Sites, Sheaves, and the Nisnevich topology

Transcription

1 Stes, Sheaves, and the Nsnevch toology Bran Wllams Pretalbot Bundles and schemes The noton of a sheaf on a toologcal sace X s a famlar one. Gven a vector bundle E X one can consder, for each oen X, the vector sace of sectons over denoted Γ E ( ). We can always restrct functons, so ths gves us a functor Γ E ( ) : (X ) o Vect. In defnng ths functor we haven t used any of the localty that goes nto defnng a vector bundle, however. So there s stll some juce left over. Ths functor enjoys a very nce roerty wth resect to nclusons and ntersectons of oen sets. Ths can be stated roughly as follows. If I have two sectons defned on searate oen sets that agree on a smaller oen set, one should be able to fnd a sngle secton that restrcts to each on the orgnal oens. Consder a resheaf on X,.e. a functor F : (X ) o Set. We say F s a sheaf on X f for all oens X and covers { } of the followng holds: f s F( ) and s j F( j ) satsfy s j = s j j then there exsts a secton s F( ) such that s = s and s j = s j. Moreover, we demand ths hold for all ars, j. Ths s exactly the condton exressed above, so that the local secton functor of a vector bundle s a sheaf on X. Remark. When one says sheaf n the classcal sense they usually have the target category of sets n mnd. A vector bundle gves a vector sace valued sheaf on X, but we ddn t utlze that extra structure above. I.e. glung condtons take lace n the doman category. When sheaves have more structure, say values n abelan grous, we can talk about sheaf cohomology. For a fxed sace X the functor Γ X : Shv(X ) Ab s left exact. Snce the category of sheaves of abelan grous has enough njectves we can consder the rght derved functors. The sheaf cohomology of X wth coeffcents n F s defned to be H (X,F) := R Γ X (F). sual homologcal algebra yoga says that short exact sequences of sheaves get sent to long exact sequences n cohomology. In fact, when X s nce enough (say, when one can choose a toologcal bass consstng of contractble saces) then H (X,Z) H (X ;Z). On the left-hand sde Z denotes the constant sheaf Z. The rght-hand sde s ordnary cohomology. We have seen that we can thnk about schemes over k as functors Y : Rng k Set. A natural queston to ask s: do all functors of ths form deserved to be called schemes? The answer s no, and t turns out that schemes should be thought of as sheaves n an arorate sense. Let s consder the followng. Suose A s a k-algebra such that A = Aa for some (a ) n A. Ths should be thought of as a cover of the k-algebra A by closed subalgebras. Now, f s Y (A[a 1 ]) and s j Y (A[a 1 ]) agree n Y (A[a 1 a 1 ]) then there should be a j j unque s X (A) that mas to s,s j. 1

2 Remark. I m lyng a lttle bt here. Geometrcally, t s necessary to know that schemes can t get too bg. To avod ths, one also demands that there exsts a cover va affnes for a cosheaf on Rng k. Ths motvates the followng. Gven a category C, we defne resheaves on C to be functors F : C o Set. Then a sheaf should be a resheaf that obeys glung wth resect to some dstngushed class of oen covers n C. We ll do ths rgorously now. 2 Toologes In kndergarten we defned a toology on a set X to be a famly of subsets called oen sets that are closed under arbtrary unons and fnte ntersectons. We re gong to do somethng smlar for a category C. The man dea here s that one should thnk of coverngs as a famly of morhsms n C. A toology on C s an assgment Cov( ) where Cov( ) s a famly of collectons of morhsms {{ } } satsfyng the followng axoms: () If V s an somorhsm then {V } Cov( ). () If { } Cov( ) and V s any arrow, then V exst for all and moreover the rojectons { V V } Cov(V ). () If { } Cov( ) and for each, {V j } Cov( ) then {V j }, j Cov( ). A category together wth a toology s called a ste. A toology allows us to talk about glung. A resheaf F : C o Set on C s sad to be a sheaf (wth resect to a rescrbed toology) ff for all coverngs { } the natural sequence s an equalzer. F( ) F( ) F( j ) Remark. When one s concerned wth homotoy theory t wll make sense to take resheaves wth values n saces or smlcal sets. In ths settng the above defnton doesn t ft. One needs to relace the above dagram wth one that extends to the rght, and demand that F( ) be a homotoy lmt for such a dagram. Johan and Ben wll talk about ths later on, so t won t really concern us for now. Examle 2.1. Consder the category (X ) of oens on X. We choose a toology on n whch { } s a coverng ff =. Then sheaves n ths toology concde wth our usual defnton of a sheaf on a sace. Examle 2.2. Identfy Rng o k, j = Aff k gven by the functor Sec( ). The Zarsk toology s gven as follows. A famly of morhsms {Sec A Sec A} s a coverng ff for each there s a a A such that A = A[a 1 ] and the ma Sec A Sec A s the functor Sec aled to the natural homomorhsm of k-algebras A A[a 1 ]. Moreover, A = Aa. Ths defnes a toology on Rng o. Sheaves wth resect to ths toology whch addtonaly satsfy the condton that they can be covered by k affnes are recsely schemes. Examle 2.3. Just as easly we can defne a toology on the category of schemes over k, denoted Sch k. Namely a famly of morhsms of schemes over k, { } s a coverng ff each ma s an oen mmerson and the mages of the s combne to cover (.e. jontly surjectve). Excercze: Check that ths concdes wth the defnton for affne schemes,.e. commutatve k-algebras. It turns out that we ddn t lose much n consderng just affnes n ths case. That s, there s an equvalence of categores Shv Zar (Rng o k ) Shv Zar (Sch k ). 2

3 Snce we re good mathematcans we should also consder relatve versons of the above examles. I ll leave t to the reader to formulate ths recsely. There s also a noton of a small ste assocated to a fxed scheme S. For the Zarsk toology ths s the category of all schemes over S equed wth an oen mmerson X S. We gve ths category the Zarsk toology as above, and denote the ste by S Zar. A resheaf (n the classcal algebra geometrc sense) n the Zarsk toology on a fxed scheme S s a functor F : S Zar Set. A sheaf n the Zarsk toology s one that satsfes Zarsk descent. Whle we ve seen the Zarsk toology s amazngly useful for geometry, t sometmes doesn t suffce for algebrac toology. For nstance, n toology fbre bundles are characterzed by beng locally trval. In geometry, algebrac morhsms we d lke to call fbre bundles may not be Zarsk locally trval. Ths has unwanted mlcatons ractcally n the comutaton of sheaf cohomology. For nstance, a theorem of Grothendeck says that f X s an rreducble scheme then H (X,F) = 0 for all constant sheaves and all > 0. To fx some of these ssues above many toologsts use the étale toology on Sch k. In ths toology a famly { } s called a coverng ff each ma s étale and of locally fnte resentaton. We now ntroduce the man toology we wll be concerned wth. Consder a morhsm of schemes : X. For each x X and u 1 {x} there s a ma of resdue felds k(x) k(u) whch should be thought of as a feld extenson. We say s comletely decomosed at x ff there exsts u over X such that the above ma s an somorhsm. Note that gven such an somorhsm there s an nduced ma whch gves the dotted arrow n the dagram O,u k(u) k(x) Sec k(x) X. As usual, ths should be thought of as some tye of local equvalence. We are now ready to defne Nsnevch coverngs. We say a famly { X } s a Nsnevch coverng of X ff t s a fnte set of mas, each X s étale of fnte tye, and for each x X there exsts such that X s comletely decomosed at x. Exercse: For a fxed scheme S check that the Nsnevch coverngs form a toology on Sch /S. Moreover, one can show that so long as C Sch /S s a full subcategory such that for all étale mas of fnte tye : X and ull-back squares V Y X. comuted n Sch /S have the roerty that V C; then C also nherts the Nsnevch toology. Note that every Zarsk coverng s a Nsnevch coverng, and every Nsnevch coverng s an étale coverng. In artcular the Nsnevch toology s subcanoncal, meanng reresentable resheaves are sheaves. 3 Nsnevch Excson Let s take a ste back and thnk about excson and the Mayer-Vetors roerty. Suose we have oen sets,v nsde some toologcal sace. In nce enough stuatons the square V V V s a (homotoy) ush-out. Then homology, thought of as a functor h : Saces Ch has the roerty that the natural ma h ( ) h ( V ) h (V ) h ( V ) 3

4 s a quas-somorhsm. In fact, homology sends arbtrary ush-out squares to ush-out squares. It s actually more convenent to consder functors that send ush-out squares to ull-back squares. Such functors are called excsve. One of the most aealng roertes of the Nsnevch toology s that sheaves are comletely characterzed by a sort of Mayer-Vetors, or excson, roerty. To state ths roerly we need some defntons. Consder mas : X V : n Sch /S, and form the ull-back X V V X. (1) Such a square s called an elementary dstngushed square f s an oen mmerson, s étale, and the nduced ma 1 (X ( )) X ( ) s an somorhsm of schemes over S. Examle 3.1. Certanly Zarsk squares are elementary. An examle of one that s not of ths form s the followng. Consder : A 1 {x} A 1. Suose that x s nonzero and has a square n k. Then the squarng ma A 1 {0, x} A 1 s étale and the condton 1 ({x}) {x} s satsfed. Thus s elementary. PB A 1 {0, x} A 1 {x} A 1. Prooston 3.1. Suose the square (1) above s a elementary dstngushed square. Then the nduced dagram n Shv(C Ns ) s a ush-out. Proof. Let F be a sheaf wth mas u : F and v : V F such that u X V = v X V. We must show there s a unque extenson X F that agrees wth u, v when restrcted to,v resectvely. Yoneda says that sheaf mas X F concde wth F(X ). The ar { X, V X } form a Nsnevch coverng, so that F(X ) F( ) F(V ) F( X ) F( X V ) F(V X ) F(V X V ) s an equalzer dagram. Here (u, v) = ( 1 u, 1 u, 1 v, 1 v) and 1 (u, v) = ( 2 u, 2 v, 2 u, 2 v) where we abuse 2 notaton and denote by the arorate rojecton. By naturalty t suffces to check that (u, v) = 1 (u, v). 2 As u, v agree on X V they agree on V X. So t suffces to show u 1 = u 2 and v 1 = v 2. As u s a monomorhsm, the frst equalty s aarent, so we focus on the latter. Note frst that 1, 2 : V X V V are the obvous rojectons. Consder the two mas : V V X V X V X V : π where the frst s the dagonal and the second s the rojecton onto the second factor. The dagonal s an oen mmerson. Moreover the second ma fts nto the ull-back dagram X V X V π V X V so that π s obtaned from an oen mmerson va base change, hence t s an oen mmerson. Thus {,π} form a Nsnevch cover for V X V. By the sheaf condton t thus suffces to show that v 1 = v 2 and v 1 π = v 2 π. The frst equalty s clear. The second one follows from a smle dagram chase. There s a cool corollary of ths result. If C s as above then we can consder a Nsnevch sheaf of abelan grous on C. Namely a functor F : C o Ab. Now, for a scheme X C we can consder the free abelan sheaf Z[X ] on hom C (, X ). That s, Z[X ] sends to the free abelan grou on the set hom C (, X ). If F s another Nsnevch sheaf recall that H (X,F) s defned as the rght derved functor of the global secton functor. By adjontness then we have X Ext (Z[X ],F) = H (X,F) 4

5 for all sheaves F. Now consder the sequence Z[ X V ] Z[ ] Z[V ] Z[X ] nduced by the obvous mas. From the above rooston the sequence s exact. Thus we get a long exact sequence n cohomology H (X,F) H (,F) H (V,F) H ( X V,F) H +1 (X,F). We wll need the followng techncal lemma. Lemma 3.2. Let : X be an étale morhsm of Noetheran schemes that s comletely decomosed at every generc ont of X. Then has a ratonal secton. Proof. Here s a sketch. Wthout loss of generalty suose X s rreducble. Let x X be the unque generc ont. Snce s Noetheran s of fnte tye. Then ratonal sectons concde wth ars u a ont of lyng above x and s a secton of the nduced ma of stalks O X,x O,u. Snce s étale we get a secton O,u O h. As X s X,x Noetheran we have O X,x s an Artnan local rng, and so s comlete. Thus O X,x O h an so. Postcomoston X,x wth of ths secton wth the nverse of ths so rovdes the desred local secton. Prooston 3.3. Let : X be Nsnevch wth, X Noetheran. Then there exsts a sequence of closed subsets such that for each the ma Z n+1 Z n Z 1 Z 0 = X has a secton. 1 (Z Z +1 ) Z Z +1 Proof. By the above lemma, there s a dense oen subset X X such that 1 (X ) X has a secton. Set Z 1 = X X. As 1 (Z 1 ) = Z 1 X we see that 1 (Z 1 ) Z 1 s Nsnevch. Proceed ad nfnum, to get a sequence Z 1 Z 2 of closed subsets wth the above desred roerty. Ths sequence must stablze snce X s Noetheran. We arrve at the man result of ths secton. Prooston 3.4. A resheaf F on a full subcategory C Sch /S consstng of Noetheran schemes s a sheaf n the Nsnevch toology ff the followng two condtons hold: (1) F() = { }. (2) For every dstngushed square (*) the nduced dagram s a ull-back square. F(X ) F( ) F(V ) F( X V ). Proof. The above lemmas mly one drecton, we rove the other. Suse F s a resheaf and the above two condtons are satsfed. Frst we reduce to the case where the Nsnevch coverng conssts of a sngle ma. Indeed, suose { X } s a Nsnevch coverng. Then

6 s an elementary dstngushed square. Thus by (1),(2) above the ma F( 1 2 ) F( 1 ) F( 2 ) s an somorhsm. By nducton we see F( ) F( ) s an somorhsm. Thus t suffces to show the sheaf condton for the sngle ma X. Henceforth relace by. We wsh to rove that the sequence F(X ) F( ) s an equalzer sequence. By Prooston we have an nteger n and a fltraton Z n+1 Z n Z 1 Z 0 = X F( X ) (2) of closed subsets such that the restrcton of to each dfference Z Z +1 has a secton. We roceed by nducton on n. If n = 1 then the sequence (2) s slt-exact, so we are done. For general n set X := X Z n and consder the ull-back dagram X X q X X. Then, we see that q s Nsnevch and by assumton there exsts closed subsets Z of X for 0 n, such that = Z n Z n 1 Z 1 Z 0 = X and such that q : q 1 (Z Z +1 ) Z Z +1 has a secton. Namely, take Z := X Z. By nducton the sequence F(X ) F(X X ) F((X X ) X (X X )) F(X X X ) (3) s exact. Choose a secton s of 1 (Z n ) Z n. We need the followng Fact: s s an oen mmerson. Gven ths, set V := s(z n ) 1 (X ) whch s oen n. The nduced ull-back square X X V X V X V hence F(X ) F(X ) F(V ) F(X X V ) (4) s a ull-back square. Fnally, we wll deduce the exactness of (2). Note that the followng natural square commutes: F(X ) F(X ) F(V ) F( ) F(X X ) F(V ). From (3) we see that the rght vertcal arrow s mono. From (4) we see the the to horzontal arrow s mono. Hence the left vertcal arrow s mono. Note that ths s the frst ma n (2). Now, suose u F( ) has the roerty that 1 (u) = 2 (u) n F( X ). nder the bottom ma n the above square u (w, v) for some w F(X X ) and v F(V ). We see that w mas to the same element under the two rght-most mas n the sequence (3). Thus by exactness of that sequence there s x F(X ) mang to w. Now x and v have the same mages n F(X X V ) so that by (4) the ar (x, v) comes from an element x F(X ). By commutatvty of the square above we see that x u. Moreover, ths element s unque by the above remarks. 6

7 Remark. The above result holds n more generalty, whch we brefly mentoned earler. Consder resheaves valued n saces F : C o Saces. Then one can stll ask f such a functor s a sheaf n the Nsnevch toology. We already noted what ths means, and the only queston s how the above condtons should change. Condton (1) should be modfed to F() and condton (2) should be modfed to requre that the square s a homotoy ull-back square. Remark. One can use ths theorem to deduce varous descent roertes n the Nsnevch toology. Ben wll talk about ths for algebrac K-theory. 4 Motvc Thom saces Suose E s a quas-coherent sheaf on a scheme X over S. To t we can assocate the free commutatve O X -algebra Sym(E) (.e. a sheaf of O X -algebras). Then take the assocated scheme over X : V(E) := Sec Sym(E) X. It s routne to check that V(E) s a vector bundle over X. Moreover, the ma above s a A 1 -homotoy equvalence over X. It s clear that V defnes a functor QCoh X Sch /X. Lettng X vary we get a contravarant functor Mod Sch /S where Mod /S s the category of ars (X,E) where X s a scheme over S and E s a quas-coherent sheaf on X. In fact, f Imm denotes the category wth objects closed mmersons : Z X and mas the obvous ull-back dagrams, then we get a contravarant functor V : Mod e /S Imm. Ths follows snse an emorhsm E E of sheaves nduces a closed mmerson V(E ) V(E). Gven a closed mmerson : Z X we can consder the assocated oen mmerson X (Z) X. Let X /(X (Z)) denote the Nsnevch sheaffcaton of the onted resheaf hom /S (, X )/hom /S (, Y ). Ths defnes a functor Q : Imm Shv Ns (C). We call the comoste Th : Mod /S V Imm Q Shv Ns (C). the motvc Thom sace functor. In homotoy theory, one of the key results that allows one to go from geometry to algebra s the tubular neghborhood lemma. For nstance the constructon of wrong way, or Gysn mas. It says that gven a ncely embedded submanfold M M one can fnd a negborhood N ε homeomorhc to a subsace of ts normal bundle ν M. More exlctly t rovdes a homeomorhsm of ars (D(ν M ), S(ν M )) = (N ε, N ε ). The left-hand sde s the defnton of the Thom sace Th(ν M ). A key facet of the Nsnevch toology s that we have a verson of ths tubular neghborhood lemma. Theorem 4.1. Let : Z X be a closed mmerson and let ν Z be the conormal sheaf. Moreover assume S s Noetheran and that X, Z are smooth, fnte tye over t. Then there s an somorhsm n the onted homotoy category hoshv Ns (C). Th(ν Z ) = X /(X (Z)) There s no obvous ma from the left-hand sde to the rght-hand sde. The somorhsm s nduced by a sequence of zgzags of A 1 -equvalences. 7