ V and hence n or d( b ) ≥ 0 , which implies that o rd( b ) ≥ 0 and thus b ∈ V .  Example 8. 3. A discr ete valuation monoid is a v alua tion monoid w ho se v a lue monoid is isomorphic to Z ∪ {∞} with its canonical ordering. In this ca se, a lifting of the ge ne r ator 1 ∈ Z to an element π in the discrete v aluation mono id V is a generator o f the maximal ideal of V a nd every non-zer o elemen t o f V + may written uniquely as uπ n for n ∈ Z and u ∈ U ( V ). Let’s call such a n element a u niformizi ng p ar ameter . Observe tha t if R is a discrete v aluation ring, then ( R, × ) is a dis crete v a lua tion monoid and the no tio n o f a uniformizing par ameter has its usual meaning. If V is a discrete v aluation monoid, its v aluation map induces a surjection π : V + \ 0 ։ Z ; wr ite M = ker π . A choice of unifor mizing para meter t is equiv alent to a section of π and identiﬁes V + \ 0 = M ×  t ± 1  . Under this identiﬁcation, π is the ev ident pro jection. Th us, every discrete v aluatio n monoid V is iso morphic to U ( V ) ∗ ∧ h t i , where h t i is the free ab elian mono id on one generator and U ( V ) is the group o f units of V . An y element of the form u ∧ t with u ∈ U ( M ) is a uniformizing parameter. 28 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL R emark 8 .3.1 . It is well-kno wn that a v aluation ring is No etherian if and only if it is a discrete v aluation r ing; see [31, § VI.10, Thm. 16] for a pro of. The same argument shows tha t a v aluation monoid is ﬁnitely generated if and only if it is a discr ete v aluatio n mo noid with a ﬁnitely ge ne r ated group of units. Deﬁnition 8.4. A map f : Y → X o f monoid schemes satisﬁes the valuative criterion for pr op erness if for every v aluation monoid V and comm utative square (8.4a) MSpec ( V + ) / /   Y f   MSpec( V ) / / : : X there is a unique map MSp ec( V ) → Y ca using bo th triangles to co mm ute. W e say f satis ﬁe s the valuative criterion of sep ar ate dness if each such square has at most one completion. A map Y → X of monoid schemes of ﬁnite type is sa id to be pr op er if it satisﬁes the v a luativ e criterion for pr oper ne s s. R emark 8.4.1 . W e a re not certain what the corr ect deﬁnition o f “prop er” is for monoid schemes not of ﬁnite type. (Recall from Remark 3.4 .1 that “separated and universally clo sed” is clear ly not the cor rect deﬁnition.) Given any mor phism f : MSp ec( V ) → X , any aﬃne op en U ⊂ X containing f ( m ) (where m is the unique c lo sed p oint of MSpec( V )) will contain the image of MSpec( V ). Hence the v aluative criterio n of prop erness and sepa ratedness are lo cal on the base: if Y | U → U s atisﬁes o ne of these c r itera for every U in a cov ering o f X , then so do es Y → X . It is immediate from Deﬁnition 8.4 that the class of maps satisfying the v alua - tive criterion of prop erness (res p., separ atedness) is closed under comp osition and pullback. Prop osition 8.5. A ﬁnite morphism b etwe en monoid s chemes satisﬁes the valua- tive criterion of pr op erness. Pr o of. Suppose Y → X is ﬁnite a nd co nsider a commutativ e square (8.4a) with V a v aluation mono id. W e may a s sume Y → X is a map of aﬃne schemes, say given by a map of monoids A → B . Then the s quare (8.4a) is a s socia ted to the squar e V + B o o V O O A. o o O O of monoids. The image of B in V + is ﬁnite over V , but V is closed under ﬁnite extensions in V + , by Lemma 8.2. It follows that the map B → V + actually la nds in V , which gives the diag onal ma p we seek.  Corollary 8. 6. Close d immersions satisfy t he valuative criterion of pr op erness. Construction 8.7. T o prov e Theore m 8.9 be lo w, we need a technical construction: Let V b e a v aluation monoid with gro up completion V + and v alue monoid (Γ , + ). Recall that to ta lly or dered groups a re nece s sarily torsio n- free, and hence, for any ﬁeld k , the ring k [Γ] is an integral domain by Lemma 5.10. TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 29 F or an element α = X γ a γ γ of k [Γ] (wher e for this ring we have rewritten Γ using · ins tead of + no tation), deﬁne ord( α ) = min { γ ∈ Γ | a γ 6 = 0 } . (F or α = 0, set o rd(0) = ∞ .) It is easily veriﬁed that o r d : ( k [Γ] , × ) → Γ is a monoid map such that ord( α ) = ∞ if and only if α = 0 . It follows that we g et a n induced map of p o in ted group co mpletio ns ord : ( k (Γ) , × ) → Γ where k (Γ) denotes the ﬁeld of fra c tions of k [Γ]. Mo reov er, the comp osition V + → ( k (Γ) , × ) ord − → Γ coincides with the origina l v a luation map ord : V + → Γ. Finally , the pair ( k (Γ) , o rd) is a v alua tio n in the usual ring-theor e tic sense. T o prov e this, it rema ins to show ord( α + β ) ≥ min { o rd ( α ) , o rd( β ) } for all α, β ∈ k (Γ). One easily reduces to the cas e when α, β ∈ k [Γ], w he r e it is o b vious from the deﬁnition of or d. Prop osition 8.8. Given a valuation monoid V , with p ointe d gr oup c ompletion V + and value monoid Γ , let o rd b e the valuation map on the ﬁeld k (Γ) given in Construction 8.7, and let R ⊂ k (Γ) denote the asso ciate d valuation ring. Then t he squar e of aﬃne monoid schemes MSpec( k (Γ) , × ) / /   MSpec ( V + )   MSpec( R , × ) / / MSpec( V ) is a pushout squar e in t he c ate gory of monoid schemes. Pr o of. F or any mono id scheme T , supp ose mor phisms f : MSp e c( V + ) → T and g : MSp ec( R, × ) → T a re given causing the evident squar e to commute. Let t ∈ T be the image of the unique close d po int of MSp ec( R, × ) under g , and let U ⊂ T be a n y aﬃne o p en subscheme of T cont aining t . Then g factors thr ough U . Since MSpec( k (Γ) , × ) → MSp ec( V + ) is a bijection on underlying sets (each is a one- po in t s et), the unique p oint of MSpec( V + ) a lso la nds in U and hence f to o factors through U . W e may th us a s sume T = U is aﬃne. That is, it suﬃces to prove V / /   ( R, × )   V + / / ( k (Γ) , × ) 30 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL is a pullback square in the ca tegory of p oin ted monoids. But this is ev iden t since V + / / ord   ( k (Γ) , × ) ord   Γ = / / Γ commutes, V = { α ∈ V + | ord( α ) ≥ 0 } a nd R = { β ∈ k (Γ) | ord( β ) ≥ 0 } .  Recall that a map of (classica l) k -s c hemes Y k → X k , where k is a ﬁeld, is said to satisfy the valuative criterion of pr op erness (resp., sep ar ate dness ) if every solid arrow squar e Spec ( F ) / /   Y k   Spec ( R ) ; ; / / X k has a unique (resp., at most one) co mpletion making b oth triang les commute, whenever R is a v alua tion r ing (which is necessa rily a k -algebra ) a nd F is its ﬁeld of factions. Theorem 8.9. L et f : X → Y b e a morphism of monoid schemes and let k b e a ﬁeld. The morphism f k : X k → Y k satisﬁes the valuative criterion of pr op erness (r esp., sep ar ate dness) if and only if f satisﬁes the valuative criterion of pr op erness (r esp., sep ar ate dness). Pr o of. By Theorem 5.2, for an y lo cal k -algebr a R , there is a natural adjunction isomorphism Hom k (Spec ( R ) , X k ) ∼ = F X (Spec R ) = Hom MSch (MSpec ( R, × ) , X ) . Now supp ose R is a v aluation ring with ﬁeld of fractions F . Then V = ( R , × ) is a v aluation mono id with V + = ( F , × ). Since R and F are loc a l, a commutativ e square of the for m Spec F / /   Y k   Spec R / / X k corres p onds via adjunction to a commut ative squa re o f mono id schemes given by the solid ar rows in the diag ram (8.10) MSp ec( V + ) / /   Y   MSpec( V ) 9 9 / / X . If Y → X satisﬁes the v aluative criterion o f prop erness (res p. separatedness), there exists a unique (resp., a t most one) mor phis m of monoid schemes MSpec ( V ) → Y represented by the dotted arrow ab ov e that causes b oth triang les to commut e. Again by adjunction, this g iv es a unique map Spec ( R ) → Y k causing b o th triangles to commute in the ﬁrst sq ua re. TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 31 Conv ersely , say a squar e (8.10) is g iv en. B y Co ns truction 8.7, ther e is a v aluation ring R with ﬁeld of fractions F = k (Γ) and morphisms MSp ec( R, × ) → MSp ec V and MSp e c ( F, × ) → MSpec V + ﬁtting into a commutativ e diagra m (8.11) MSpec ( F, × ) / /   MSpec( V + ) / /   Y   MSpec ( R, × ) 3 3 / / MSpec ( V ) / / X . By Prop osition 8.8 , the left-hand squa re is a pushout sq uare in the categor y of monoid schemes. Using adjunction as a bov e, if Y k → X k satisﬁes the v aluative criterion of pr o pernes s (resp., separatedness ), ther e exists a unique (resp., at mo st one) map re pr esen ted by the dotted a rrow in (8.1 1) that cause s the outer tw o tr i- angles to commute. Since the left-hand square is a pushout, it follows immediately that ther e exists a unique (res p., at most one) arr o w MSp ec( V ) → Y causing b oth triangles in (8.1 1) to comm ute.  Corollary 8.1 2 . F or any ﬁeld k , a morphism b etwe en monoid schemes of ﬁnite typ e Y → X is pr op er if and only if Y k → X k is pr op er. Pr o of. Merely observe that Y k and X k are No etherian, and apply the v alua tiv e criterion of the prop erness theorem [16, I I.4.7].  R emark 8.12 .1 . Say f : Y → X satisﬁes the v aluative cr iterion of prop erness. If Y k is quasi-co mpact, EGA I I(7.2.1) implies that f k is prop er. Corollary 8.13. A morphism b etwe en monoid schemes of ﬁ nite typ e is pr op er if and only if it satisﬁes t he valuative criterion of pr op erness of Deﬁnition 8.4 for al l discr ete valuation monoids. Pr o of. If f : X → Y satisﬁe s the criterion o f Deﬁnition 8 .4 for a ll discrete v aluation monoids, then, for an y ﬁeld k , its k -realization f k : X k → Y k satisﬁes the v aluative criterion of pro pernes s for all D VRs. This follows, using adjunction, from the fact that MSp ec( R, × ) is a discr ete v a luation monoid if R is a DVR. Since X k and Y k are No etherian and f k has ﬁnite type, it follows that f k is prop er (see [16, E x . I I.4.11]). The result now follows from Co r ollary 8.12.  Corollary 8.14. A pr oje ctive morphism Y → X b etwe en monoid schemes of ﬁnite typ e is pr op er. In p articular, if X is a monoid scheme of ﬁnite typ e and X Z is t he blow-up along an e qu ivari ant close d subscheme Z , t hen the map X Z → X is pr op er. Pr o of. Using Prop osition 5.11 and Remark 7.1.1, we see that if k is a ﬁeld, then Y k → X k is a pro jective morphism of k -sc hemes and hence is prop er. F or the second assertion, reca ll that X Z → X is pro jective a nd X Z has ﬁnite type.  R emark 8 .15 . In fact, a pro jective morphism of arbitrary mo noid schemes satisﬁes the v aluative criterio n of pro perness . W e sketc h the pro of o f this fact. First o ne observes that, by Corollar y 8 .6, it suﬃces to chec k that for any monoid scheme X and n ≥ 1 the pr o jection P n X → X satisﬁes the criterio n. Second, one reduces further to showing that if V is a v alua tio n monoid then any section MSpec ( V + ) → P n V + of the canonica l pr o jection extends to a se c tio n MSp ec( V ) → P n V of P n V → MSpec( V ). Third, one observes tha t for an aﬃne scheme MSp ec A a section of P n A → MSpec A is deter mined by an equiv alence class of n -tuples ( b 0 , . . . , b n ) of 32 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL elements of A such tha t at least one of the b i is nonzero, mo dulo the co ordinate- wise action of U ( A ). Finally , one prov es tha t if b = ( b 0 , . . . , b n ) determines a section MSpec( V + ) → P n V + as ab ov e, then m ultiplying the b i by an a ppropriate p ow er of a uniformizing parameter we obtain a n eq uiv alent tuple b ′ with b ′ i ∈ V for a ll i . Thus the section extends to MSp ec( V ). Recall from 4.1 that a monoid scheme of ﬁnite type is toric if it is s e pa rated, connected, to r sionfree and normal. By Theorem 4.4, there is a faithful functor from fans to toric mono id schemes. Corollary 8.16. L et φ : ( N ′ , ∆ ′ ) → ( N , ∆) b e a morphism of fans. Then the asso ciate d morphism of t oric monoid schemes X ′ → X is pr op er if and only if φ has the pr op erty that for e ach σ ∈ ∆ , φ − 1 R ( σ ) is a union of c ones in ∆ ′ . Pr o of. This follows fro m the well-known fac t that if k is a ﬁeld, then X ′ k → X k is prop er if and only if φ has the stated pr o perty (se e [10, p. 39]).  Corollary 8. 17. Every pr op er map b et we en monoid s chemes of ﬁ n ite t yp e is sep- ar ate d. Pr o of. By Theorem 8.9 and the V alua tive Cr iterion o f Separatedness Theorem for No etherian schemes, then the k -r ealization of a pr oper map b etw een monoid schemes of ﬁnite type is se parated if k is a ﬁeld. Now use Pro position 5.13.  9. P ar tiall y cancella tive torsion free monoid schemes A monoid A is p ctf if it is isomorphic to a monoid of the form B /I where B is a cancellative torsion free monoid (i.e., a cancellative monoid whose group completion is torsion fre e ) a nd I is an ideal. A monoid scheme is p ctf if all o f its stalks a re. Prop osition 9.1. W e have: (1) If a p ct f monoid is ﬁnitely gener ate d, then it is isomorphic to A/I wher e A is a ﬁ nitely gener ate d torsion fr e e c anc el lative monoid. (2) Al l submonoids and lo c alizations of a p ctf monoid ar e p ctf. In p articular, for a monoid A , MSp ec( A ) is p ctf if and only if A is p ctf. (3) If A is a p ctf monoid and p is a prime ide al, then A/ p is a c anc el lative torsionfr e e monoid. (4) An op en subscheme of a p ctf m onoid scheme is p ctf. (5) An e quivariant close d subscheme of a p ctf monoid scheme is p ctf. Pr o of. Say A = B / I with B cancellative and tors ion free. Pick elements b 1 , . . . , b m in B that map to a g enerating set of A and let B ′ be the submonoid of B they generate. Then A = B ′ / ( I ∩ B ′ ), proving the ﬁrst asser tion. F or the s econd, say A = C / I with C cancellative and torsion free. If B is a submonoid of A , let B ′ denote the inv erse image of B in C and s et I ′ = I ∩ B ′ . Then B = B ′ /I ′ , and so B is p ctf. The assertion concerning lo calizations holds since S − 1 ( C /I ) ∼ = S − 1 C /S − 1 I . The rema ining assertion of par t (2) is clear . If A = B /I then A/ p = B / p ′ for some prime idea l o f B , so (3) follows from the elementary obser v ation that if A is cance lla tiv e and torsio nfree then so is A/ p . Assertion (4) is lo cal and follows from (2); hence (5) is lo cal, and is then easy .  Prop osition 9.2. The blow-up of a p ct f monoid scheme along an e quivariant close d subscheme is p ctf. TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 33 Pr o of. Let Y → X b e the blow-up of a p ctf mo noid scheme X along an equiv a riant closed subscheme. Since the question is lo cal on X , we may assume that X is aﬃne, say X = MSp ec( A ) with A p ctf. Then Y is MPro j( A [ I t ]) for an ideal I . F or ea c h s ∈ I , we get an aﬃne op en subset of Y given by the mo no id { f s n | f ∈ I n , n ≥ 0 } . This is a submonoid of A [ 1 s ] a nd hence is p ctf. The co llection of such op en subsets as s v a ries ov er all element s of I form an op en cover of Y . Thus Y is p ctf.  Prop osition 9.3. L et X = ( X , A ) and Y = ( Y , B ) b e m onoid schemes and let f : Y → X b e a morphism. Ther e is a unique close d su bscheme Z of X which is minimal with r esp e ct to t he pr op ert y t hat f factors thr ough Z ⊂ X . If U ⊂ X is an aﬃne op en subscheme of X , then Z ∩ U is the aﬃne scheme MSpec( C ) , wher e the monoid C is the image of B ( U ) → A ( U × X Y ) . I n p articular, if X is of ﬁnite typ e then s o is Z . Pr o of. If f factors through tw o diﬀer en t closed subschemes W 1 and W 2 of X , then it factors through W 1 × X W 2 , which is (canonically isomor phic to) a closed subscheme of X (see E xample 3.2). So, we deﬁne Z to be the inv erse limit taken ov e r the partially ordered s et o f clo sed subschemes W of X such that f factor s throug h W . F or the lo cal description of Z , we may a ssume tha t X = U = MSp ec( B ) is aﬃne. An y closed subscheme o f X has the form W = MSpec ( D ) with B → D a surjection of mo noids. Then f facto rs through W if and only if B → A ( Y ) factors through D , that is , if a nd only if B → C factors as B → D → C ; in o ther words, if and only if Z ⊆ W .  Deﬁnition 9. 4 . The subs cheme Z of 9.3 is called the scheme-the or et ic image of f . If f is an op en immersion, we write Y for Z and (b y abuse) call it the closur e of Y . Prop osition 9.5. L et Y b e a monoid scheme and supp ose U ⊂ Y is an op en subscheme that is p ctf. Then the scheme-the or etic image U of U in Y is p ctf. Mor e over, if Y is sep ar ate d, then U is s ep ar ate d. Pr o of. The ﬁrst ass e rtion is lo cal on Y and so we may assume Y = MSpe c ( B ) for a monoid B a nd U = MSp ec( S − 1 B ) for a multiplicativ e subset S . Then U is the aﬃne scheme asso ciated to the image B of B → S − 1 B . The monoid S − 1 B is p ctf by ass umption a nd 9 .1(4), and hence so is B by 9.1(2). The second asser tion is just the obser v ation that a closed subs cheme of a sepa - rated scheme is also separ a ted by Lemma 3.4.  10. Bira tional morphisms A morphism p : Y → X of monoid schemes is bir ational if there is a n op en dense subscheme U o f X such that p − 1 ( U ) is dense in Y and p induces a n is o morphism from p − 1 ( U ) to U . Prop osition 10. 1 (Birational maps) . L et p : ( Y , B ) → ( X , A ) b e a map b etwe en monoid schemes of ﬁn ite t yp e, Then p is bir ational if and only if the fol lowing c onditions hold: (1) p maps the generic p oints of Y bije ctively onto the generic p oints of X (2) A p oint y ∈ Y is generic if (and only if ) p ( y ) ∈ X is generic (3) for e ach generic p oint y ∈ Y the induc e d map A ( p ( y )) → B ( y ) on st alks is an isomorphism. 34 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL Pr o of. If p is birationa l a nd U is a s in the deﬁnition a bov e, then U contains all o f the g eneric po in ts of X and p − 1 ( U ) co n tains all the generic p oints of Y as well as every p oin t of y that ma ps to a generic p oint of X . The conditions are then clearly satisﬁed. Conv ersely , take U to b e the (dense op en) set of g eneric p oints of X . By hy- po thesis, p − 1 ( U ) is the set of generic p oints of Y and the map p : p − 1 ( U ) → U is bijectiv e. Hence p − 1 ( U ) is op en and dense. Since the map p − 1 ( U ) → U is bijectiv e and induces a n is omorphism on all stalks , it is an isomorphism.  Corollary 10.2. If p : X ′ → X is a pr op er map of toric monoid schemes that is bir ational, then p is given by a map of fans φ : ( N ′ , ∆ ′ ) → ( N , ∆) such that φ : N ′ ∼ = − → N and the image of ∆ ′ under the isomorphism φ R is a s u b division of ∆ . Conversely, any such map φ induc es a pr op er bir ational map of monoid schemes. Pr o of. F rom 4 .4(2), p comes fro m a morphism of fans s uc h tha t φ : N ′ ∼ = − → N , and such a morphism is a sub division by Coro llary 8.1 6. Conv e r sely , if p is induced by a morphism of fans φ : ( N ′ , ∆ ′ ) → ( N , ∆) such that φ R is a sub division of ∆, then p k is prop er by [1 0, § 2.4]; hence p is pr oper by Theorem 8.9.  Example 10.3. If X is a monoid scheme of ﬁnite t yp e, let X η denote the equiv ar i- ant closur e of a gener ic p oin t η (in the sens e of 2.9). Then e a c h X η has a unique generic p o in t, na mely η . If X is p ctf, then ea c h X η is cancellative and torsio nfree by 9.1(3), a nd hence p c tf. If X is reduced, the mor phism ` η X η → X is birational. Prop osition 10. 4. If Y → X is a bir ational map and X ′ → X is a morphism such that X ′ is of ﬁnite typ e and every generic p oint of X ′ maps to a generic p oint of X , then t he pul lb ack Y × X X ′ → X ′ is bir ational. Pr o of. The po set underlying Y × X X ′ is given by the pullback of the underlying po sets (by 3.1). Since Y → X is bira tional, a p oint ( y , x ′ ) in Y × X X ′ is generic if and only if x ′ is a generic p oint of X ′ , a nd in this case y a nd x ′ map to the s ame po in t x of X , which is generic. Hence the map Y × X X ′ → X ′ is a bijection on sets of generic p oints. W riting A ′ , A a nd B for the stalk functors of X ′ , X and Y , the map on generic stalks is o f the form A ′ ( x ′ ) → A ′ ( x ′ ) ∧ A ( x ) B ( y ). This is an isomorphism, since the map A ( x ) → B ( y ) is a n is omorphism.  Deﬁne the height o f a p oint x in a monoid scheme X to be the dimension of A x ; i.e., it is the largest in teger n such that there exis ts a strictly decre asing chain x = x n > · · · > x 0 in the p oset underlying X . W e write this as ht ( x ) o r ht X ( x ). F or example, if X = X ( N , ∆) is the monoid s c heme as socia ted to a fan, then ht ( σ ) = dim( σ ) for each cone σ ∈ ∆. Here dim( σ ) refers to the dimension of the real vector subspace of N R spanned by σ . Lemma 10. 5. Supp ose p : Y → X is a pr op er, bir ational map of sep ar ate d p ctf schemes of ﬁnite t yp e. Then for any y ∈ Y , we have ht Y ( y ) ≤ ht X ( p ( y )) . Pr o of. Suppose h t Y ( y ) = m , so that we hav e a chain o f p oints y = y m > · · · > y 0 in Y . Clearly y 0 m ust b e minimal, and thus generic. Le t η = p ( y 0 ), and deﬁne X η to be the equiv ariant closur e of { η } in X . As po in ted out in Example 10 .3 , X η is cancellative and tor sionfree. The pullback Y η = X η × X Y is an equiv a riant clo sed subscheme of Y containing y 0 as its unique generic p oint , and hence ea c h y i . By Prop osition 9.1(5), Y η is also p ctf, and Y η → X η is birational by Prop osition 10.4. TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 35 Let Y ′ denote the equiv ar ian t closure of y 0 in Y η . By Example 10.3, Y ′ → Y η is birational, Y ′ contains all the y i and Y ′ is cancellative and tor sionfree. Repla c ing X and Y by X η and Y ′ , we ma y assume that bo th X and Y a re connected, cancellative and torsionfree. Hence the norma lization maps X nor → X and Y nor → Y exis t a nd are homeomorphis ms (by 1.6.1), and bo th X nor and Y nor are tors ionfree. Since Y → X is birational, it induces a birational mo rphism Y nor → X nor . The map Y nor → Y is ﬁnite by 1 .7 and hence prop er by 8 .5. Thus Y nor → X and hence Y nor → X nor are prop er. Thus we may assume that X and Y are separ ated, normal and torsionfre e. By Pr opos ition 4.5 and Coro llary 4.6, w e have reduced to the ca se where Y → X is a pro per bir ational ma p of toric monoid schemes, given by a map o f fans φ : ( N ′ , ∆ ′ ) → ( N , ∆). The birationa l hypothesis means that φ : N ′ → N is an iso morphism. B y Cor ollary 10.2, the prop er hyp o thesis means that ∆ ′ is a sub div ision of ∆. Since φ ( σ ) is the smallest cone in ∆ co n taining the imag e of σ under φ R and since height corres ponds to dimension of cones, the result is now clear.  11. Resolutions o f singular ities for toric v arieties The pur pose of this section is to establish some pro perties for mo noid schemes that are analog ous to thos e known to hold for arbitra ry v a rieties in characteristic zero. These prop erties will b e used in Section 12 to prove that certain presheaves of sp ectra satisfy the a nalogue of “smo oth cdh descent” for monoid schemes. Theorem 11.1. L et X b e a sep ar ate d c anc el lative p ctf monoid scheme of ﬁnite typ e. Then ther e is a bir ational pr op er m orphism Y → X such that Y is smo oth. Pr o of. W e may assume that X is connected. Since the nor ma lization ma p is prop er birational by Prop ositions 6.3 and 8.5, we may assume that X is norma l. Since X is p ctf it is torsionfree by P rop osition 9 .1(3). By P ropo sition 4.5, X is toric and X ∼ = X (∆) for s o me fan ∆. There ex ists a sub division ∆ ′ of ∆ such that X (∆ ′ ) is smo oth, and it follows from Cor ollary 10.2 that the mor phism X (∆ ′ ) → X (∆) is prop er birational.  Let N b e a free a belian gro up of ﬁnite rank. Recall (from [10, page 34], e.g .) that a cone in N R is called simplicial if it is g enerated by linearly indep enden t vectors, and that a fan is s implicial if every cone in it is simplicial. W e will nee d the notion of the baryc en tric sub division of a simplicial fan ∆ in N R : F or a s implicial cone σ in N R of dimensio n d , let v 1 , . . . , v d be the minimal lattice p oin ts a long the one- dimensional fac es of σ , a lso c a lled the rays o f σ . F or each no n-empt y subset S of { 1 , . . . , d } , let v S = P i ∈ S v i . The b aryc entric sub division of σ , which we wr ite as σ (1) , is deﬁned as the collection of 2 d cones given as the span of vectors of the form v S 1 , . . . v S e , where 0 ≤ e ≤ d and S 1 ⊂ · · · ⊂ S e is a chain of pr oper subsets of { 1 , . . . , d } . It is clea r that if τ is a fa c e of σ , then the set of co nes in σ (1) that a re contained in τ form the fan τ (1) . It follows that ∆ (1) := n σ (1) | σ ∈ ∆ o is again a s implicial fan. W e inductively deﬁne ∆ ( i ) = (∆ ( i − 1) ) (1) for i ≥ 2. 36 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL Lemma 11.2. If ∆ ′ is any sub divisio n of a simplicial fan ∆ in N R , then for i ≫ 0 , the fan ∆ ( i ) is a sub division of ∆ ′ . Pr o of. It suﬃces to show that a n y ray of ∆ ′ , that is, any 1-dimensional cone of ∆ ′ , is a ray of some ∆ ( i ) . Given a p ositive integer co m bination v = P n i v i of the vertices in a cone, we may reo rder the vertices to ass ume the n i are in decr easing order. Then v is in the cone of ∆ (1) spanned by the v S i , wher e S i = { 1 , . . . , i } , and (if v 6 = v 1 ) w e can write v = P n ′ i v S i with P n ′ i < P n i . The result follows by induction on P n i .  Lemma 11.3. If ∆ is a smo oth fan, then for al l i ≥ 1 , the toric monoid scheme X (∆ ( i ) ) is obtaine d fr om X (∆) via a se quenc e of blow-ups along smo oth c enters. Pr o of. W e may ass ume i = 1. If ∆ is smo oth alrea dy , then ∆ (1) is a ls o smo oth. In gener al, the fan ∆ (1) is o bta ined from ∆ v ia a ser ies of steps o f the following sort: s ta rting with a smo oth fan ∆, w e form a sub division ∆ ′ by picking a cone σ , letting v 1 , . . . , v d be the minima l lattice po in ts alo ng its rays, and deﬁning ∆ ′ to be the sub division of ∆ given by inser tio n o f the ray spa nned by v 1 + · · · + v d . By Example 7.5, X (∆ ′ ) → X (∆) is the blow-up along the smo oth, closed equiv ariant subscheme deﬁned by x 1 = · · · = x d = 0 .  Theorem 11.4. F or a morphism π : Y → X b etwe en sep ar ate d c anc el lative p ctf monoid schemes of ﬁnite typ e, assume X is smo oth and π : Y → X is pr op er and bir ational. Then ther e ex ists a se quenc e of blow-ups along smo oth close d e quivariant c enters, X n → · · · → X 1 → X 0 = X , such t hat X n → X factors thr ough π : Y → X . Pr o of. By Theore m 1 1 .1, there is a prop er bir ational morphism Z → Y with Z smo oth. W e may therefore assume that Y is s mooth. W e may also assume that X and Y ar e connected, so that they hav e unique generic p oints. Thu s, b y Co rollary 10.2, Y → X is given b y a mor phis m ( N ′ , ∆ ′ ) → ( N , ∆) of fans that is an is omorphism of lattices and s uc h tha t ∆ ′ is a sub division of ∆. Lemmas 11.2 and 11 .3 complete the pro of.  12. cd structures on monoid schemes. Let M p ctf denote the categor y of monoid schemes of ﬁnite t y pe that ar e separated and p ctf. In this sectio n, we will b e concer ned with cartesia n squar es o f the form (12.1) D / /   Y p   C e / / X . Deﬁnition 12.2. An abstr act blow-up is a car tesian sq uare of monoid schemes of ﬁnite type of the form (12.1) suc h that p is prop er, e is an equiv ariant c losed immersion, a nd p maps the op en complement Y \ D isomor phica lly onto X \ C . The square with Y = ∅ a nd C = X red is such a square. TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 37 Prop osition 12. 3. If X is of ﬁnite typ e, C is an e quivariant close d subscheme of X and p : Y → X is t he blow-up of X along C , t hen the r esu lting c artesian squar e is an abstr act blow-up. If X b elongs to M p ctf , so do Y , C and D . Pr o of. By Corolla ry 8.14, p is prop er. As noted in Deﬁnition 7.4, p maps Y \ D isomorphica lly to X \ C (b ecause D = C × X Y ). The second asser tio n follows fro m Prop ositions 9.1 and 9.2.  Prop osition 12 .4. Supp ose an abstr act blow-up squar e (12 .1) is given with X in M p ctf . L et ¯ Y b e the scheme-the or etic image of Y \ D in Y , and deﬁne ¯ D = C × X ¯ Y . Then D / /   Y p   C e / / X is an abstr act blow-up squar e in M p ctf . Pr o of. By P rop osition 9 .1, X \ C and hence Y \ D is p ctf, and so by Pro positio n 9.5, Y is p ctf as well. Since equiv ariant clo sed subschemes o f pctf sc hemes ar e p ctf, C and D also be lo ng to M p ctf . The map Y → X is a co mposition o f prop er maps and hence is pr oper . Finally , ¯ Y \ ¯ D = Y \ D .  Recall from [2 7, 2.1] that a cd structur e o n a categor y C is a collection of dis- tinguished commutativ e s quares in C . If C has an initial ob ject ∅ , any cd structur e deﬁnes a to polog y: the smallest Grothendieck top ology such tha t for each distin- guished squar e (1 2 .1) the sieve gener ated b y { p, e } is a covering sieve (and the empt y s iev e is a cov ering of the initial ob ject). The coverings { p, e } are called elementary . Deﬁnition 12 .5. T he blow-up cd stru ctur e on M p ctf is given b y the collection o f all abstra ct blow-up squares with X, Y , C , D a ll belo ng ing to M p ctf . The Zariski cd structu r e on M p ctf is given by all cartesian squa res a sso ciated to a covering of X by tw o op en subschemes. The cdh top ology on M p ctf is the top ology g e nerated by the union of these tw o cd structures. F ollowing [27, 2.3, 2.4], we say that a cd s tructure is c omplete if C has an initial ob ject ∅ a nd any pullback of an element ary cov ering contains a sieve which can b e obtained by iterating e lemen tary cov erings. W e say that a c d structure is r e gular (see [27, 2.1 0]) if each distinguished s q uare (12.1) is a pullbac k, e is a monomorphism and the morphis m o f sheav es (12.6) ρ ( D ) × ρ ( C ) ρ ( D ) ∐ ρ ( Y ) → ρ ( Y ) × ρ ( X ) ρ ( Y ) is onto, wher e ρ ( T ) denotes the sheaﬁﬁcatio n o f the preshea f repres en ted b y T . Theorem 12.7. The blow-up and Zariski cd st ructur es on M p ctf ar e c omplete and r e gular. Pr o of. The completeness pr oper t y for Zar iski squar es is clear since they a re pre- served by pullback, and the reg ularity prop erty is even cleare r . F or the blow-up cd 38 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL structure, consider an abstra ct blow-up square D / /   Y p   C e / / X . Let X ′ → X b e any morphis m in M p ctf and cons ider the square inv olving X ′ , C ′ , Y ′ and D ′ formed by pullback. The scheme Y ′ might no t b e long to M p ctf , but the scheme-theoretic image Y ′′ of Y ′ \ D ′ in Y ′ do es by Pr opos ition 12.4. The resulting square involving C ′ , X ′ , Y ′′ and D ′′ := C ′ × X ′ Y ′′ is a n abstra ct blow-up by the same result, and hence by [2 7, Lemma 2.4] the blow-up cd structur e is complete. F or the reg ula rit y pr operty , we need to show that (12.6) is onto. Ev ery o b ject admits a cov ering in this to polog y b y aﬃne, ca nc e lla tiv e monoids, and it suﬃces to prov e sur jectivit y of the map given b y the underlying presheav es ev alua ted at such an aﬃne cancellative U . Tha t is, say f : U → Y , g : U → Y are given with p ◦ f = p ◦ g . W e need to prov e either f = g or they bo th factor throug h D a nd coincide a s maps to C . Let u be the unique gener ic po in t o f U . If either f ( u ) or g ( u ) lands in Y \ D , then they b oth must land ther e. Since Y \ D ∼ = X \ C , it follows that f and g coincide generically . B ut since U is ca ncellativ e, it follows f = g on all of U . (T o see this, one may w ork lo cally: If h, l : A → B are tw o maps of monoids with B cancellative a nd the comp ositions of h, l with the inclusion B ֒ → B + coincide, then h = l .) Otherwise, w e hav e that the gener ic p oint , a nd hence every p oint, of U is mapped by b oth f and g to po in ts in the closed s ubset D of Y . Again using that U is ca ncellativ e, it fo llo ws that f , g factor through D ֒ → Y . (This is also pr o ven by working lo cally .) Finally , the co mpositions of thes e ma ps f , g : U → D with D → C coincide since C → X is a c lo sed immersion.  W e deﬁne the st andar d density st ructur e on M p ctf as follows: The set D i ( X ) consists of those o pen immersio ns U ⊂ X s uc h that ev ery p oint in X \ U has height at least i . It is clear that this satisﬁes the axioms r e quired of a density structure of ﬁnite dimension in [27, 2.20]. A cd structur e is said to b e b ounde d for a given density structure if a n y distin- guished sq uare has a reﬁnement which is reducing for the density s tr ucture in the sense of [27, 2 .21]. Theorem 12. 8. The blow-up and Zariski cd st ructur es on M p ctf ar e b oth b ounde d for the standar d density stru ctur e. Pr o of. T o see that the blow-up cd structure is b ounded, we need to show that any abstract blow-up squar e (12.1) in M p ctf has a reﬁnement that is reducing for D ∗ . Consider the squar e obtained by repla cing Y b y the monoid scheme-theoretic ima g e of Y \ D (in the sense of Deﬁnition 9.4), and D by the pullbac k. This is also an abstract blow-up squa re, a nd it reﬁnes (12.1). This r eﬁnemen t has the features that p − 1 ( X \ C ) is dens e in Y , Y maps birationally onto the scheme-theoretic image of X \ C in X , and D do es no t co n tain any g eneric points of Y . T o s how that this squa r e is reducing, we assume given C 0 ∈ D i ( C ) , Y 0 ∈ D i ( Y ) and D 0 ∈ D i − 1 ( D ). Deﬁne X ′ to b e the op en subscheme X \ Z of X , where Z ⊂ X is the eq uiv ariant closur e (in the sense of 2.9) of the unio n of the images of each of C \ C 0 , D \ D 0 and Y \ Y 0 in X . W e need to show tha t X ′ belo ngs to D i ( X ) TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 39 and that the pullback o f the o riginal sq uare (12 .1) alo ng X ′ ֒ → X gives an abstract blow-up sq ua re. If y ∈ Y is a p oin t of height a t lea st i , then p ( y ) has heig ht at least i in the scheme-theoretic image of X \ C , by Lemma 10.5. Hence p ( y ) has height at least i in X itse lf (since a clo s ed immersio n is an injection o n underlying p osets). If d ∈ D has height at least i − 1, then its height in Y is at least i (since D contains no generic po ints of Y ) and hence its image in X has height a t least i to o. Since C is an equiv ariant clos e d subscheme, if c ∈ C has heig h t a t le ast i , it has height at least i in X . Recall that Z ⊂ X is the equiv ariant closure of the union o f the imag es of ea c h of C \ C 0 , D \ D 0 and Y \ Y 0 in X . Each of these imag e s cons is ts of p oints of heig h t at least i and hence every p oin t in Z has heig ht at lea st i in X b y Re ma rk 2.9.1. Therefore X ′ belo ngs to D i ( X ) and the pullback o f the ab o ve square along X ′ ֒ → X gives an abs tr act blow-up squar e that prov es our original sq uare is reducing. The argument in the previous paragr aphs applies mutatis mutandis to show that every Zar iski sq uare is reducing .  Corollary 12. 9. L et S b e a pr eshe af of ab elian gr oups on M p ctf ; let t b e either the Zariski or the cdh -top olo gy, and write a t S for t he she aﬁﬁc ation with r esp e ct to t . If X ∈ M p ctf is of dimension d , t hen H n t ( X, a t S ) = 0 for n > d. Pr o of. Immediate from Theor em 1 2.8 and [27, Thm. 2.26].  The categ ory of sp ectra we use in this pap er will not b e critical. In or der to min- imize technical issues, we will use the terminology that a s p e ctrum E is a sequence E n of simplicia l sets tog ether with b onding maps b n : E n → Ω E n +1 . W e say that E is an Ω -sp e ctrum if all b onding maps are weak equiv alences. A map of sp ectra is a strict map. W e will use the mo del structure on the categ ory of sp ectra deﬁned in [3]. Note that in this mo del structure, every ﬁbrant s pectrum is an Ω-sp ectrum. Giv en a Gro thendiec k top ology , the ca tegory of co n trav ariant functors F from M p ctf to sp ectra ( pr eshe aves o f sp ectra) has a closed mo del structure, in which a mo r phism φ : F → F ′ is a coﬁbra tion when F ( X ) → F ′ ( X ) is a coﬁbra tion for every monoid scheme X in M p ctf ; φ is a weak equiv ale nc e if it induces isomo rphisms betw een the sheav es o f stable homotopy gro ups (see [20], [21]). W e write H cdh ( − , F ) for the ﬁbrant replacement of F using this mo del structure for the cdh top ology , as in [7]. A pr e sheaf of sp ectra F on M p ctf satisﬁes the Mayer-Vietoris pr op erty for so me family C of c a rtesian squa res if F ( ∅ ) = ∗ and the application of F to each member of the family gives a homo top y cartesia n squar e of sp ectra. Prop osition 12. 10. L et F b e a pr eshe af of sp e ctr a on M p ctf . Then t he c anonic al map F ( X ) → H c dh ( X, F ) is a we ak e quivalenc e of sp e ctr a for al l X if and only if it has the Mayer-Vietoris pr op erty for every abstr act blow-up squar e and every Zariski squar e of pctf monoid schemes. Pr o of. By Theo rems 1 2.7 and 12.8, the cdh cd -structure is complete, regular and bo unded. Now the asser tion follows from [7, Theor em 3.4].  Given Pr opos ition 12.1 0 , the deﬁnition of cdh -descent given in [7, 3.5] b ecomes: 40 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL Deﬁnition 12.11 . Let F b e a pr e s heaf of sp ectra on M p ctf . W e say that F satisﬁes cdh desc ent if the canonica l map F ( X ) → H cdh ( X, F ) is a weak equiv alence of sp ectra for all X . R emark 12.11 .1 . W riting H zar ( − , F ) for the ﬁbra n t replacement with resp ect to the mo del structure for the Zaris ki top ology , we obtain the no tion of Zaris k i de s cen t. The pro of of Pr opos ition 12.10 applies to show that F satisﬁes Za riski de s cen t if and only if it has the May e r-Vietoris pr oper ty fo r every Za riski squa r e. It follows that cdh -descent implies Zaris ki descent. It is us e ful to r estrict to the full sub c ategory S o f smo oth monoid schemes (see Deﬁnition 6.4). By Pr o positio n 6.5, these ar e the cancellative, torsionfree, separated monoid schemes o f ﬁnite type who se k -re alizations are smo oth for any co mm utative ring k . (This condition is indep enden t of k , by 6.5.) Deﬁnition 12.12. W e deﬁne the smo oth blow-up cd stru ctur e on S to consis t of squares (1 2 .1) such that X is smo oth, e is the inclus ion of an equiv a riant , s mooth closed subscheme and Y is the blow-up of X a lo ng C . (Thes e assumptions ensure, by (7.7), that Y and D are also s mooth.) The Zariski cd s tr ucture is given by all cartesian square s in S asso ciated to a cov ering o f X b y tw o op en subschemes. W e deﬁne the scdh top ology on S to b e the Grothendieck topo logy asso ciated to the union of the smo oth blow-up cd -structure and the Za riski cd -s tructure on S . F or a preshea f of s p ectra on S , we deﬁne H scdh ( − , F ) just as H cdh was deﬁned ab o ve. W e s a y such a presheaf F satisﬁes scdh descent if the ca no nical ﬁbra n t replacement map F ( X ) → H scdh ( X, F ) is a weak equiv alence for all X ∈ S . Prop osition 12.13. The smo oth blow-up cd -structu r e and the Zariski cd structu r e on S ar e r e gular, b ounde d, and c omplete. Conse quently, a pr eshe af of sp e ctra deﬁne d on S satisﬁes scdh desc ent if and only if it has the Mayer-Vietoris pr op erty for e ach smo oth blow-up squar e and e ach Zariski squar e in S . Pr o of. That the smo o th blow-up cd - s tructure is co mplete can b e proved exac tly as V o evodsky did for smo oth k -schemes in [28, Lemma 4.3], replacing resolution of singula r ities by our Theore m 1 1.4. Regularity is proved exactly as in Theorem 12.7 for the non-smo oth case. The pro of that the smo oth blow-up cd -str ucture is bo unded works exactly as in The o rem 12.8, keeping in mind that open subschemes of smo oth monoid schemes are smo oth. The pro of that the Za riski cd -structure is complete, r egular and b ounded is a gain the s ame as in the non-smo oth category . It follows that the scdh top olog y is gener ated by a complete, regular, b ounded cd -structure and so [7, Theorem 3 .4] applies to prov e the second as sertion.  Prop osition 12.14. F or any X ∈ S and any pr eshe af of sp e ctr a F deﬁne d on M p ctf , we have a we ak e quivalenc e H c dh ( X, F ) ∼ − → H sc dh ( X, F | S ) . Pr o of. In this proo f we write F cdh for the restriction of the presheaf H cdh ( − , F ) to S . By Pr opo sition 12 .10, F cdh satisﬁes the May er-Vietor is pro perty for smo oth blow- up and Zar iski sq uares. Therefor e F cdh satisﬁes scdh descent (Deﬁnition 12 .12). TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 41 By Theorems 1 1.1 a nd 1 1.4, ev ery cov ering sieve for the cdh top ology on M p ctf has a reﬁnement co n taining a sieve gener ated b y a cov er consisting of ob jects of S . It follows that F | S → F cdh is an scdh -lo cal weak e q uiv alence. Ther e- fore H scdh ( − , F | S ) → H scdh ( − , F cdh ) is an ob jectwise w eak equiv alence (see [7, page 561 ]). T ogether, the tw o ob jectwise weak equiv alences exhibited in the pro of give the a s sertion.  13. Weak c dh k descent Throughout this section, we ﬁx a commutativ e ring k . Deﬁnition 13.1. Let X k be a scheme of ﬁnite type over k and assume Z k ⊂ X k is a closed subscheme. W e say Z k is r e gularly emb e dde d in X k if the sheaf of ideals deﬁning Z k is lo cally generated by a regular sequence — that is, if for all x ∈ Z k , the kernel I x of O X k ,x → O Z k ,x is gener ated by a O X k ,x -regular sequence of elements. Deﬁnition 13. 2. A presheaf o f s pectra F deﬁned on M p ctf has we ak cdh k desc ent if F ha s the May er-Vietor is prop erty for each ca rtesian square D / /   Y p   C e / / X in M p ctf satisfying one of the following conditions: (1) It is member o f the Zaris ki cd structure. (2) It is a ﬁnite abstra c t blow-up — i.e., it is a member o f the abstra ct blow up cd structure having the additional pro perty that p is a ﬁnite morphism. (3) C is an equiv ariant clos ed subscheme, Y → X is the blow-up of X along C , and C k is a regula rly embedded closed subscheme of X k . R emark 13.2 .1 . Theorems 13.3 and 14.3 b elo w s uggest (but do not pr o ve) that the deﬁnition of weak cdh k descent is actually indep enden t of the choice of k . Since a smo oth blow-up square is an ex ample of a blow-up alo ng a regularly embedded subscheme, Prop ositions 12.13 and 12 .14 imply the following theorem. Theorem 13. 3. If F is a pr eshe af of sp e ctr a on M p ctf that satisﬁes we ak c dh k desc ent, t hen F satisﬁes scdh desc en t. That is, t he c anonic al map F ( X ) → H c dh ( X, F ) is a we ak e quivalenc e for every smo oth m onoid scheme X . The main go al of this pape r, realized in the next se c tion, is to esta blish a pa rtial generaliza tion of Theorem 13 .3 to all schemes in M p ctf . The goal of the r est of this s e ction is to establish some tec hnical prop erties needed in the next. W e ﬁrst int ro duce a slightly stronger notion than that of weak cdh k descent. Recall from [EGA IV, 6.10.1] that given a closed subscheme C k of a k -scheme X k , de ﬁned by a n idea l sheaf I , X k is s aid to b e normal ly ﬂat along C k if the restriction of ea c h I n / I n +1 to C k is ﬂat. R emark 13.3 .1 . Her e is a mono id-theoretic co ndition o n a s heaf I of ideals on a monoid s c heme ( X , A ) which guar a n tees that, for all k , the k -realization of X is normally ﬂat along the k -r e alization of the equiv ariant closed submono id C deﬁned 42 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL by I : at each po int x of C , under the natur al action of the monoid A x /I x on each of the p ointed se ts L n = I n x /I n +1 x , each L n is a b ouquet of copies of A x /I x . W e do not know if this condition is necessa ry . W e will say that a ca rtesian square of schemes in M p ctf , D / /   Y p   C e / / X , is a nic e blow-up squar e if C is an equiv aria n t closed subscheme of X , Y is the blow-up of X a long C and there exists a car tesian squa re in M p ctf of the fo r m (13.4) C e / /   X   B / / Z such that Z is cancellative, X → Z is the nor malization of Z and B is an equiv ariant closed smo oth subscheme o f Z such that Z k is nor mally ﬂat along B k . Deﬁnition 13.5. A pres heaf of spec tr a o n M p ctf satisﬁes we ak+nic e cdh k desc ent provided it s atisﬁes weak cdh k descent and, in addition, it has the Mayer-Vietoris prop erty for all nice blow-up s quares in M p ctf . Prop osition 13 .6. If F is a pr eshe af of sp e ctr a on M p ctf that satisﬁes cdh desc ent, then F satisﬁes we ak+nic e cdh k desc ent for any c ommu tativ ring k . Pr o of. This is immedia te fro m Pr o positio n 1 2.10, since ea c h of the sq ua res ap- pea ring in the deﬁnition o f weak+nice cdh k descent is a member of the cdh cd structure.  W e will need the following technical r esult ab out lo cal domains. Reca ll that if I is an ideal in a commutativ e ring R then a n ideal J ⊆ I is called a r e duction of I if J I n − 1 = I n for so me n > 0; a minimal r e duction of I is a reduction which contains no other reductio n o f I . Lemma 13. 7. L et R b e a no etherian lo c al domain with inﬁ nite r esidue ﬁeld k , let p b e a prime ide al, and assume R is n ormal ly ﬂat along R / p . Le t J b e a minimal r e duction of p that is gener ate d by h := ht ( p ) = ht ( J ) elements. (Given R and p with these pr op erties, such a J ex ist s by [18, 5.2 , 5.3] .) L et ˜ R b e the normalization of R and assume ˜ R is Cohen-Mac aulay. Then J ˜ R is a r e duct ion of p ˜ R gener ate d by h elements and Sp ec( ˜ R/J ˜ R ) is r e g- ularly emb e dde d in Sp ec( ˜ R ) . Pr o of. W e have that J p n − 1 ˜ R = p n ˜ R , and so the ﬁr s t a ssertion is clear. Since R ֒ → ˜ R is an integral extension of do mains, w e hav e h = ht ( J ) = h t ( J ˜ R ). F or any maxima l ideal ˜ m of ˜ R , we hav e that J ˜ R ˜ m is a heig h t h ideal gener a ted by h elements in the lo cal r ing ˜ R ˜ m . Since ˜ R ˜ m is Cohen-Maca ulay by assumption, these generator s necessar ily form a regular sequenc e .  The following is the evident analo gue of the notion of weak cdh k descent for presheav es of sp ectra on the ca teg ory of k -schemes. TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 43 Deﬁnition 13.8. F or a comm utative ring k , let S ch/k b e the categor y of separated schemes essentially o f ﬁnite type over k . A pres heaf of sp ectra deﬁned on S ch/k satisﬁes we ak cdh desc ent if it has the May er-Vietoris pr oper t y for each ca rtesian square D / /   Y p   C e / / X of schemes s a tisfying one of the following conditions: (1) e and p are op en immersions whose ima ges cov er X . (2) It is a ﬁnite a bstract blow-up — i.e., e is a closed immersion, p is ﬁnite, and p maps Y \ D isomorphica lly onto X \ C (3) e is a r egular closed immersion and p is the blow-up of X a long C . Lemma 13.9. Assume k is a c ommutative r e gular n o etherian domain c ontaining an inﬁnite ﬁeld and G k is a pr eshe af of sp e ctr a on S ch/k that satisﬁes we ak cdh desc ent. L et G b e t he pr eshe af of sp e ct ra on M p ctf deﬁne d by G ( X ) := G k ( X k ) . Then G satisﬁes we ak+nic e cdh k desc ent on M p ctf . Pr o of. Since the k -realizatio ns of the squar es in volv ed in the deﬁnition of w eak cdh k descent for M p ctf (Deﬁnition 1 3.2) are s quares inv olved in the deﬁnition of weak cdh des cen t for S ch/ k (Deﬁnition 13 .8), it follows tha t G satisﬁes weak cdh k descent. Say X , Y , C, D , Z , and B are as in the deﬁnition o f a nice blow-up s q uare. W e need to prove that the square (13.10) G k ( X k ) / /   G k ( C k )   G k ( Y k ) / / G k ( D k ) is homotopy cartesian. Let R b e any lo cal r ing of Z k and let p b e the pr ime ideal of R cutting o ut B k lo cally . Let V = Sp ec( ˜ R ˜ m ) wher e ˜ R is the nor malization of R and ˜ m is any of the maximal ideals of ˜ R . Then, since X k is the norma lization of Z k by Prop osition 6 .1, V is the sp ectrum of a lo cal ring of X k , a nd for v arious choices of R a nd ˜ m , every lo cal ring of X k arises in this manner. By Cor ollary 5.4, C k = X k × Z k B k , so the closed subscheme V × X k C k of V is cut o ut by q = p ˜ R ˜ m . As X is the nor malization o f the sepa rated cancellative, torsionfree monoid scheme Z , P rop osition 6.1 implies that X k is a to ric v ariety . By [19], all tor ic s c hemes ov er k ar e Cohen-Macaulay; hence so a r e X k and V . By Lemma 13.7, q = p ˜ R ˜ m admits a reduction I ⊂ q such that Sp ec( ˜ R ˜ m /I ) ֒ → V is a regula r em b edding. Since V × X k Y k is the blow-up o f V k along V × X k C k (b y Prop osition 7.7), and the exc e ptio nal divisor is V × X k D k (b y 5.4), the pro of of [18, 5.6] (with K H replaced by G ) gives that G k ( V ) / /   G k ( V × X k C k )   G k ( V × X k Y k ) / / G k ( V × X k D k ) 44 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL is homo top y car tesian. Since G k satisﬁes the May er -Vietoris prop erty for Zaris ki cov ers a nd the V occur ring here is an ar bitrary lo cal scheme of X k , the pro of of [18, 5.7] (with K H r eplaced by G k ) shows that (13.10) is homotopy car tesian.  Example 13.11 . Let K H denote W eib el’s homotopy algebra ic K -theor y [2 9]. W e may view K H as a presheaf of sp ectra on S ch/k . By abuse o f notation, we also write K H fo r the presheaf of sp ectra on M p ctf deﬁned by K H ( X ) = K H ( X k ). By [26], [29, 4.9] a nd [25], K H satisﬁes weak cdh desc en t on S ch/k (1 3.8) ; by Lemma 13.9, K H satisﬁes weak+nice cdh k -descent o n M p ctf . 14. Main Theorem In this section, w e prove our ma in theorem (Theo r em 14.3), which gives a co n- dition for F to satisfy cdh descent on M p ctf . W e will nee d the Biers tone-Milman Theorem, which we extract from the embedded version [1, Thm. 1.1]. Theorem 14.1. L et X b e a sep ar ate d c anc el lative torsionfr e e monoid scheme of ﬁnite typ e, emb e dde d as a close d subscheme (se e Deﬁ n ition 2.5) in a smo oth toric monoid scheme M (se e D eﬁnition 4.1). F or any c ommut ative ring k c ontaining a ﬁeld, ther e is a se quen c e of blow-ups along smo oth e quivariant c enters Z i ⊂ X i , 0 ≤ i ≤ n − 1 , Y = X n → · · · → X 0 = X such t hat Y is smo oth, and e ach ( X i ) k is normal ly ﬂat along ( Z i ) k . Pr o of. Since nor mal ﬂatness is sta ble under ﬂat extensio n of the ba se, and k is ﬂat ov er a ﬁeld, we may assume that k is a ﬁeld. Let ¯ k denote the a lgebraic clos ure of k , a nd let T b e the tor us acting on M ¯ k . The Bier stone-Milman Theo rem ([1, Thm. 1.1 ]) tells us that we can ﬁnd a s equence of blow-ups M n → · · · → M 0 = M ¯ k of smo oth toric ¯ k -v arieties , the blow-up of M i being taken alo ng a s mooth T -inv ariant center N i , with the following prop erties. Setting X ′ 0 = X ¯ k , we inductively deﬁne Z ′ i = N i ∩ X ′ i ; then Z ′ i is a smo oth equiv ariant k -v ariety , X ′ i is nor mally ﬂat a long Z ′ i , and X ′ i +1 is the strict transfor m of X ′ i . The ¯ k -realization functor from fans to (nor mal) tor ic ¯ k -v arieties (and eq uiv ariant morphisms) is well known to b e an equiv alence. It follows that each of the N i and M i and the mo rphisms b etw een them come from fans, a nd hence by Theor em 4.4 are ¯ k -realizations of toric monoid schemes (which by abuse of notation, w e will call N i and M i ), and mo rphisms of such. Inductively we deﬁne monoid schemes X i and Z i , starting from X 0 = X and Z 0 = N 0 ∩ X , to b e the blow-up of the monoid scheme X i − 1 along Z i − 1 in the sense of 7.4. By Prop osition 7.7 and Coro llary 5.4, Z ′ i = ( Z i ) ¯ k and X ′ i = ( X i ) ¯ k . In particular, ( X n ) ¯ k = Y is a smo oth toric v ar iet y and therefor e the monoid scheme X n is smo oth b y Prop osition 6.5. Finally , faithfully ﬂa t descent implies that ( X i ) k is normally ﬂa t a long ( Z i ) k if and only if ( X i ) ¯ k is nor mally ﬂat along ( Z i ) ¯ k .  Theorem 14. 2. Supp ose G is a pr eshe af of sp e ctr a on M p ctf satisfying we ak+nic e cdh k desc ent for some c ommut ative ring k c ont aining a ﬁ eld. If G ( X ) ≃ ∗ for al l X in S then G ( X ) ≃ ∗ for al l X in M p ctf . Pr o of. W e pro ceed b y induction o n the dimension of X . Given X , let x 1 , . . . , x l be its gener ic p oint s, and let Y i = { x i } eq be their equiv ariant closur es (see Le mma 2.9). W e hav e a cover X = Y 1 ∪ · · · ∪ Y l by equiv ar ian t clos ed subschemes each o f TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 45 which is cancellative by Exa mple 10.3. Moreov er, each Y i × X Y j is equiv ariant and closed, hence p ctf. Since G has the May er-Vietor is pro p erty for closed covers, and G v a nishes on the Y i × X Y j for all i 6 = j by the induction hypothes is , we get G ( X ) = Y i G ( Y i ) . W e may thus assume that X is cancellative. (This also es ta blishes the base cas e dim( X ) = 0 , s ince in that case the Y i are in S .) Since G s atisﬁes May er-Vietor is for op en covers, we may assume X is a ﬃne. In particular, we may as sume X can b e embedded in a smo oth toric mo noid scheme, for exa mple, by choosing a surjection from a free abe lia n monoid ont o A whe r e X = MSp ec( A ). This will allow us to apply the Bierstone-Milma n Theo rem 1 4 .1 to obtain a sequence of blow-ups along smo oth monoid schemes Z i , Y = X n → · · · → X 0 = X . W e claim that G ( X i ) ≃ G ( X i +1 ) for all i . Since G ( Y ) ≃ ∗ , this will ﬁnish the inductive s tep and hence the pro of of the theo rem. T o simplify the notation, ﬁx i and write Z for Z i ⊂ X i and X Z for X i +1 , the blow-up of X i along Z , s o that our goal is to pr o ve that G ( X i ) → G ( X Z ) is a weak equiv a le nc e . Let ˜ X denote the normalizatio n ( X i ) nor of X i and set ˜ Z = Z × X i ˜ X . W r ite ˜ X ˜ Z for the blow-up of ˜ X along ˜ Z . By natura lit y of blow-ups (see 7.4), there is a commutativ e square ˜ X ˜ Z / /   X Z   ˜ X / / X i (that need not b e cartesia n). Since the map ˜ X → X i is ﬁnite, the map ˜ X ˜ Z → X Z is also ﬁnite, by Lemma 7.6 . Applying G gives a commutativ e square of sp ectra G ( ˜ X ˜ Z ) G ( X Z ) o o G ( ˜ X ) O O G ( X i ) . o o O O T o prove that the r igh t-hand vertical arr o w is a weak equiv alence, it suﬃces to prov e the other three ar e. The ﬁnite map ˜ X → X i is an iso morphism on the generic p oints. Consider the eq uiv ariant closure E ⊂ X i of the ﬁnitely man y height 1 p oin ts of X i ; by Remark 2 .9.1, every p oin t in E has height ≥ 1 in X i , so E is the complement of the generic p oint o f X i . Since E is p ctf, G ( E ) ≃ ∗ b y our inductive ass umption. Since the pullback ˜ E := E × X i ˜ X is an eq uiv ariant closed subscheme of ˜ X , it is pctf by P rop osition 9.1, and hence G ( ˜ E ) ≃ ∗ as w ell, by induction. Using the ﬁnite abstract blow-up s quare in volving X i , ˜ X , E a nd ˜ E , we have a weak equiv alence G ( X i ) ≃ − →G ( ˜ X ) . The map ˜ X ˜ Z → X Z is also ﬁnite and bira tional, a nd so the same a rgument shows G ( X Z ) ≃ − →G ( ˜ X ˜ Z ) . 46 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL is a weak equiv alence. Finally , obser v e that ˜ Z × ˜ X ˜ X ˜ Z / /   ˜ X ˜ Z   ˜ Z / / ˜ X is a nice blow-up squa re, b ecause the b ottom row may be co mpared with Z → X i and ( X i ) k is normally ﬂa t along Z k . Because G ha s descen t for nice blow-up squares, and G ( ˜ Z ) ≃ G ( ˜ Z × ˜ X ˜ X ˜ Z ) ≃ ∗ by the induction h y pothesis, we get a w eak equiv alence G ( ˜ X ) ≃ − →G ( ˜ X ˜ Z ) . It follows that G ( X i ) ≃ G ( X Z ), as cla imed. This completes the pr o of.  W e now state and prov e the main theorem of this pap er, which gives a pa rtial generaliza tion of Theor em 13.3 to all ob jects in the catego ry M p ctf . Theorem 14.3. L et F k b e a pr eshe af of sp e ct r a on S ch/k for some c ommutative r e gular no etherian r ing k c ontaining an inﬁn ite ﬁ eld, and deﬁne F to b e the pr eshe af of sp e ctr a on M p ctf deﬁne d by F ( X ) = F k ( X k ) . If F k satisﬁes we ak cdh desc ent on S ch/k , then F satisﬁes cdh desc ent on M p ctf . Pr o of. Let G be the homo top y ﬁb er of F → H cdh ( − , F ) — i.e., for all X in M p ctf , G ( X ) is the homotopy ﬁb er of F ( X ) → H cdh ( X, F ). By Lemma 13.9 a nd Prop o- sition 13.6, b oth F and H cdh ( − , F ) sa tisfy weak+nice cdh k descent, and hence G satisﬁes weak+nice cdh k descent to o. Theorem 13.3 gives that G ( X ) ≃ ∗ for all X ∈ S . Now we a pply Theor e m 14.2 to co nc lude G ( X ) ≃ ∗ for all X in M p ctf .  The following corollary is the Theorem announced in the intro duction. Corollary 14.4. Assum e k is a c ommut ative r e gular no etherian ring c ontaining an inﬁnite ﬁeld and let F k b e a pr eshe af of sp e ctr a on S ch/ k t hat satisﬁes the Mayer- Vietoris pr op erty for Zariski c overs, ﬁ nite abstr act blow-up squar es, and blow-ups along r e gularly emb e dde d su bschemes. Then F k satisﬁes the Mayer-Vietoris pr op erty for al l abstr act blow-up squar es of toric k -schemes obtaine d fr om sub dividing a fan. Pr o of. By Deﬁnition 13.8, F k satisﬁes weak cdh descent on S ch/ k . By Theorem 14.3, F satisﬁes cdh descent in M p ctf . Now use Pr opos ition 12 .10.  Corollary 14.5. L et k b e a c ommutative r e gular no etherian ring c ontaining a ﬁeld. The pr eshe af of sp e ctr a K H on M p ctf , deﬁne d as K H ( X ) = K H ( X k ) , satisﬁes cdh desc ent. Mor e over, b oth natur al m aps K H ( X ) → H c dh ( X, K H ) ← H c dh ( X, K ) ar e we ak e quivalenc es for al l X in M p ctf . Pr o of. W e ﬁr s t reduce to the case when k is of ﬁnite t yp e ov er a ﬁeld. W e can express k as a ﬁlter ed colimit of rings k i , all regular o f ﬁnite type over a ﬁeld (b y Popes c u’s theore m [24, 2.5]). The functor K H is the homotopy colimit of the co rresp onding functors deﬁned by k i -realiza tio n. By Pro positio n 12.10, we can chec k descent b y sho wing that certain squares of monoid schemes ar e tra nsformed b y K H into homotopy co- cartesian squares of sp ectra (a squa re o f sp e ctra is ho mo top y cartesian if and only if it is ho motop y co -cartesian); since ho motop y co limits o f TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 47 homotopy co- c artesian sq ua res are homotopy co-c artesian, we may assume tha t k is of ﬁnite t yp e ov er its ﬁeld of constants. Now if the r e gular ring k do es not co n tain an inﬁnite ﬁeld, it is smo oth over the (per fect) ﬁeld o f constants it co n tains and hence stays regular under base change from its ﬁeld of co nstan ts to any algebr aic extension. W e can therefore apply the standard transfer a r gumen t and may assume that k c o n tains an inﬁnite ﬁeld. By Example 1 3.11 and Theor em 1 4.3, K H sa tisﬁes cdh descent on M p ctf . F or any X in M p ctf , consider the commutativ e square of sp ectra: K ( X ) / /   K H ( X )   H cdh ( X, K ) / / H cdh ( X, K H ) , where K is algebraic K -theory , rega rded as a pre sheaf of sp ectra on S ch/k and hence on M p ctf . Since K H satisﬁes cdh des cen t, the rig h t-hand vertical map is a weak equiv alence for all X . This is the ﬁrst a ssertion of the cor ollary . If X is s mo oth, then the top ho rizontal map is a w eak equiv alence b y [29] (since X k is smo oth ov er k hence r egular b y 6.5). By ﬁbra n t replacement and P ropo sition 12.14, the b ottom map is also a weak equiv ale nc e for all X in S . By induction on dim( X ) and Theorem 11.1, this implies that H cdh ( − , K ) → H cdh ( − , K H ) is a lo cal weak equiv alence and, as observed (for a ny site) in [7, p. 561], this implies that H cdh ( X, K ) → H cdh ( X, K H ) is a weak equiv alence for a ll X in M p ctf .  R emark 14 .6 . It follows from Corolla ries 12 .9 and 14 .5 and a cdh -descent arg umen t that if X ∈ M p ctf is of dimension d and k is a commutativ e regula r ring containing a ﬁeld, then K H n ( X k ) = 0 for n < − d (cf. [18, Thm. 8.1 9]). The a nalogous statement for K -theory is a lso tr ue, at least if X is cancellative and tor sion-free. Indeed for aﬃne X , K n ( X k ) = 0 for n < 0, b y [15, Thm. 1 .3]; the general case follows from this by a Zariski descent a rgument , using 12 .9. In orde r to apply Co rollary 14.5 to the relation b et ween K -theory and topolo gical cyclic homolog y , we need to recall so me terms. Fix a prime p and a commutativ e regular ring k of characteristic p . T o each scheme X essentially o f ﬁnite type ov er k , there is a pr o-sp e ctrum { T C ν ( X, p ) } ∞ ν =0 and the cycloto mic trace is a compatible family of morphisms tr ν : K ( X ) → T C ν ( X, p ). Deﬁne F ν k to be the presheaf of sp ectra on S ch /k given as the homotopy ﬁb er of K ( X ) → T C ν ( X, p ). Then Geisser and Hesselholt observe in the pro of of [11, Thm. B] that each F ν k takes ele mentary Nisnevich squares and regula r blow-up squares to homoto py cartesia n square s of pro-sp ectra. F ollowing Geisser-Hess elholt [11], a strict map of pro-sp ectra { X ν } → { Y ν } is said to b e a we ak e quivalenc e if fo r every q the induced map { π q ( X ν ) } → { π q ( Y ν ) } is an isomorphis m of pro- a belian gro ups . A square dia gram of stric t maps of pro- sp ectra is said to b e homotopy c artesian if the canonica l map from the upp er left pro-sp ectrum to the level-wise homotopy limit of the other terms is a weak eq uiv- alence. Given a clas s C of squar e s w e will say that a pro-pr esheaf of sp ectra satis ﬁe s the pro- a nalogue o f C -descent if it sends each square in C to a ho mo top y cartesia n square of pro-s pectra . 48 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL Deﬁne {F ν } to b e the pro -presheaf of sp ectra on M p ctf given a s the family o f homotopy ﬁb ers o f the maps K ( − ) → T C ν ( − , p ). That is, F ν ( X ) = F ν k ( X k ) is the homotopy ﬁb er of K ( X k ) → T C ν ( X k , p ) for each X a nd ν . Prop osition 14.7. Assu me k is a c ommutative r e gular no etherian ring c ontaining an inﬁnite ﬁeld of char acteristic p > 0 . Then { F ν } satisﬁes cdh desc ent on M p ctf in t he sense that {F ν } → { H ( − , F ν ) } is a we ak e quivalenc e of pr o-sp e ctr a. Pr o of. Fix ν and let G ν be the homotopy ﬁb er of F ν → H cdh ( − , F ν ). It suﬃces to prov e that for each X and q the pro-ab elian g roup { π q G ν ( X ) } is pro-zer o. W e will do so by modifying the pro of of Theo r em 14.3. F or ea c h ν , H cdh ( − , F ν ) satisﬁes w eak+ nice cdh k descent by Prop osition 13.6. By [12, Thm. 1] and [13, Thms. B, D], { F ν k } sends ﬁnite abstract blow-up s q uares to homotopy cartes ian squares o f pro- spectra . Thus { F ν k } satisﬁes the pro-ana logue of weak cdh descent (Deﬁnition 13.8). In the pro of o f Le mma 13.9, the reduction ideals used a re reduction idea ls on aﬃne ne ig h b orho ods of the max imal ideal m of R . By the arg umen t used in the pro of of [11, Thm. 1.1 ], the pro of o f our Lemma 13.9 now applies mutatis mu t andis to show that the pr o-presheaf of sp ectra F ν satisﬁes the Mayer-Vietoris prop ert y for nice blow-up squares . It now follows tha t {G ν } satisﬁes the pro-a na logue of weak+nice cdh k descent. F or each ν , F ν satisﬁes Zariski descent and also has the May er-Vietor is pro perty for regula r blow-ups, so F ν satisﬁes scdh descent by 12.13. B y deﬁnition, this means that for each smo oth X the sp ectrum G ν ( X ) is cont rac tible. Now the pro of of Theorem 14 .2 applies verb atim to ﬁnish the pro of.  Corollary 14.8. Assu me k is any c ommutative r e gular no etherian ring of char- acteristic p > 0 . F or any monoid scheme X in M p ctf , the fol lowing squar e of pr o-sp e ctr a is homotopy c artesian. K ( X ) / /   K H ( X )   { T C ν ( X, p ) } / / { H c dh ( X, T C ν ( − , p )) } . Pr o of. By a standard transfer argument as in Coro llary 14 .5, we may assume that k contains an inﬁnite ﬁeld. By Pr opo s ition 14.7, the homoto p y ﬁb er {F ν ( X ) } of the left vertical ma p is weakly equiv a len t to { H cdh ( X, F ν ) } . By Corolla ry 14 .5 , this coincides up to weak equiv alence with the homotopy ﬁb er of the right vertical map.  R emark 1 4 .9 . As explained in Remar k 14.6, if k is a n y commut ative regular ring containing a ﬁeld, a nd X ∈ M p ctf is cancellative a nd tor s ion-free then K n ( X k ) = 0 for n < − dim X . T o extend this result to a ll X ∈ M p ctf it would s uﬃce to prov e that the b ottom horizontal map in the diagr am in Corolla ry 14.8 induces an isomorphism (r esp. an epimo r phism) of homotopy groups in deg r ees n < − dim( X ) (resp. n = − dim( X )). Geisser and Hesselholt pr o ved the analog ue s tatemen t for schemes essentially of ﬁnite type o ver a ﬁeld of p ositive characteristic which admits resolution of singularities ([11, Thm. C]). Adapting their metho ds to our situatio n seems rather hard. TORIC V ARIETIES , MONOID SCHEMES AND cdh DESCENT 49 Ac kno wledgements. The authors would like to thank the re feree for a c areful reading, for sugg e sting the notion of a monoid p oset and for the cur ren t pro of o f Lemma 5.5. References [1] E. Bi erstone and P . Milman, Desi ngularization of toric and binomial v arieties J. A lg. Ge om. 15 (2006), 443–486. [2] Benjamin A. Blander, Lo cal pr o jectiv e model structures on simplicial preshea ve s, K -The ory 24 (2001), 283–301. [3] A. K. Bousﬁeld a nd E. M. F ri edlander, H omotop y theory of Γ-spaces, spectra, and bi s implicial sets, In Ge ometric applic ations of homotopy t he ory (Pr o c. Conf., Evanston, Il l., 1977), II , v olume 658 of L e ctur e Notes in Math. , pages 80–130. Springer, Berlin, 1978. [4] A. Connes and C. Consani, Schemes ov er F 1 and zeta functions, Comp os. Math. 146 (2010), 1383–141 5. [5] A. Connes and C. Consani, On the notion of geometry o ver F 1 , J. Alg. Ge om. 20 (2011), 525–557. [6] A. Connes, C. Consani and M. Mar colli, F un with F 1 , J. Numb er The ory 129 (2009), 1532– 1561. [7] G. Corti ˜ nas, C. Haesemey er, M . Sc hl i c h ting and C. W eib el, Cyclic homology , cdh -cohomology and negative K -theory , A nn. Math. 167 (2008), 549–573. [8] A. Deitmar, F 1 -sche mes and toric v arieties, Beitr¨ age Algebr a Ge om. 49 (2008), 517–525. [9] D. Eisen bud and J. Har ris, The Ge ometry of Schemes , Grad. T exts in Math. 197, Springer V erlag, 2000. [EGA 0 I ] A. Grothendiec k and J. Dieudonn ´ e, Chapitre 0, ´ Elemen ts de g´ eom´ etrie alg´ ebrique I, Springer, 1971. [EGA IV] A. Grothendiec k, ´ Elemen ts de g´ eom´ etrie alg´ ebrique IV (2 eme Pa rtie), Publ. Math. IHES 24 (1964). [10] W. F ulton, Intr o duction to T oric V arieties , Annals of Math. Studies 131, Pr inceto n Uni v. Press, 1993. [11] T. Geisser and L. H esselholt, On the v anishi ng of negativ e K - groups, Math. Ann. 348 (2010), 707–736. [12] T. Geisser and L. Hesselholt, Bi -relativ e algebraic K -theory and topological cyclic homology , Invent. Math. 166 (2006), 359–395 . [13] T. Geisser and L. Hesselholt, On relative and bi-relative algebraic K -theory of r ings of ﬁnite c haracteristic, J. AMS 24 (2011), 29–49. [14] R. Gil mer, Commutative Semig r oup R ings , The Universit y of Chicago Press, C hi cago , 1984. [15] J. Gubeladze, K -theory of aﬃne toric v arieties, Homolo gy, homotopy and applic ations 1 (1999) 135–145. [16] R. Hartshorne, Algebr aic Ge ometry , Springer V erlag, 1977. [17] A. Hattori, Theory of multi-fans, Osaka J. Math. 40 (2003), 1–68. [18] C. Haesemey er, Descent properties of homotop y K -theory , Duke Math. J. 125 (2004), 589–620. [19] M. Hochster, Ri ngs of in v arian ts of tori, Cohen-Macaulay rings generated b y monomials, and polytop es. A nn. of Math. 96 (1972), 318–337. [20] J. F. Jardine, Sim plicial preshea ves, J. Pur e Appl. A lgebr a , 47 (1987), 35–87. [21] J. F. Jardine, Gene r alize d ´ etale c ohomolo gy the ories , Birkh¨ auser V erlag, Basel, 1997. [22] K. Kato, T oric singulari ties, A mer. J. Math. 116 (1994), 1073–109 9. [23] J. L´ opez Pe˜ na and O. Lorscheid, Mapping F 1 -land: An ov erview of geometries o ver the ﬁeld with one element, prepri n t (2009). Av ailable at ar X iv:0909.0069. pp. 241-265 in Non- c ommutative ge ometry, arithmetic and r elate d topics , 2011. [24] D. Popescu, General N´ eron desingularization and approximation, Nagoya Math. J. 104 (1986), 85–115. [25] R. W. Thomason, Les K -group es d’un sch ´ ema ´ eclat ´ e et une formule d’intersect ion exc´ e- den taire, Invent. M ath. 112 (1993), 195–215. [26] R. W. Thomason and T. T r obaug h, Higher algebraic K -theory of schemes and of derived categories. In The Gr othendie ck F estschrift , V olume III , v olume 88 of Pr o gr ess in Math. , pages 247–436. Birkh¨ auser, Boston, Bas´ el, Berlin, 1990. 50 G. COR TI ˜ NAS, C. HAE SEMEYER, MARK E. W ALKER, AND C. WEIBEL [27] V. V o ev odsky , Homotopy theory of simpli cial sheav es in completely decomp osable top ologies, J. Pur e A ppl. Algebr a 214 (2010), 1384–1398. [28] V. V o ev o dsky , Unstable motivic homotopy categories i n Nisnevich and cdh-top ologies, J. Pur e Appl. Algebr a 214 (2010), 1399–1406. [29] C. W eibel, H omotopy Al gebraic K -theory, Contemp or ary M ath. 83 (1989), 461–488. [30] C. W eibel, The negative K -theory of norm al surf aces, Duke M ath. J. 108 (2001), 1–35. [31] O. Zariski and P . Sam uel, Commutative algebr a. V ol. II. Repri nt of the 1960 edition. Graduate T exts in Mathematics, V ol. 29. Spr inger-V erlag, New Y ork-Heidelb erg, 1975. Dept. Ma tem ´ atica-Inst. San t al ´ o, FCE yN, Universidad de Buenos Aires, Ciudad Uni- versit aria, (1428) Buenos Aires, Argentina E-mail addr e ss : gcorti@ dm.uba.ar Dept. of Ma them a tics, University of California, Los Angeles CA 90095, USA E-mail addr e ss : chh@mat h.ucla.edu Dept. of Ma them a tics, University of Nebraska - Lincoln, Lincoln, NE 68588, USA E-mail addr e ss : mwalker 5@math.unl.e du Dept. of Ma them a tics, Rutgers University, New Brunswick, NJ 0 8901, USA E-mail addr e ss : weibel@ math.rutgers .edu

Toric varieties, monoid schemes and $cdh$ descent

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment