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Some Basics of Algebraic K-Theory

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SOME BASICS OF ALGEBRAIC K-THEORY AARON LANDESMAN 1. INTRODUCTION:ALGEBRAIC AND TOPOLOGICAL K THEORY In this paper, we explore the basics of algebraic K-theory, with a focus on the K-theory of schemes. As the purpose is to explore K theory, we take foundational results in scheme theory and commu- tative algebra for granted. We mainly follow [Wei, Sections II.6, II.7, II.8]. In this paper, we stick to the case of K0(•), and do not discuss higher K-groups. One reason for this is that K0(•) was historically the first K-group examined, and there are many interesting topics to cover before looking at higher K-groups. After defining K-theory as the Grothendieck group of coherent sheaves on a scheme, we give some examples to show subtleties which can occur in the world of algebraic K-theory, which do not occur in the land of topolog- ical K-theory, such as the failure of exact sequences to split. We then discuss the techniques of devissage´ and localization, and use these to compute the K-theory of projective space. Finally, we dis- cuss the K-theory of curves X over a field, showing that they satisfy ∼ 0 K0(X) = H (X, Z) ⊕ Pic X. Remark 1.1 (An advanced relation between algebraic and topolog- ical K-theory). Before continuing with the body of the paper, we pause to mention a very deep relationship between algebraic and topological K-theory, far beyond the scope of this paper. Loosely speaking, suppose X is a separated regular Noetherian scheme with ` invertible. Then, there exists a certain element β in the second K- group of X (depending upon `), known as the Bott element, so that if one takes the algebraic K-theory of X and inverts β, one obtains the topological K-theory. This is described in [Tho85, p. 1]. Many remarks in this paper are aimed at a reader familiar with topological K-theory, though no background in topological K-theory is assumed. 1.1. Acknowledgements. I thank Peter Kronheimer for teaching the course on topological K-theory, which prompted this paper. I thank 1 2 AARON LANDESMAN Inna Zakharevich for suggesting this topic. I thank Akhil Matthew for answering several questions. I thank Peter Landesman for help- ful comments. 1.2. Two Candidates for algebraic K-theory. In this section, we de- scribe two K-theory like groups on a scheme X, (1)G 0(X), which encapsulates all coherent sheaves, (2) and K0(X), which encapsulates all vector bundles. We see in Theorem 1.13 that these agree in nice situations. While the vector bundle theory is more true to topological K-theory, the coherent sheaf theory is sometimes nicer. For example, the coherent sheaf K-theory of a scheme agrees with the coherent sheaf K-theory of its reduction, as shown in Corollary 2.2, and so coherent sheaf K-theory only depends on the reduced scheme structure, and “does not see nilpotents.” This will motivate our definitions for algebraic K-theory. In order to define a sort of K-theory in terms of coherent sheaves, we introduce the Grothendieck group. Definition 1.2. Let A be an abelian category. Define the Grothendieck group of A, notated K0(A) as the abelian group with generators [A] for A 2 A an object, and with relations [A] = A0 + A00 where A, A0, A00 2 A are objects in an exact sequence (1.1) 0 A0 A A00 0. Warning 1.3. We will often abuse notation by writing [A] 2 K0(A) simply as A, when it is understood to lie in K0(A) and not A. Start by recalling the definition of topological K-theory. Example 1.4. In the setting of topological K-theory, if X is a topolog- ical space, and Vect(X) is the category of vector bundles on X, then K0(Vect(X)) is what one would usually call K0(X). Definition 1.5. Let X be scheme and let CohShf(X) denote the cate- gory of coherent sheaves on X. Then, define G0(X) := K0(CohShf(X)) to be the Grothendieck group of CohShf(X). To make the connection to topological K-theory, we’d like to un- derstand what the category of all coherent sheaves has to do with the category of vector bundles. Unfortunately, the Grothendieck group SOME BASICS OF ALGEBRAIC K-THEORY 3 of an abelian category is not enough to define a K-theory for vector bundles, (or equivalently, a K-theory for invertible sheaves,) since vector bundles do not form an abelian category, as we now explain. Remark 1.6. The reason that we have to define K0(X) for a gen- eral exact category instead of an abelian category is that locally free sheaves on schemes do not form an abelian category. The only ax- iom of an abelian category they fail to satisfy is that they are not closed under cokernels. This failure is already seen in the case of 1 Ak := Spec k[t], for k a field, as evidenced by the exact sequence of modules on k[t] ×t (1.2) 0 k[t] k[t] k[t]/t · k[t] 0 Of course, here the first two sheaves are the trivial sheaf, but their cokernel is nonzero and torsion, hence not locally free. Our next aim is to define K0(X), which will essentially be the ana- log of G0(X) for invertible sheaves, but first we define exact cate- gories, to set up the construction in more generality. Definition 1.7. Let A be an abelian category let C be a full subcate- gory, and let E be a collection of sequences in C (1.3) 0 B C D 0. We say (C, E) is an exact category if (1) E is the collection of all exact sequences as above which are exact in A. (2) C is closed under extensions. We say C is an exact category when E is understood. Lemma 1.8. Let X be a scheme, let CohShf(X) denote the category of coher- ent sheaves on X, and let LocFreeShf(X) denote the category of locally free sheaves on X. Then, LocFreeShf(X) ⊂ CohShf(X) is an exact subcategory. Proof. We verify the two properties of the definition of exact category. A sequence of locally free sheaves is exact if and only if it is exact as a sequence of sheaves, so the first property is satisfied. If we have a sequence of sheaves (1.4) 0 F G H 0 with F, H both locally free, we claim G is locally free. Indeed, to check this, it suffices to check it at each stalk, in which case we may assume that we are working with modules over a local ring, and 4 AARON LANDESMAN we want to show an extension of free modules is free. Indeed, this follows, since if we have (1.5) 0 M N P 0 with M and P free, then N is a free module as well, since N is gen- erated by the lifts of generators of P together with the images of the generators of M. Definition 1.9 (A generalization of Definition 1.2 to exact subcate- gories). Let C be small exact category. Then K0(C) is the abelian group generated by objects C for C 2 C, with relations [C] = [B] + [D] for each exact sequence (1.6) 0 B C D 0 with B, C, D 2 C. We can now define K0(X), for X a scheme. Definition 1.10. Let X be a scheme, and let LocFreeShf(X) denote the category of locally free sheaves on X. Then, we define K0(X) := K0(LocFreeShf(X)). Remark 1.11. One might ask why we define K0(•) in terms of exact sequences instead of just saying that [A ⊕ B] = [A] + [B]. Indeed, the problem is that not all exact sequences of coherent sheaves split. Indeed, even not all exact sequences of locally free sheaves split. As a simple example, for E an elliptic curve, over a field k, there is a nontrivial extension (1.7) 0 OE F OE 0 since extensions are classified by Ext1 and 1 ∼ 1 ∼ ExtE(OE, OE) = H (E, OE) = k 6= 0. 1.3. The Cartan map is an isomorphism in nice situations. We’d ∼ like to know whether K0(X) = G0(X). To start, we introduce a natu- ral map between them. Definition 1.12. Let X be a scheme. Define the Cartan map Cartan : K0(X) G0(X) ! SOME BASICS OF ALGEBRAIC K-THEORY 5 to be the map induced via the universal property of group comple- tion by the map of monoids LocFreeShf(X) G0(X) F 7 [F] , ! where [F] denotes the class of the coherent sheaf F in G0(X). ! At this point, we will begin to use some heavy machinery from scheme theory. Even to establish one of the most foundational the- orems, Theorem 1.13, we will need to assume the fairly technical Serre’s theorem on cohomological dimension, stating that any mod- ule on a regular ring has finite projective dimension. Theorem 1.13. Let X be a separated regular Noetherian. Then the Cartan ∼ homomorphism Cartan : G0(X) = K0(X) is an isomorphism. Proof. First, using some fairly technical scheme theory every coher- ent sheaf on a regular separated Noetherian scheme admits a sur- jective map from a locally free sheaf, and further has a finite such resolution, as follows from [BGI71, II.2.2.3, II.2.2.7.1].
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