ALGEBRAIC K-THEORY OF SCHEMES
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1 ALGEBRAIC K-THEORY OF SCHEMES BEN KNUDSEN 1. Introduction and Definitions These are expository notes on algebraic K-theory, written for Northwestern s 2014 Pre-Talbot Seminar. The primary reference is [?]. Definition 1.1. (1) A full subcategory C of an abelian category A is called exact if it is closed under extensions in A. (2) If C is exact, Q(C) is the category with the same objects as C, and in which an arrow from M to M is an isomorphism (in C) of M with a subquotient of M. (3) The K-theory space of C is The K-groups of C are K(C) := ΩBQ(C). K i (C) := π i (K(C)) = π i+1 (BQ(C)). One should check that Q is functorial for exact functors between exact categories, so that K is as well. Remark 1.2. (1) Our definition of an exact category is ugly in the same way that defining a manifold as a subset of Euclidean space is ugly, and we have adopted it for the sake of expedience. See [?] for the coordinate-free definition and a discussion of why the two are the same. (2) K-theory has blessed mathematicians with a host of routes to its construction. The one adopted here is the least general construction that is still functorial. The reader interested in a machine that accepts more general input and produces more homotopical output might consult [?]. Example 1.3. (1) Any abelian category is exact. (2) If R is a ring, the category of finitely-generated projective R-modules is exact. We denote by K(R) the K-theory of this exact category. Since projective modules are flat, R K(R) is a covariant functor on the category of rings. (3) If R is Noetherian, the category of all finitely-generated R-modules is exact, since it is abelian, and its K-theory is denoted G(R). The assignment R G(R) is functorial for flat homomorphisms. (4) Globalizing the previous examples, the category of vector bundles on a scheme X is exact, and its K-theory is denoted K(X), while the category of coherent O X -modules on X is abelian, if X is Noetherian, and its K-theory is denoted by G(X). The former is a contravariant functor on schemes, while the latter is only functorial for flat maps. Exercise 1.4. Show that the exact categories of (2) and (3) are not abelian. Exercise 1.5. It is far from obvious that the definition of K 0 (R) given in (2) coincides with the usual definition. Convince yourself that it does by drawing pictures of cells in the relevant classifying space. In the author s experience, this is the best way to make one s peace with the Q-construction. 1
2 2 BEN KNUDSEN Example 1.6. K-theory is often very difficult to compute; for example, K n (Z) is not known for all i. For an early and exceptional success story, see [?], in which it is shown (using a different but equivalent definition of K-theory) that { Z/(q i 1)Z, n = 2i 1 K n (F q ) = 0, n = 2i. 2. Comparison Theorems and Applications This section is concerned with the fundamental abstract theorems of K-theory. We do not aim for completeness, but we include some proofs and sketches of proofs in order, we hope, to give the reader some of the flavor of the subject. We also include applications of the theorems, the overarching theme of which is the reduction of big computations to smaller ones. The basic tool that makes everything go is the famous Theorem 2.1 (Quillen s Theorem A). Let F : D 0 D 1 be a functor. Then BF is a homotopy equivalence if and only if B(D 0 d) for all d D 1. For a proof, see [?]. Theorem 2.2 (Resolution). Let C be an exact category and D C a full subcategory closed under extensions and kernels of epimorphisms in C. Suppose that, for every object M C, there is an exact sequence 0 P n P 0 M 0 with P i D. Then the inclusion of D induces a homotopy equivalence K(D) and in particular an isomorphism on K. K(C) Proof. C is the union of the categories C n of objects with resolutions of length less than or equal to n, so that it suffices to prove that each inclusion C n C n+1 induces a homotopy equivalence. Since this inclusion satisfies the hypotheses of the theorem, and since each object of C n+1 has a length one resolution by objects of C n, it suffices to prove the theorem under the additional assumption that all objects of C admit D-resolutions of length one. Denote by Q(D) the full subcategory of Q(C) spanned by Q(D), and consider the inclusion of Q(D) into this full subcategory. For N Q(D), the category Q(D) N consists of all pairs (N 0, N 1 ) such that N i N and N 1 /N 0 D, and there is a morphism from (N 0, N 1 ) to (N 0, N 1) if and only if N 1 N 1 and N 0 N 0. It follows that both N 0 and N 1 are in D, so (N 0, N 1 ) (0, N 1 ) defines an endofunctor π on Q(D), and there is an obvious natural transformation id π. Since there is also a natural transformation 0 π, and since natural transformations induce homotopies at the level of classifying spaces, we conclude that B(Q(D) N). Thus, by Theorem A, the inclusion Q(D) Q(D) induces a homotopy equivalence on classifying spaces. Next, fix M C, and consider M Q(D), whose objects are triples of an object P D, an admissible subobject P P, and a surjection P M. This category is non-empty, since M is a quotient of an object of D by assumption. Now, within this category is the full subcategory E of objects for which P = P. Since D is closed under admissible subobjects, P always lies in D, and sending (P, P M) to (P, P M) defines a right adjoint to the inclusion of E into the comma category. Thus it suffices to
3 ALGEBRAIC K-THEORY OF SCHEMES 3 show that BE is contractible. For this, fix an object (P 0, P 0 M), and note that the arrows (P, P M) (P 0 M P, P M P 0 M) (P 0, P 0 M) are natural in P. Before applying the theorem, we recall the following result about the relationship between coherent sheaves and vector bundles on good schemes. Proposition 2.3 ([?], 6, II, 2.2). Let X be a regular, Noetherian, separated scheme. Then every coherent O X -module admits a finite resolution by vector bundles. Corollary 2.4. Let X be a regular, Noetherian, separated scheme. Then the inclusion of vector bundles on X into coherent O X -modules induces a homotopy equivalence K(X) G(X). Thus, provided we restrict our attention to this nice subcategory of schemes, it makes no difference whether we define K-theory using vector bundles or coherent sheaves. This is good news for us, for, while the former category is the category of geometric interest, the latter category, and therefore its K-theory, is much better behaved. In particular, the category of coherent sheaves on X is abelian, so that we may apply the theorems of devissage and localization discussed below in our study of K (X). Having dealt with resolutions, we consider the dual notion, filtrations. Theorem 2.5 (Devissage). Let A be an abelian category and B a full subcategory closed under subobjects, quotients, and finite products. Suppose that, for every object M A, there is a filtration 0 = M 0 M n = M with M i /M i 1 B. Then the inclusion of B induces a homotopy equivalence K(B) and in particular an isomorphism on K-theory. K(A) Proof. For M A, consider the category Q(B) M. Notice that M 1 B, so that Q(B) M 1 has a terminal object. It therefore suffices to show that each inclusion Q(B) M n Q(B) M n+1 induces a homotopy equivalence on classifying spaces. We factor this functor through a third category J, an object of which consists of N 1 M n and N 0 M n+1 such that N 0 /N 1 B. Then the inclusion of Q(B) M n into J has a left adjoint (N 0, N 1 ) (N 0 M n, N 1 ), and, similarly, the inclusion of J into Q(B) M n+1 has a right adjoint. Devissage allows us to make the following kind of reduction: Corollary 2.6. Let A be an abelian category in which every object has a finite filtration with simple quotients. Then K (A) = α K (End(M α ) op ), where {M α } is a set of representatives for the isomorphism classes of simple objects of A.
4 4 BEN KNUDSEN Definition 2.7. Let C 0 and C 1 be exact categories. A sequence F F F of exact functors from C 0 to C 1 is said to be exact if 0 F (M) F (M) F (M) 0 is an exact sequence in C 1 for every M C 0. Note that, if C is an exact category, the direct sum in C endows K(C) with the structure of an H-space (in fact, it is an infinite loop space, but we won t need this fact). Theorem 2.8 (Additivity). Let C 0 and C 1 be exact categories and F F F an exact sequence of exact functors from C 0 to C 1. Then K(F ) K(F )+K(F ) as H-space maps. Sketch proof. Let E(C) denote the category whose objects are the exact sequences of C and whose arrows are commuting diagrams. Then E(C) is an exact category, and the source, target, and quotient functors S, T, and Q form an sequence of exact functors from E(C) to C. Clearly, to give an exact sequence of functors from C 0 to C 1 is the same as to give an exact functor from C 0 to E(C 1 ). Thus it suffices to prove the theorem under the assumption C 0 = C, C 1 = E(C), F = S, F = T, and F = Q. Since the diagram C C E(C) T S Q obviously commutes, we are reduced to showing that K( ) is a homotopy equivalence. This is accomplished using Theorem A in the spirit of the proofs above. In more conceptual treatments of algebraic K-theory, the additivity theorem is of fundamental importance, but we will be content with the following humble but useful application. Corollary 2.9. Let X be a scheme. Then, for every i, K i (X) is a K 0 (X)-module. Proof. If E is vector bundle, then E ( ) is an exact functor on vector bundles, and additivity implies that C (E ( )) = (E ( )) + (E ( )) for every exact sequence 0 E E E 0. Before stating the next theorem, we recall the notion of a Serre subcategory of an abelian category. Definition Let A be an abelian category. A subcategory B A is a Serre subcategory if B is closed under subobjects, quotients, and extensions. If B A is a Serre subcategory, then there is a quotient category A/B, formed by inverting those morphisms in A whose kernel and cokernel both lie in B (see [?]). The quotient category is an abelian category. Example Let R be a Noetherian ring, s R any element, A the abelian category of finitely generated R-modules, and B A the full subcategory of modules M such that s n M = 0 for some n. Then B is a Serre subcategory, and A/B is equivalent to the abelian category of finitely generated s 1 R-modules. Example Globalizing the previous example, let X be a Noetherian scheme, U X an open subscheme, A the abelian category of coherent O X -modules, and B A the full subcategory of modules M such that M U = 0. Then B is a Serre subcategory, and A/B is equivalent to the abelian category of coherent O U -modules.
5 ALGEBRAIC K-THEORY OF SCHEMES 5 Theorem 2.13 (Localization). Let A be an abelian category and B A a Serre subcategory. Then the diagram K(B) K(A) K(A/B) is homotopy cartesian. In particular, there is a long exact sequence K 1 (A/B) K 0 (B) K 0 (A) K 0 (A/B) 0. The key result for this theorem is the following companion to Theorem A: Theorem 2.14 (Theorem B). Let F : D 0 D 1 be a functor such that, for every arrow f : d d in D 1, the induced map B(d D 0 ) B(d D 0 ) is a homotopy equivalence. Then for any d D 1, the diagram is homotopy cartesian. See [?] for a proof. B(d D 0 ) B(D 0 ) B(d D 1 ) B(D 1 ) Sketch of proof of localization. Let 0 denote a fixed zero object of A, i : B A the inclusion, and π : A A/B the canonical projection. By definition, π sends every object of B to an object isomorphic to π(0), so the functor Q(i) factors through π(0) Q(A). The theorem follows by Theorem B once we are assured that B(Q(B)) B(π(0) Q(A)) is a homotopy equivalence and that Q(π) induces the appropriate homotopy equivalences on undercategories. Proving these assertions takes some time and will be omitted. As an example of the types of algebraic applications that are possible using these theorems, we prove the following pleasant Corollary There is a long exact sequence K 1 (Q) p K 0(F p ) K 0 (Z) K 0 (Q) 0. Proof. In the localization theorem, take A to be the category of finitely generated abelian groups and B the Serre subcategory of torsion groups. Then A/B is equivalent to the category of finite-dimensional Q-vector spaces, so that K (A/B) = K (Q), and the resolution theorem applied to the inclusion of finitely-generated projective (i.e. free) Z-modules implies that K (A) = K (Z). Finitely-generated torsion modules have finite length, so K (B) = p K (F p ) by devissage. Remark The same result holds mutatis mutandis for any Dedekind domain, with the same proof. We close this section with a global application of localization, which will be extremely important in what follows. Corollary Let Z be a closed subscheme of a Noetherian scheme X, and U its complement. There is a long exact sequence G 1 (U) G 0 (Z) G 0 (X) G 0 (U) 0.
6 6 BEN KNUDSEN Proof. We apply localization with A the category of coherent O X -modules and B the Serre subcategory of coherent sheaves supported in Z. Then A/B is equivalent to the category of coherent O U -modules, so we obtain a fiber sequence G(X, U) G(X) G(U), and it remains to identify G(Z) with G(X, U). Let M be a coherent O X -module with support in Z, and let I denote the sheaf of ideals of Z. The filtration I n M IM M is finite, since X is Noetherian, and each quotient is annihilated by I, thereby defining a coherent sheaf on Z. Devissage completes the proof. 3. K-Theory of Schemes The form of Corollary?? should remind us of the familiar long exact sequence of a pair in cohomology. Motivated by this resemblance, we wish to view K-theory as a cohomology theory for schemes. The goal of this section is show that this is a reasonable thing to do by exhibiting K-theoretic analogues for the key properties of cohomology. We will close by rephrasing our findings in the high-brow language of (Zariski and Nisnevich) descent Functoriality. Just as in the topological situation, algebraic vector bundles pull back under maps of schemes. To be more precise, a map f : X Y induces an exact functor f in the opposite direction between the respective categories of vector bundles, and so there is an induced map f : K(Y ) K(X). As expected, then, K-theory is a contravariant functor on schemes, and, more generally, on pairs of schemes. Topological cohomology theories also enjoy a covariant functoriality, at least for certain maps and certain spaces. For example, if f : M N is a map between compact, oriented n-manifolds, there is a pushforward f : H (M) = H n (M) H n (N) = H (N), where the isomorphisms come from Poincaré duality. More generally, the cohomology of a non-compact oriented manifold is dual to the relative homology of its one-point compactification (the Borel-Moore homology), and this duality allows one to define a pushforward in the same way, provided f is proper, since X X + is functorial only for proper maps. To get a pushforward on ordinary cohomology, we had to assume regularity hypotheses on both the spaces M and N and the map f, and the same is true for algebraic K-theory. For example, if X and Y are regular and quasi-projective over a field, and if f : X Y is a proper map, then a pushforward f : K(X) K(Y ) exists (these are not the most general hypotheses possible). Since we will not make use of this map, we only sketch the steps in its construction, referring the reader to [?] for details. (1) Since X and Y are regular, we may replace K by G, i.e. replace vector bundles by coherent sheaves. (2) Since f is proper, the higher direct image functors R i f preserve coherence. (3) Since X is quasi-projective, every coherent sheaf on X is a quotient of an f -acyclic sheaf, i.e. one for which R i f = 0 for i > 0. (4) Together with the fact that R i f = 0 for i sufficiently large, the resolution theorem implies that G(X) is equivalent to the K-theory of the exact category of f -acylics. (5) The functor f is exact on this category and so induces a map on K-theory.
7 ALGEBRAIC K-THEORY OF SCHEMES 7 The alert reader will notice a discrepancy in the analogy between topology and algebraic geometry. On the topological side, we had to assume that X and Y were oriented manifolds, and no corresponding hypotheses seems to be present on the algebraic side. This discrepancy is not a sign that our analogy is flawed; rather, it is a clue to a deeper truth. Consider the following question: when does a proper map between manifolds admit a pushforward in topological (complex) K-theory? The appearance of Poincaré duality above was not a red herring, for the existence of such a pushforward turns out to be equivalent to the admittance of K-theory fundamental classes for M and N. It turns out that a sufficient hypothesis for the existence of such a fundamental class is that M and N are both almost complex manifolds. One says that topological K-theory is a complex oriented cohomology theory, meaning a theory in which complex vector bundles admit Thom classes. The corresponding statement on the algebraic side is that algebraic K-theory is an oriented cohomology theory (see [?]). In other words, we did not have to assume X and Y to be oriented, since, in the eyes of algebraic K-theory, they already were Homotopy invariance. Perhaps the most important property of ordinary cohomology is that it cannot distinguish between homotopy equivalent spaces. The corresponding result in the algebraic setting is the following Theorem 3.1 (Homotopy invariance). Let X be a regular, Noetherian, separated scheme and π : E X a flat map whose fibers are affine spaces. Then π : K(X) K(E) is a homotopy equivalence. Example 3.2. Take π to be the projection map of a vector bundle. A particularly relevant case is P = X A 1. The theorem will follow by globalizing the following local calculation: Theorem 3.3 (Fundamental theorem). Let R be a Noetherian ring. Then the inclusion R R[t] induces a homotopy equivalence G(R) G(R[t]). In order to prove this theorem, we will need to know some things about the K-theory of graded modules over graded rings, where graded for us will always mean non-negatively graded. For a graded ring S, denote by G gr (S) the K-theory of the exact category of finitely generated graded S-modules. Notice that this exact category carries an action of N by exact functors, with the generator acting by suspension, and G gr inherits this action. Denote this suspension operator by σ. Lemma 3.4. Let S = A S 1 be a graded Noetherian ring that is flat as an A-module. If A admits a finite S-flat resolution, then tensoring with S induces an N-equivariant equivalence G(A) N G gr (S). In particular, there are Z[σ]-module isomorphisms for all i 0. G i (A) Z[σ] = G gr i (S) Proof. The assertion is a special case of Theorem 6 of [?].
8 8 BEN KNUDSEN Proof of the fundamental theorem. We outline the main points of the argument. For details, see [?]. Introduce a new variable z with deg z = 1 and consider the graded subring R[zt, z] R[t, z]. The exact functor M M/(1 z)m realizes the abelian category of finitely generated R[t]-modules as the quotient of the abelian category of finitely generated graded R[zt, z]-modules by the Serre subcategory of z-torsion modules. By devissage, this latter category has the same K-theory as the subcategory of modules actually annihilated by z, which is equivalent to the category of finitely generated R[zt]-modules. By localization, then, there is a fiber sequence G gr (R[zt]) G gr (R[zt, z]) G(R[t]). Since both R[zt] and R[zt, z] satisfy the hypotheses of the preceding lemma, we obtain an exact sequence G n (R) Z[σ] G n (R) Z[σ] G n (R[t]) for each n. One checks that the left hand map is multiplication by 1 σ, which is injective with cokernel G n (R), completing the proof. Remark 3.5. Let g be a finite-dimensional Lie algebra over a field k and U(g) its universal enveloping algebra. Using the Poincaré-Birkhoff-Witt filtration on U(g), a modification of the same argument shows that K(k) K(U(g)). Lemma 3.6 (Noetherian induction). Let X be a Noetherian topological space, and let P be a property of closed subsets of X such that, for any closed Z X, P holds for Z whenever P holds for all proper closed subsets of Z. Then P holds for X. Proof. It is important to note the condition is vacuously true for Z =, so the empty set has property P. Assume that X does not have property P. To avoid a contradiction, there is a proper closed subset Z 1 X such that Z 1 also does not have property P. Iterating, we obtain a chain Z n Z 1 X of closed subsets of X, all of which fail to have property P. Since X is Noetherian, this chain has a minimal element. In other words, there is a closed subset Z N X that does not have property P and which contains no proper closed subset that does not have property P. This is a contradiction. Lemma 3.7. Let X α be a filtered diagram of schemes whose arrows are all flat and affine. Then the natural map colim G(X α ) G(lim X α ) is an equivalence. The proof of the lemma is that the category of coherent sheaves on X is essentially the limit of the categories of coherent sheaves on the X α. Of course, this statement cannot literally be true, since the relevant categories do not in fact form a commuting diagram, the issue being that certain composites of functors are canonically isomorphic rather than equal. This obstacle is not a serious one, and the reader interested in how to get around it may wish to consult [?]. Proof of homotopy invariance. For a closed subset Z X with complement U = X \ Z, consider the induced map of localization sequences G(Z) G(X) G(U) G(E Z ) G(E) G(E U ).
9 ALGEBRAIC K-THEORY OF SCHEMES 9 By the five lemma, if any two vertical arrows is an equivalence, then so is the third. Let P be the property π : G(Z) G(E Z ) is an equivalence. Since the empty set has property P, we may assume by Noetherian induction that all proper closed subsets Z X also have property P. Moreover, we may assume that X is irreducible, for if X = Z 1 Z 2, then localization for the pair (Z 1, Z 1 Z 2 ) and the five lemma imply that Z 2 \(Z 1 Z 2 ) has property P, and the same considerations applied to the pair (X, Z 1 ) then prove the theorem for X. Finally, we may assume that X is reduced, since the inclusion of any subscheme with nilpotent sheaf of ideals induces an equivalence on G-theory by devissage. Take the colimit over Z of the above diagram. The term appearing on the upper right is now the intersection over all non-empty open U X. First, this intersection is nonempty, since it contains any generic point of X. Second, any point x of this intersection must be a generic point, for if x X, then X \ x was one the of the open sets used to form the intersection, so that x must have been excluded. Since X was assumed irreducible, it has a unique generic point, which is precisely the intersection in question. By the above lemma concerning filtered limits, we obtain the diagram colim Z G(Z) G(X) G(Spec k(x)) = G(k(x)) colim Z G(E Z ) G(E) G(E X Spec k(x)) = G(k(x)[t 1,..., t n ]). Since the righthand arrow is an equivalence by the fundamental theorem, and since K and G coincide, the proof is complete. A similar argument establishes the following algebraic analogue of a familiar result in topological K-theory. Theorem 3.8 (Projective bundle theorem). Let X be a regular, Noetherian, separated scheme, E a vector bundle of rank r +1 over X, and π : P(E) X the associated projective bundle. Then the map {M i } i π (M i ) O( i) induces a homotopy equivalence r i=0 K(X) K(P(E)). Proof. Using localization as in the previous argument, we may assume that X = Spec k, V = k x 0,..., x r, and P(E) = P r k = Proj(S), where S = k[x 0,..., x r ]. As in the affine case, one may produce a sheaf of modules on P r k from any graded S-module, and the graded global sections functor Γ (F ) := i Z Γ(F (i)) does the opposite; however, in the projective case, these functors are not quasi-inverse. Instead, Γ induces an equivalence between sheaves of modules on projective space and the quotient of the category of graded S-modules by the Serre subcategory of bounded modules (see [?], II.5). Since the K-theory of this latter category is equivalent to G gr (k) by devissage, the upshot for us is that there is a fiber sequence G gr (k) i G gr (S) G(P r k ) G(k) N G(k) N G(P r k ),
10 10 BEN KNUDSEN where i is induced by restriction along the augmentation. As before, we are reduced to determining the map h in the exact sequences G n (k) Z[σ] h G n (k) Z[σ] G n (P r k ). Consider the Koszul resolution 0 S( r 1) Λ r+1 V S( 1) V S k 0. Tensoring this exact sequence against a graded k-module M produces a resolution of M by graded S-modules functorially in M, and so we may view this sequence as an exact sequence of exact functors from graded k-modules to graded S-modules. Since the term on the right is just the inclusion, the additivity theorem implies that i is given by multiplication by r+1 ( 1) i [S( i) Λ i V ] K 0 (k). i=0 But notice that the action of σ is given by tensoring with S( 1), so h must be given by multiplication by so that r+1 r+1 ( ) r + 1 ( σ) i [Λ i V ] = ( σ) i = (1 σ) r+1, i i=0 i=0 G (P r k) = Replacing G by K completes the proof. G (k)[σ] (1 σ) r+1. With more technology in play, this isomorphism can be upgraded to an isomorphism of rings, further confirming our intuition Local-to-global properties. Proposition 3.9 (Excision). Let U, V X be open subschemes covering the Noetherian scheme X. Then the obvious inclusions induce a homotopy equivalence G(X, V ) G(U, U V ). Proof. The result is immediate from the stronger claim that the exact categories of coherent O X -modules vanishing on V and coherent O U -modules vanishing on U V are equivalent. Now, consider the respective localization sequences for the pairs (X, U) and (U, U V ). There is a commutative diagram G(X, V ) G(X) G(V ) G(U, U V ) G(U) G(U V ). whose rows are fiber sequences. Since the left hand map is an equivalence, it follows that the righthand square is homotopy Cartesian, from which we derive the
11 ALGEBRAIC K-THEORY OF SCHEMES 11 Proposition 3.10 (Zariski Mayer-Vietoris sequence). Let U, V X be open subschemes covering the Noetherian, separated, regular scheme X. Then the diagram is homotopy Cartesian. K(X) K(V ) K(U) K(U V ) We say that K-theory is a Mayer-Vietoris presheaf. 1 Seemingly innocuous reformulation is extremely important, in light of the following Theorem 3.11 (Brown-Gersten). A simplicial presheaf satisfies Zariski descent if and only if it is a Mayer-Vietoris presheaf. The reader will find a refresher on the homotopy theory of simplicial presheaves in the appendix. The theorem will follow from the following Proposition Let X be a finite-dimensional Noetherian space and F a Mayer- Vietoris presheaf on X such that π F ( ) = 0. If (π F ) + = 0, then π F = 0. Proof. Let Y be an open subset of X and α π q (F (X)). Since X is Noetherian, there is a maximal open U X such that α U = 0. We wish to show that U = X, so assume otherwise. We say that a subset Z X is bad for (Y, U, q, α) if it can tell the difference between U and X, in the sense that Z Y but Z U =, and Z is simply bad if it is bad for some (Y, U, q, α). If there are no bad sets for (Y, U, q, α), then the closure in X of an irreducible component of X \ U must have non-empty intersection with U, which is impossible, so that X \ U must be empty, and we are done. Thus we may assume that there are bad sets, and, since X is finite dimensional, we may take Z to be a maximal bad set. Now, by assumption, all of the stalks of π F are zero. In particular, this holds for the stalks at any point of Z, so we can find an open set V X such that V Z and α V = 0. Since F is a Mayer-Vietoris presheaf, there is an exact sequence π q+1 F (U V ) π q F (U V ) π q F (U) π q F (V ) from which we conclude that α U V = β for some β π q+1 F (U V ). Since β = 0, there is a maximal open subset W U V such that β W = 0. Obviously, Z W =. In fact, we claim that Z is an irreducible component of X \ W. Before proving this, let s see how it implies the proposition. Take V to be the complement in V of the union of all of the irreducible components of X \ W other than Z. The claim is that α U V = 0, which contradicts the maximality of U. To see this, notice that U V W U V, since U V (X \ W ) U Z =. Therefore, by naturality of, since β W = 0, we have α U V = (β U V ) = ((β W ) U V ) = 0. It remains to prove that Z is an irreducible component of X \ W, so assume otherwise. Then Z is properly contained in an irreducible component Z. By the maximality of Z, 1 We are now thinking of K-theory as valued in simplicial sets via the functor of singular chains. In honor of the unimportance of this distinction, our notation will not reflect it.
12 12 BEN KNUDSEN Z is not a bad set, so we must have Z U. Since Z Z and Z V, we also have that Z V. Since Z is irreducible, we conclude that Z U V. But then Z is bad for (U V, W, q + 1, β), which is a contradiction. Proof of the Brown-Gersten theorem. Suppose first that F is a simplicial presheaf satisfying Zariski descent. Without loss of generality, we may assume that F is in fact fibrant. Let ιu denote the representable presheaf of sets associated to U, regarded as a presheaf of simplicial sets. We claim that the square ιx ιv ιu ι(u V ) is homotopy cocartesian. This claim is enough, since mapping into the presheaf F produces the square F (X) F (V ) F (U) F (U V ), which must be homotopy cartesian, since mapping into a fibrant object takes homotopy pushouts to homotopy pullbacks. To see that the aforementioned square is homotopy cocartesian, note first that, since all simplicial presheaves are cofibrant in the local injective model structure, and since the maps ι(u V ) ιu and ι(u V ) ιv are levelwise inclusions of sets, they are cofibrations in the local injective model structure. Therefore, we may compute the relevant homotopy pushout as the ordinary pushout, and we are reduced to showing that the canonical map ιu ι(u V ) ιv ιx is a weak equivalence in the local injective model structure, which amounts to the induced map (ιu ι(u V ) ιv ) + = (ιu) + ι(u V ) + (ιu) + (ιx) + being an isomorphism of sheaves, since these presheaves are all discrete (the second pushout is taken in sheaves, and we used that sheafification is a left adjoint). The universal property of the pushout for (ιx) + now follows from the definition of a sheaf. Conversely, suppose that F is a Mayer-Vietoris presheaf and define a simplicial presheaf G by G(U) = hofiber(f (U) H(U; F )), where H is a fibrant replacement functor for the local injective model structure. Since F is Mayer-Vietoris by assumption and H( ; F ) is Mayer-Vietoris by the previous argument, G is also Mayer-Vietoris. Since the map F H( ; F ) is a local weak equivalence, the stalks of π G all vanish, so that π G = 0 by the previous proposition. The claim now follows by considering the long exact sequence of the fibration G(U) F (U) H(U; F ). Corollary If X is Noetherian, separated, and regular, then K-theory satisfies Zariski descent on X.
13 ALGEBRAIC K-THEORY OF SCHEMES Nisnevich descent. Exchanging Zariski for Nisnevich, the preceding results go through after a brief change in language. Definition The pullback square U Y Y U is upper distinguished if f is étale, U is open in X, and f Y \UY is an isomorphism onto X \ U. A presheaf is a Mayer-Vietoris presheaf for the Nisnevich topology provided it sends upper distinguished squares to homotopy pullback squares. The corresponding result here is Theorem A presheaf satisfies Nisnevich descent if and only if it is a Mayer- Vietoris presheaf for the Nisnevich topology. For a proof, see [?], for example. Proposition If X is Noetherian, separated, and regular, then K-theory is a Mayer- Vietoris presheaf for the Nisnevich topology. Proof. The proof follows from the diagram X f G(X \ U) G(X, U) G(X) G(U) G(Y \ U Y ) G(Y, U Y ) G(Y ) G(U Y ). Corollary If X is Noetherian, separated, and regular, then K-theory satisfies Nisnevich descent on X. Appendix A. Homotopy Theory of Presheaves Let (C, τ) be a site and Psh(C) := Fun(C op, sset) the category of simplicial presheaves on C. Given a simplicial presheaf F and an integer n 1, we obtain a sheaf of groups (resp. n = 0, sets) (π n F ) +, where ( ) + denotes τ-sheafification. In the following, F, G Psh(C) and ϕ : F G is a map of presheaves. Definition A.1. The map ϕ is (1) a global weak equivalence if ϕ c : F (c) G(c) is a weak equivalence of simplicial sets for all objects c of C; (2) a local weak equivalence if (π n ϕ) + : (π n F ) + (π n G) + is an isomorphism of sheaves for all n; (3) a cofibration if ϕ c : F (c) G(c) is a cofibration of simplicial sets for all objects c of C; (4) a fibration if ϕ has the right lifting property with respect to any ψ that is simultaneously a cofibration and a local weak equivalence. Theorem A.2 (Jardine). There is a model structure on Psh(C) with (co)fibrations as defined above and with weak equivalences the local weak equivalences.
14 14 BEN KNUDSEN This model structure is called the local injective model structure. Let F H( ; F ) denote a functorial fibrant replacement in this model category, and write H n (c; F ) = π n H(c; F ). Definition A.3. The presheaf F satisfies descent if the map F H( ; F ) is a global weak equivalence. Example A.4. Let X be a topological space, and take C = Open(X) with τ the usual Grothendieck topology. Let A be a sheaf of abelian groups on a space X. Define K(A, n) to be the simplicial presheaf U K(A(U), n). Then H q (X; A) = H q (X; K(A, 0)), and satisfying descent corresponds to the property of being flasque. For more on this example, see [?]. References [1] K. Brown and S. Gersten. Algebraic K-theory as generalized sheaf cohomology. [2] P. Gabriel. Des catégories abéliennes. [3] R. Hartshorne. Algebraic Geometry. [4] A. Grothendieck. Séminaire de géométrie algébrique du Bois Marie. [5] R. Jardine. Simplicial presheaves. [6] I. Panin. Oriented cohomology theories of algebraic varieties II. [7] D. Quillen. Higher algebraic K-theory. [8] D. Quillen. On the cohomology and K-theory of the general linear groups over a finite field. [9] R. W. Thomason and T. Trobaugh. Higher algebraic K-theory of schemes and of derived categories. [10] C. Weibel. An Introduction to Homological Algebra. [11] C. Weibel. The K-Book: An Introduction to Algebraic K-Theory.
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