ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0.

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1 ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0. ANDREW SALCH During the last lecture, we found that it is natural (even just for doing undergraduatelevel complex analysis!) to associate a divisor class group to a variety, or perhaps even a scheme, not just a commutative ring; for example; we expect that if X is the Riemann sphere, then we want to know that is some kind of K-theory for complex analytic varieties with the property that the reduced K 0 -group K 0 (X) is isomorphic to Z, since, given a divisor f on the Riemann sphere (i.e., a Z-linear combination of points on the Riemann sphere), it is possible to find a meromorphic function on the Riemann sphere whose divisor is f if and only if the sum of the coefficients of f is zero. That sum of coefficients lies in Z and is clearly capable of being any element in Z, so Z is (isomorphic to) the obstruction group to a given divisor on X being the divisor of a meromorphic function, so Z ought to be the divisor class group of X. Of course, we also want whatever K-theory groups we construct to also have the right localization properties so that the exact sequences (the Weil divisor exact sequence, and the fractional ideal exact sequence) we saw in the last lectures occur as localization sequences in the K-groups (at least for nonsingular algebraic curves, and other schemes whose ring of sections in every open set is a Dedekind domain; for nonsingular schemes of dimension > 1, the sections in open sets are of global dimension > 1 and hence fail to be Dedekind, so the relationship between divisors and K-groups is not as simple as in the Dedekind case; the relationship is instead essentially what one computes by the Gersten spectral sequence, which is where we are headed. For schemes with singularities, even just curves with singularities, the relationship between K-groups and divisors is even more oblique, since for schemes with singularities, the Gersten spectral sequence only converges to the G-theory groups of the scheme, not necessarily the K-groups!). We will do all of this, but to accomplish this, we need some general machinery: a version of G 0 and K 0 which apply not just to modules over a ring, but rather to the more abstract categories that occur as (for example) the categories of modules over the structure ring sheaf of a scheme. That means we need to talk about abelian categories and exact categories. The work we do on this kind of abstract framework pays off later: this same material is used in Quillen s Q-construction that produces the higher algebraic K-groups, so when we get to K n for n > 1 we will be using these same ideas. This level of generality also allows us to handle algebraic K-theory of some things that are even more general than schemes, like algebraic spaces and algebraic stacks G 0 of abelian categories. 1. Abelian categories and exact categories. Date: January

2 2 ANDREW SALCH Definition 1.1. A category with biproduct is a category C with finite coproducts and finite products, and such that, for every finite set of objects X 1,..., X n of C, the natural map n n X i i=1 i=1 is an isomorphism. Then this finite coproduct, equivalently finite product, is called the biproduct in C. (Motivating example: the direct sum in the category of modules over any ring.) If C is a category with biproduct, then the hom-sets in C naturally have the structure of commutative monoids, as follows: if X, Y are objects of C and f, g hom C (X, Y), then we let f + g be the image of ( f, g) under the composite hom C (X, Y) hom C (X, Y) hom C (X, Y Y) hom C (X, Y Y) X i π hom C (X, Y), where π is the map induced by the natural codiagonal map Y Y Y. (All the maps here which I am describing as natural come from universal properties of products and/or coproducts.) We say that C is additive if: for every pair of objects X, Y in C, the commutative monoid hom C (X, Y) has inverses, i.e., it is an abelian group, and We say that C is abelian for every pair of objects X, Y of C and every map f : X Y, the map f has both a kernel (i.e., equalizer of f and the zero map) and cokernel (i.e., coequalizer of f and the zero map) in C. if, for every morphism f in C, the natural map coker(ker f ) ker(coker f ) is an isomorphism. (Example: if you unravel what this condition says in the category of abelian groups, it s the fundamental homomorphism theorem from elementary group theory!) Example 1.2. If R is a ring, the category of R-modules is abelian. If R is a Noetherian ring, the category of finitely generated R-modules is abelian. If X is a topological space, the category of sheaves of abelian groups on X is abelian. If X is a scheme, the category of O X -modules is abelian. If X is a Noetherian scheme, the category of coherent O X -modules is abelian. (A review of these definitions is below, in these notes.) If R is a ring, the category of projective R-modules is usually not abelian: it usually doesn t have kernels. (An exception: when R is semisimple, e.g. R a field, then every R-module is projective, so the projective R-modules form an abelian category in that case.) (So abelian categories are a reasonable setting for algebraic G-theory, but we want a very slightly different setting for dealing with algebraic K-theory, and that s what exact categories are for.) Definition 1.3. Let C be a small abelian category. Then the degree zero algebraic G- theory group of C, written G 0 (C), is defined to be the Grothendieck group completion of the monoid of isomorphism classes of objects of C, modulo the equivalence relation in

3 ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0. 3 which we set [Y] = [X] + [Z] whenever there exists a short exact sequence in C. Note that, when C is (a skeleton of) the category of finitely generated modules over a Noetherian ring R, then this definition of G 0 (C) agrees with the definition of G 0 (R) we already gave. Also note that we could just as well have defined G 0 (C) to be the free abelian group generated by the set of isomorphism classes of objects of C, modulo the same equivalence relation; this would get us an isomorphic G 0 (C). Why the smallness condition in the definition of G 0 (C)? It s so that the monoid of isomorphism classes we get in the end actually has a set of elements, and not a proper class! There are some games you can play here with large cardinals to avoid assuming smallness, but this kind of thing is rarely a useful generalization, since G 0 (C) will typically be trivial (by an Eilenberg swindle, as we talked about in class) if no finiteness condition on C is assumed K 0 of exact categories. Definition 1.4. An exact category consists of a pair (C, E), where C is an additive category and E is a class of sequences of the form in C, such that there exists an abelian category A and a faithful, full functor F : C A satisfying the two conditions: E is class of all short exact sequences in A which are in the image of F, and if is a short exact sequence in A and both X and Z are in the image of F, then Y is isomorphic to an object in the image of F. The motivating example of an exact category is, as you might guess, the category of finitely generated projective R-modules, where R is a Noetherian ring; the abelian category A in that case can be taken to be the category of all finitely generated R-modules, or it can be taken to be the category of all R-modules. There is an intrinsic definition of an exact category, which is sometimes useful when you have a given additive category C with some notion of sequences that look like they should be short exact sequences, but you have no obvious choice of abelian category to embed C into. Here are the intrinsic conditions; I think the equivalence of this definition with the above definition is called the Gabriel-Quillen theorem. Definition 1.5. Let C be an additive category, and let D be a collection of sequences in C of the form (1.2.1) 0 X i Y j Z 0. We refer to a map i that occurs in a sequence in D as an admissible monomorphism, and we refer to a map j that occurs in a sequence in D as an admissible epimorphism. (It follows from the conditions below that i is indeed a monomorphism and j is indeed an epimorphism.) We say that the pair (C, D) is an exact category if each of the following conditions is satisfied:

4 4 ANDREW SALCH D is closed under isomorphisms. D contains all sequences of the form 0 X i X Y j Y where i is the inclusion as the first coproduct summand and j is projection to the second coproduct summand. If 0 X i Y j Z 0 is in D, then i is the kernel of j, and j is the cokernel of i. (So, in particular, kernels of admissible epimorphisms exist in C, and cokernels of admissible monomorphisms exist in C.) A composite of two admissible epimorphisms is an admissible epimorphism. A composite of two admissible monomorphisms is an admissible monomorphism. If 0 X Y Z is in D and T Z is a map in C, then the pullback Y Z T in C exists, and the canonical map p : Y Z T T has the property that there exists some map i : W Y Z T such that 0 W i Y Z T p T 0 is in D. If 0 X Y Z is in D and X T is a map in C, then the pushout Y X T in C exists, and the canonical map i : T Y X T has the property that there exists some map p : Y X T W such that 0 T i Y T X p W 0 is in D. If g : Y Z is a map in C which has a kernel in C, and f : T Y is a map in C such that g f is an admissible epimorphism, then g is an admissible epimorphism. If f : X Y is a map in C which has a cokernel in C, and g : Y T is a map in C such that g f is an admissible monomorphism, then f is an admissible monomorphism. You can write these intrinsic definitions in a more compact way, but I wrote them out, above, in a way that s suitable for actually verifying them in practical cases (because I just copied and pasted it from a paper where that s what I was doing). Definition 1.6. Let (C, E) be an exact category, and suppose that C is small. By the degree zero algebraic K-theory of (C, E), written K 0 (C, E), we mean the Grothendieck group completion of the monoid of isomorphism classes of objects of C, modulo the equivalence relation in which we set [Y] = [X] + [Z] whenever there exists a short exact sequence in E. Again, we could just as well have defined K 0 (C, E) to be the free abelian group generated by the isomorphism classes of objects of C, modulo the same equivalence relation.

5 ALGEBRAIC K-THEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0. 5 Example 1.7. If C is an abelian category, let E be the class of all short exact sequences in C. Then K 0 (C, E) agrees with G 0 (C) as defined in the previous section. As a special case: if R is a Noetherian ring, let C be the category of finitely generated R-modules and let E be the class of all short exact sequences in C. Then K 0 (C, E) G 0 (R). If R is a Noetherian ring, we can instead let C be the category of finitely generated projective R-modules and let E be the class of all split exact sequences in C. Then K 0 (C, E) K 0 (R). 2. G 0 and K 0 of schemes. Now here are some relevant definitions from algebraic geometry. Hartshorne s textbook is a good reference for the algebro-geometric foundations here, but this material is standard enough that you find it in many sources. Definition 2.1. If R is a commutative ring and M an R-module, we write M for the O Spec R -module associated to M. That is, if r R and U r is the open subset U r = Spec R[r 1 ] of Spec R, we let M(U r ) be M R R[r 1 ]; since the open subsets of Spec R of the form U r for a basis for the Zariski topology on Spec R, this definition of M extends in at most one way (actually exactly one way) to arbitrary open subsets of Spec R, using the sheaf axiom. If X is a scheme, recall that an O X -module F is said to be quasicoherent if X admits an open cover {U i } i I with the following properties: for each i I, the scheme U i is affine, and for each i I, the O Ui -module F (U i ) is isomorphic to M for some Γ(U i )- module M. Recall that a scheme X is said to be a Noetherian scheme if X is quasicompact and admits an open cover by affine schemes whose rings of global sections are all Noetherian. (Equivalently: X is quasicompact and, for every open affine subset Y of X, the ring of global sections Γ(Y) of Y is Noetherian.) If X is a Noetherian scheme, recall that an O X -module F is said to be coherent if X admits an open cover {U i } i I with the following properties: for each i I, the scheme U i is affine, and for each i I, the O Ui -module F (U i ) is isomorphic to M for some finitely generated Γ(U i )-module M. The category Coh(X) of coherent O X -modules is defined as the full subcategory of the category of O X -modules generated by the coherent O X -modules. (That is, Coh(X) is the category whose objects are the coherent O X -modules, and whose morphisms are arbitrary morphisms of O X -modules.) If X is Noetherian, then Coh(X) is abelian. All the algebraic geometry in this class is very much optional! I will cover it because it is not that difficult, and it is really appealing how algebraic K-theory carries and organizes important information from algebraic geometry, but if you find scheme theory uninteresting or impenetrable, you are welcome to only ever consider affine schemes for the duration of the whole semester; remember that the category of affine schemes is equivalent to the opposite category of commutative rings, and under this equivalence, the quasicoherent O X -modules (for X a scheme) are equivalent to the Γ(O X )-modules, i.e., the category of R-modules is equivalent to the category of O Spec R -modules. That is, algebraic geometry over affine schemes is essentially just a rephrasing of the commutative algebra you have

6 6 ANDREW SALCH been doing for several years now, and you are welcome to only work with the commutative algebra versions of these ideas, if you prefer not to ever have to deal with schemes! Definition 2.2. Let X be a Noetherian scheme. An O X -module F is called free if F is isomorphic to a direct sum of copies of O X. The rank of F is then the cardinality of that set of copies of O X. An O X -module F is called locally free if there exists an open cover {U i } i I of X by subschemes such that each F Ui is a free O Ui -module. An O X -module F is called locally free if there exists an open cover {U i } i I of X by subschemes such that each F Ui is a free O Ui -module. At each point x X, the rank of F at x is then the cardinality of a basis for the free (O X ) x -module F x. (Here (O X ) x is the local ring of O X at the point x, i.e., it is the colimit of O X (U) over all open neighborhoods U of x; and similar for F x.) An O X -module F is called a vector bundle if F is locally free and of finite rank at each point. The category of vector bundles over X is defined as the full subcategory of the category of O X -modules generated by the vector bundles. That is, the category of vector bundles over X is the category whose objects are vector bundles over X, and whose morphisms are morphisms of the underlying O X -modules. Definition 2.3. Let X be a Noetherian scheme. G 0 (X) is defined as G 0 (C), where C is the category of coherent O X -modules. K 0 (X) is defined as K 0 (C, E), where C is the category of vector bundles over X, and E is the set of short exact sequences of O X -modules whose objects are all vector bundles. Theorem 2.4. (Serre.) Let R be a Noetherian commutative ring. Then a finitely generated R-module M is projective if and only if the coherent O X -module M is a vector bundle. Consequently K 0 (R) K 0 (Spec R).