SOME BASICS OF ALGEBRAIC K-THEORY

AARON LANDESMAN

1. INTRODUCTION:ALGEBRAICANDTOPOLOGICAL 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 theory and commu- tative 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- examined, and there are many interesting topics to cover before looking at higher K-groups. After defining K-theory as the 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 . 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 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 , 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 theory is more true to topological K-theory, the coherent theory is sometimes nicer. For example, the 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 . Define the Grothendieck group of A, notated

K0(A) as the with generators [A] for A ∈ A an object, and with relations [A] = A0 + A00 where A, A0, A00 ∈ A are objects in an (1.1) 0 A0 A A00 0.

Warning 1.3. We will often abuse notation by writing [A] ∈ 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 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 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 . 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 , in which case we may assume that we are working with modules over a local , 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 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 ∈ C, with relations [C] = [B] + [D] for each exact sequence

(1.6) 0 B C D 0 with B, C, D ∈ 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 , 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 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 of group comple- tion by the map of

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 , 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 admits a sur- jective map from a locally free sheaf, and further has a finite such , as follows from [BGI71, II.2.2.3, II.2.2.7.1]. It only re- mains to verify that the Cartan homomorphism is an isomorphism. This follows from the next theorem, Theorem 1.14 in the case that A = LocFreeShf(X) and B = CohShf(X).  Theorem 1.14 (The Resolution Theorem). Suppose A ⊂ B is an in- clusion of additive categories and B is an abelian category so that every object B ∈ B has finite A dimension, meaning that B has a finite resolu- tion P•, with Pn ∈ A. Then, the inclusion A B yields an isomorphism K0(A) K0(B). Proof. We construct a map inverse to the Cartan→ map, given by → φ : K0(B) K0(A) i B B 7 (−1) Pi , → i X B → where P• is a resolution of B in A. Once we check that this gives a well defined map, it is clearly an inverse. So, we only need verify this map is independent of the choice of projective resolution and is additive. To verify independence of projective resolution, suppose we had B another resolution of B, call it Q• . Then, we can construct a projec- B B B tive resolution R• of B mapping to both Q• and P• by taking the 6 AARON LANDESMAN projective resolution completing the diagram

B R• B

(1.8) ∆

B B P• ⊕ Q• B ⊕ B B B where ∆ denotes the diagonal map. The two induced maps R• Q• B B and R• P• are both quasi , and so we have equality i B i B i B i(−1) Pi = i(−1) Ri = i(−1) Qi as elements of K0(A). → To complete→ the proof, we only need check that φ is additive. PHowever, supposeP we haveP an exact sequence (1.9) 0 C C0 C00 0 0 0 of elements of B. Then, take a projective resolution P• of C and 0 a projective resolution P• of C mapping to P•. Observe that the 0 00 cone(P• P•) is a complex quasi-isomorphic to C (meaning the 00 complex with C in degree 0). Therefore, φ is indeed additive.  Example→ 1.15. Although we see that for separated regular Noether- ∼ ian schemes X, we have an isomorphism K0(X) = G0(X), via the Cartan map, this is not always the case. In fact, the Cartan map can even fail to be an isomorphism for zero dimensional schemes, as we now show. As a simple example, it will fail to be an isomorphism for Y := n ∼ Spec Z/p . First, we can compute that K0(Y) = Z, with generator represented by the class of the module corresponding to Z/pn, as locally free sheaves over a point are the same as free modules. ∼ However, we claim that G0(X) = Z with generator corresponding to the module Z/p. If we show this, it will imply that the Cartan map is not an isomorphism, as the class of the sheaf corresponding to Z/pn is n times the class of the sheaf corresponding to Z/p, using induction and the exact sequence

(1.10) 0 Z/pk Z/pk+1 Z/p 0.

To see G0(Y) is isomorphic to Z, note that every finitely generated module over Z/pn has size pk for some k. Then, we can send any such module to the element k ∈ Z. This is a well defined, additive map. Since it is injective, and 1 lies in the image, this map defines an isomorphism. Admittedly, although the Cartan map did not induce an isomor- phism, we still have that K0(Y) is abstractly isomorphic to G0(Y). For SOME BASICS OF ALGEBRAIC K-THEORY 7 an example where these two groups are not isomorphic, see Exam- ple 3.14. n Remark 1.16. The computation of G0(Spec Z/p ) was essentially done by taking a filtration of every Z/pn module by modules whose subquotients were Z/p. This is one of the simplest examples of devissage,´ a useful technique for computing K-theory by breaking a module up into smaller pieces, which we now discuss in section 2

2. DEVISSAGE´ In this section, we discuss the technique of devissage,´ which we are able to prove right away in Theorem 2.1. Theorem 2.1. Suppose B ⊂ A are small abelian categories with B an exact subcategory of A closed under subobjects and quotient objects. Suppose further that each A ∈ A has a filtration

A = A0 ⊃ A1 · · · ⊃ An = 0 and Ai/Ai+1 ∈ B. Then, the inclusion B ⊂ A induces ∼ K0(B) = K0(A). Proof. We define an inverse map

φ : K0(A) K0(B) n−1 A →7 Ai/Ai+1 i=0 X where A = A0 ⊃ · · · An = 0 is→ a filtration of A with Ai/Ai+1 ∈ B. We have to check that this map is well defined and is additive. It is then clear that this defines an inverse to the natural inclusion, since n−1 A = i=1 Ai/Ai+1 in K0(A). First, we check that this function is well defined. Suppose we had P two such filtrations of A, one given by Ai, and the other given by Bj. Then, we can construct a refinement of Ai, call it Ai,j, given by

Ai,j := (Ai ∩ Bj) + Ai+1 and a refinement of Bj, call it Bj,i, given by

Bj,i := (Ai ∩ Bj) + Bj+1. Note that ∼ Ai ∩ Bj ∼ Ai,j/Ai,j+1 = = Bj,i/Bj,i+1. Ai ∩ Bj+1 + Ai+1 ∩ Bj 8 AARON LANDESMAN

Therefore, two filtrations have equivalent refinements in the sense that all subquotients of the Bj,i refinement have a corresponding iso- morphic subquotient in the Ai,j refinement. It follows that the map is well defined. To conclude, we only need check the map is additive. Indeed, to check additivity, we want to show that for an exact sequence

π (2.1) 0 A A0 A00 0 we have φ(A0) = φ(A) + φ(A00). Choosing a filtration of A0, call it 0 00 00 0 Ai, we get a corresponding filtration in A given by Ai := π(Ai), 0 and a corresponding filtration on A given by Ai := A ∩ Ai. Note that Ai/Ai+1 ∈ B by the assumption that B is closed under subob- 00 00 jects and Ai /Ai+1 is in B by the assumption that B is closed under quotient objects. Then, additivity follows from exactness of

0 0 00 00 (2.2) 0 Ai/Ai+1 Ai /Ai+1 Ai /Ai+1 0.

 Corollary 2.2. If X and Y are two Noetherian schemes with isomorphic ∼ reductions, then G0(X) = G0(Y). ∼ Proof. It suffices to show G0(X) = G0(Xred). Apply devissage´ Theo- rem 2.1 to the category of coherent sheaves on X and Xred. Let I de- note the sheaf of Xred in X. We see that since I is nilpotent, and X is Noetherian, we know that there is some n for which In = 0. For any coherent sheaf F on X, F has a filtration F ⊂ I · F ··· In · F = 0. The subquotients of this filtration are killed by I and hence define coherent sheaves on Xred. Therefore, by Theorem 2.1, the natural inclusion G0(Xred) G0(X) is an isomorphism.  n ∼ Remark 2.3. This gives another proof that G0(Z/p ) = G0(Z)/p, although it is quite→ related to the proof given above in Example 1.15.

3. LOCALIZATION In this section, we discuss another fundamental technique in al- gebraic K-theory, known as localization, which essentially yields an excision exact sequence relating the K-theory of X and a closed sub- scheme Z to that of X \ Z. This might be called localization as an open subset of schemes corresponds to localization on the level of rings, see Remark 3.9. SOME BASICS OF ALGEBRAIC K-THEORY 9

3.1. Definition of localization. Definition 3.1. A Serre subcategory of an abelian category A is an abelian subcategory B ⊂ A which is closed under subobjects, quo- tients, and extensions. Definition 3.2. Let A be a small category with B ⊂ A a Serre sub- category. The quotient category or localization A/B is defined as the category whose objects are the same as those of A. To define its morphisms we say that a morphism f : A1 A2 in A is a B-iso if ker f ∈ B and coker f ∈ B. A morphism in B between A1 and A2 is an equivalence class → f 0 g A1 − A − A2 ∼ with f a B-iso, where the equivalence← → relation ∼ says that two mor- 0 0 00 00 f 0 g f 00 g phisms A1 − A − A2, A1 − A − A2 are equivalent if there is a commutative diagram of maps in A. ← → ← A0 →

A A A (3.1) 1 2

A00 with A0 A A00 both B-iso’s. f 0 g h 00 k If A1 − A − A2, A2 − A − A3 are two maps, their composi- tion is by← definition→ 0 0 00 00 ← →A1 A ← A ×→A2 A A A3. Note here that fiber products exist in any abelian category because ← ← →∼ → f,−g products and kernels do, and A ×f,B,g C = ker(A ⊕ C −− B). Also, note that base change of B-iso’s are B-iso’s, since the kernel and cok- ernel of a base change are isomorphic to the cokernel and→ cokernel of the original map. Furthermore, compositions of B-iso’s are B-iso’s. Finally, we define the natural map loc : A A/B A 7 A to be the localization map. → → 10 AARON LANDESMAN

Lemma 3.3. Let A ∈ A a small abelian category with B ⊂ A a Serre subcategory. We have loc A = 0 in A/B if and only if A ∈ B. Further, for f : A B, loc f is an isomorphism if and only if f is a B-iso. Proof. First, if A ∈ B, then loc A = 0, since we may take the mor- → id phism B − B 0. We see this defines an isomorphism in A/B, since we have a commutative diagram ← → B

B B 0 (3.2)

0 where all maps are B-iso’s. Conversely, if A ∈/ B, then the 0 map A A cannot be equivalent to the identity, since if we had B-iso’s g f A − C − 0, then we would have that C ∈ B, and so A ∈ B as well. Next,→ we show that loc f is an isomorphism if and only if it is a B-iso.← We→ know that f is a B-iso if and only if its kernel and cokernel are equivalent to 0, using the above. Since a map is an isomorphism if and only if its kernel and cokernel are 0, we see loc f is an isomor- phism if and only if f is a B-iso.  Lemma 3.4. Let B ⊂ A be a Serre category of a small abelian category. Then, A/B is abelian. Proof. It is fairly straightforward to verify that A/B is an abelian cat- egory. For example, direct sums are given by the images of the cor- responding direct sums in A under the localization map, the 0 object is the image of the zero object. The kernel of a map f g A1 − A − A2 is ker g and the cokernel is coker g. It takes some diagram chasing to verify these properties, which← we omit.→  Lemma 3.5. Let B ⊂ A be a Serre category of a small abelian category. Then, the localization map loc : A A/B is exact. Proof. We will just verify that cokernels are preserved under localiza- tion. The other verifications are→ similarly complex diagram chases, and we omit them. SOME BASICS OF ALGEBRAIC K-THEORY 11

Suppose we have a map f : A B, and let C := coker f so we have a right exact sequence → (3.3) A B C 0. Now, suppose we have a map loc B loc D in the localized category with the composition loc A loc B loc D equal to 0. Such a map loc B loc D can be written as a pair→ of maps → η → B − E D, → with η a B-iso. To show loc C = coker loc f, we want to construct a map loc C loc D making the← appropriate→ diagram commute. Let F := A ×B E. From the definition of composition in a localized cat- egory, this→ means that the composition F E D is equivalent to 0. So, using the for morphisms in the localized category, we have a diagram → → F

A G σ D (3.4) τ 0 H

We conclude that τ = 0 ◦ σ = 0, and so there is some B-iso G F so that the composition G F E D is 0 in A. Let E/G := coker(G F E). Then, by the universal property of cokernels,→ there is a map E/G D. We→ also→ have→ a map E/G C by the uni- versal property→ → of cokernels for f. We want to show that g : E/G C is a B-iso so that we→ obtain a map loc C loc D in the→ localized cat- egory with the composition from A being 0 in the localized category.→ To check g is a B-iso, note we have a diagram→ 0 G E E/G 0 (3.5) h k g t A B C 0. First, we know h : G A and k : E B are both B-iso’s by construc- tion. From the snake lemma, there is a surjection coker k coker g, which implies that coker→ g lies in B→. To conclude the proof, we only → 12 AARON LANDESMAN need verify that ker g ∈ B. For this, observe that we have a sur- jection A ker t, with t : B C, (it is a surjection since ker t is the cokernel of the map ker f A, using the fact that the kernel of the cokernel→ is the cokernel of→ the kernel, one of the axioms for an abelian category). Hence, by surjectivity→ of A ker t, we get a sur- jection coker(G A ker t) coker(A ker t). Since B is closed under quotients, we obtain coker(A ker t) ∈→B. Now, we have a diagram → → → → 0 G E → E/G 0 (3.6) h k g t 0 ker t B C 0. Using the snake lemma, we have a sequence, (3.7) ker k ker g coker(ker t G) which is exact in the middle. Since the right and left terms and in B, and B is closed under subobjects, quotients, and→ extensions, we obtain that ker g ∈ B.  3.2. The universal property of localization. Proposition 3.6 (Universal Property of localization). Suppose A is an abelian category and B is a Serre subcategory. Then, for T : A C any exact with T(B) = 0 for all B ∈ B, the functor T 0 : A/B C → loc A 7 T(A) is the unique making → → A loc A/B (3.8) T T 0 C commute. Proof. Uniqueness is immediate from the definition. We only need check T 0 is well defined and exact. We first check T 0 is well defined. To check T 0 is well defined, sup- ∼ ∼ pose that loc A1 = loc A2. We want to check T(A1) = T(A2). Since ∼ loc A1 = loc A2, we have a map f g A1 − A − A2

← → SOME BASICS OF ALGEBRAIC K-THEORY 13 ∼ ∼ where both f and g are B-iso’s. But then, this means that [A1] = [A] = ∼ [A2] as elements of coker(A B), and hence T(A1) = T(A2), by the universal property of cokernels. 0 To complete the proof, we→ only need verify exactness of T . Sup- pose we have an exact sequence in A/B.

(3.9) 0 loc A0 loc A1 loc A2 0

f g The second map can be represented as A1 − A − A2 with f a B-iso. Using exactness of the localization functor, Lemma 3.5, applied to ← → (3.10) 0 ker g A A2 coker g 0 we have that loc coker g = 0 so coker g ∈ B. This also implies ∼ ∼ loc ker g = loc A0. We also obtain that T(ker g) = T(A0), using that T(B) = 0, and so ∼ T(A1) = T(A) ∼ = T(A2) + T(ker g) + T(coker g) ∼ = T(A2) + T(A0) + 0, as desired.  Theorem 3.7. Suppose A is an abelian category and B is a Serre subcate- gory. Then the sequence

loc (3.11) K0(B) K0(A) K0(A/B) 0 is exact. Proof. From Proposition 3.6, the functor T 0 : A/B coker(A B) loc A 7 [A] is well defined and exact. The→ universal property→ of cokernels gives the inverse to this functor. →  3.3. Applications of the localization exact sequence. We now give some consequences of the localization exact sequence. Lemma 3.8. Suppose X is a Noetherian scheme and Z is a closed subscheme and let U := X \ Z. Let CohShf(Z, X) denote the coherent sheaves on X supported on Z. Then, CohShf(X)/CohShf(Z, X) =∼ CohShf(U). Further, we have an exact sequence

(3.12) G0(Z) G0(X) G0(U) 0. 14 AARON LANDESMAN

Proof. For the first statement, both CohShf(X)/CohShf(Z, X) and CohShf(U) satisfy the same universal property, where the universal property of localization of categories is given in Proposition 3.6. The localization exact sequence gives

(3.13) K0(CohShf(Z, X)) G0(X) G0(U) 0 ∼ so it suffices to show K0(CohShf(Z, X)) = G0(Z). This follows from devissage´ Theorem 2.1, since any sheaf supported on Z is killed by some power of the of Z in X.  Remark 3.9 (Reason for the name of localization). In particular, if ∼ ∼ X = Spec R, Z = Spec R/f, so that U = Spec Rf, we have that the localization of CohShf(X)/CohShf(X, Z) is the coherent sheaves on the localization Spec Rf. ∼ 1 Proposition 3.10. Let X = Spec R be an affine scheme. Let i : AX X be the natural projection. Then, i induces 1 ∼ → G0(AX) = G0(X). 1 ∗ Proof. Flatness of AX X tells us that the functor i : CohShf(X) 1 ∗ 1 CohShf(AX) is exact, and hence descends to a functor i : G0(AX) G0(X). We want to check→ this is an isomorphism. → First, we prove injectivity. Define a map → π : CohShf(Spec R[t]) CohShf(Spec R)

M 7 M/tM − annM(t). → where we are identifying the module M with the corresponding co- herent sheaf. This is an exact functor→ because given an exact se- quence of R modules (3.14) 0 A B C 0 we get an exact commutative diagram of R[t] modules 0 A B C 0 (3.15) ×t ×t ×t 0 A B C 0.

The kernels of the vertical maps correspond to annM(t) while the cokernels correspond to M/tM. So, the snake lemma guarantees π is exact, and so it induces a map ∗ π : G0(Spec R[t]) G0(Spec R).

→ SOME BASICS OF ALGEBRAIC K-THEORY 15

We have that π∗i∗(M) = π∗(M[t]) = M − 0 = M, so i∗ is an injection. To complete the proof, we will show i∗ is a surjection. ∼ Suppose that G0(R) 6= G0(R[t]). Let J maximal among all ideals ∼ satisfying G0(R/J) 6= G0((R/J)[t]). We know Spec R/J must be re- duced using Corollary 2.2, hence R/J is an integral domain. Let p1, ... , pn be the generic points of Spec R. Let S be the set of nonzero divisors. This is the complement of all associated primes of Spec R (which are the same as the generic points of Spec R, since R is re- ∼ n duced). This implies that Spec RS = Spec ⊕i=1 κ(pi), where κ(p) de- ∼ n notes the fraction field of p. We also have Spec RS[t] = ⊕i=1κ(pi)[t]. By an inductive application of Lemma 3.8, we have n ∼ n G0(⊕i=1κ(pi)) = ⊕i=1G0(κ(pi)) n ∼ n G0(⊕i=1κ(pi)[t]) = ⊕i=1G0(κ(pi)[t]) We obtain an exact sequence (3.16) 0 (R sR) (R) ⊕ (κ(p )) 0 −lims∈S G0 / G0 G0 i

→ 0 (R sR[t]) (R[t]) ⊕ (κ(p )[t]) 0 −lims∈S G0 / G0 G0 i ∼ The fundamental theorem of modules over a PID ensures Z = G0(Spec )κ(pi)[t], where the→ generator corresponds to the structure sheaf. Therefore, we have that the right vertical maps are isomorphisms. Further, our inductive hypothesis guarantees that the left vertical maps are isomorphisms. So, the five lemma implies i∗ is surjective, as de- sired.  Lemma 3.11. We have n ∼ n+1 G0(Pk ) = Z Pn with generators given by O n ... O k (n). Pk , , n ∼ Proof. First, we claim that G0(Ak ) = Z, for k a field. Indeed, this follows by induction on n from Proposition 3.10, and the fact that ∼ G0(Spec k) = Z. n−1 n n ∼ Next, consider a hyperplane Pk ⊂ Pk . We will show G0(Pk ) = Zn+1. The localization exact sequence for schemes from Lemma 3.8 yields

n−1 n n (3.17) G0(Pk ) G0(Pk ) G0(Ak ) 0. 16 AARON LANDESMAN

n This proves that G0(Pk ) is generated by at most n + 1 elements, so it suffices to exhibit n + 1 generators. n ∼ n Using that G0(Pk ) = K0(Pk ) by Theorem 1.13, it suffices to exhibit n n + 1 generators of K0(Pk ). Now, consider the function n n φ : K0(Pk ) × K0(Pk ) Z n i n ∨ (F, G) →7 h (Pk , F ⊗ G ). i=0 X This is well defined and additive in→ exact sequences because the Eu- ler is additive in exact sequences, and invertible sheaves are flat (this is why we are defining this function on K0(X) instead of G0(X)), so tensoring with them preserves exact sequences. Φ Φ := φ(O n (i) O n (j)) Now, define the matrix by i,j Pk , Pk . Observe that this is an upper triangular matrix with 1’s on the diagonal. There- O n ... O n (n) fore, the elements Pk , , Pk are independent. But, since we n showed above K0(Pk ) has at most n + 1 generators, it must have n + 1 O n ... O n (n) rank precisely . Further, the elements Pk , , Pk must n ∼ n+1 generate a sublattice of K0(Pk ) = Z . Finally, since Φ has determi- 1 O n ... O n (n) nant , this sublattice must be the whole lattice, so Pk , , Pk n ∼ n are generators for K0(Pk ) = G0(Pk ).  Remark 3.12. At first, one might guess that the K-theory of a very ∼ nice space is simply G0(X) = Pic X ⊕ Z. For instance, this will be ∼ true if G0(X) = K0(X) and all vector bundles can be written as a sum of line bundles, such as on P1. In fact, in section 4, we will see this holds for all regular curves. However, it emphatically does not hold, even for Pn with n > 1, since Pic Pn =∼ Z. Remark 3.13. Here is one connection between algebraic and topo- 1 ∼ 2 1 ∼ logical K-theory. Observe that P = S , and we just saw G0(P ) = Z ⊕ Z. We also similarly saw K(S2) =∼ Z ⊕ Z, so at least in this simplest case, the two agree, as vector bundles on both are determined by rank and degree (where degree means the first ). Example 3.14. Let X denote the affine plane with the doubled ori- gin. That is, X is a gluing of A2 and A2 along the complement ∼ of the origin in both planes. We will show G0(X) = Z ⊕ Z but ∼ K0(X) = Z. This gives an interesting, but nonseparated, example where the Grothendieck group of the generated by vector bundles does not agree with the Grothendieck group of all coherent sheaves. SOME BASICS OF ALGEBRAIC K-THEORY 17 ∼ First, we show G0(X) = Z ⊕ Z using Theorem 3.7 for the inclusion 2 2 ∼ A X. Since G0(A ) = Z from Proposition 3.10, we have an exact sequence → (3.18) 0 Z G0(X) Z 0 ∼ so G0(X) = Z ⊕ Z. The first map is injective since restriction defines of sheaves defines a one sided inverse from X to A2. ∼ 2 Second, we calculate K0(X). We have that LocFreeShf(X) = LocFreeShf(A ), since locally free sheaves on a regular space are in bijection with Cartier divisors, and either of the inclusions A2 X induce an iso- morphism on Cartier divisors. Therefore, ∼ 2 ∼ 2 ∼ ∼ 2 K0(X) = K0(A ) = K0(A ) = Z = G→0(A ) using Theorem 1.13 and Proposition 3.10. This same example works for the affine n-plane with the doubled origin, for all n ≥ 2.

4. ALGEBRAIC K THEORYFORREGULARCURVES In this section, we show that if X is a curve over a field, then ∼ ∼ 0 G0(X) = K0(X) = Pic X ⊕ H (X, Z). Essentially, the K-theory of X is determined by line bundles and rank, which is as nice as one could hope for it to be. This statement holds true more generally for any 1 dimensional Noetherian separated regular scheme, although we don’t prove it here. For this proof, we start by recalling the classification of invertible sheaves (projective modules) over a . Theorem 4.1. Let P be a finitely generated over a 1 di- mensional regular domain R. Then, P =∼ L ⊕ Rf, with L rank 1 projective module and f = rk P − 1. That is, finitely generated projective modules are classified by their rank and determinant. Proof. Omitted, see [Bas68, Chapter IV].  ∼ Corollary 4.2. We have an isomorphism rk ⊕ det : K0(Mod(R)) = Z ⊕ Pic R. Proof. First, the map is well defined because determinant and rank are additive in exact sequences. Second, the map is surjective be- cause R⊕a−1 ⊕ L is a projective module of rank a and determinant L. Finally, the map is injective by Theorem 4.1.  Proposition 4.3. Let X be a smooth curve over a field. Then, ∼ 0 K0(X) = H (X, Z) ⊕ Pic X. 18 AARON LANDESMAN

Proof. First, using Theorem 3.7, we may reduce to the case that X is connected, in which case we have H0(X, Z) =∼ Z, so we only need ∼ show K0(X) = Z ⊕ Pic X. Observe that since X is regular and con- nected, it is also integral. Second, if X is affine, we are done by Corollary 4.2. Note that any curve has an embedding into a projective regular curve by [Vak10, Theorem 17.4.2]. Further, if C is a projective curve, then for any closed point p on X, we have that X \ p ⊂ X is an affine . One way to see this is to use Riemann Roch: There is some n so that O(np) gives an embedding, (if we take n > 2g this will hold,) C Pt, for some t. Further, under this embedding, the hy- perplane tangent to C at p only intersects C at p (and it intersects with multiplicity→ n). It follows that C \ p is a closed subscheme of At = Pt \ Pt−1, and is hence affine, as it is the preimage of an affine set under a closed embedding, which is an affine map. Since we have just shown X is either projective or affine, and we have dealt with the case X is affine, it suffices to deal with the case that X is projective. As mentioned above, X \ p is affine. In this case, we have a commutative diagram, where all rightward sequences are exact

K0(p) K0(X) K0(R) 0

(4.1) i∗ K0(p) K^0(X) K^0(R) 0

f h t g 0 Z PicX PicR 0. where K^0(X) denotes the image of det : K^0(X) Pic X. Exactness of the top row follows from Theorem 3.7, while exactness of the second row follows from, among other things, the snake→ lemma applied to 0 0 Z Z 0 (4.2)

0 K0(p) K0(X) K0(R) 0 where the kernels of the vertical maps are 0 and the cokernels of the vertical maps product the second row. Right exactness of the third row is the standard excision exact sequence from algebraic and left exactness holds because we have a one sided inverse given ∼ by restricting a sheaf on X to p. Further K0(p) = Z, and all squares SOME BASICS OF ALGEBRAIC K-THEORY 19 in the above diagram commute. The only one requiring some veri- fication is the bottom left square. Indeed, the generator of K0(p) is Op, so we only need check it commutes for Op. We have h ◦ i∗Op = h(OX − OX(−p)) = OX(p). We also have g ◦ f(Op) = gO(p) = O(p). Therefore, we have that f is an isomorphism. We also have that t is an isomorphism by Corollary 4.2. Additionally, we know i∗ is in- jective, because both f and g are injective, hence g ◦ f = i∗ ◦ h is injective, so i∗ is injective. This implies that we have a commutative diagram

0 K0(p) K^0(X) K^0(R) 0 (4.3)

0 Z PicX PicR 0 where the left and right vertical maps are isomorphisms. Hence, the middle vertical map is an isomorphism as well, by the snake lemma. ∼ ∼ It follows that K^0(X) = Pic X, so K0(X) = Z ⊕ Pic X, as desired. 

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