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noetherian scheme vector bundle coherent sheaf Euler characteristic Hilbert polynomial

Algebraic K-theory and Algebraic Cycles

We saw in the last chapter that both the Euler characteristic and the Hilbert polynomial are invariants of a coherent sheaf that are additive in exact sequences. In this chapter, we want to introduce algebraic K-theory, which is the natural setting for such invariants. In addition, we explore the relationships between vector bundles, coherent sheaves, and algebraic cycles.

Algebraic K-theory

Definition 5.1. Let X be a noetherian scheme. Define K_{0}(X) to be the quotient of the free abelian group generated by isomorphism classes of vector bundles on X by the relations: [F] = [E] + [G] whenever there is an exact sequence of vector bundles 0 → E → F → G → 0. Define G_{0}(X) in the same way, by replacing “vector bundle” with “coherent sheaf” everywhere in the previous sentence.

Remark 5.2. An operation on coherent sheaves that is additive on short exact sequences defines a group homomorphism with domain G_{0}(X). The two examples we have seen so far are the Euler characteristic

χ : G_{0}(X) → Z,

and the Hilbert polynomial

p : G_{0}(X) → Q[z].

In the same way, any operation on locally free sheaves that is additive on short exact sequences defines a homomorphism with domain K_{0}(X). For example, tensor product of locally free sheaves induces a ring structure on K_{0}(X), since the assignment E ↦ F ⊗ E preserves exact sequences. For the same reason, G_{0}(X) is a K_{0}(X)-module. Moreover, there are always natural homomorphisms

ι_{X} : K_{0}(X) → G_{0}(X).

If X is nonsingular, then ι_{X} is an isomorphism. (See @@ 375 @@ or, more generally, @@ 40 @@.)

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2 Chapter 5: Algebraic K-theory and Algebraic Cycles

direct image

higher direct image

Serre’s Finiteness Theorem projection formula delta functor intersection pairing

Remark 5.3. If f : X → Y is any morphism of noetherian schemes, then pulling back vector bundles is an exact functor defining a natural ring homomorphism f^{∗} : K_{0}(Y) → K_{0}(X). If f is proper, then we can also define a natural group homomorphism f_{∗} : G_{0}(X) → G_{0}(Y) as follows. Note that this takes more care, first of all because the direct image of a vector bundle need not be a vector bundle. The hypothesis of properness assures us that all the higher direct images R^{i}f_{∗}(F) of a coherent sheaf F are themselves coherent. (This is a sheaf-theoretic version of Serre’s Finiteness Theorem.) However, we still need to be careful because f_{∗} is not an exact functor. As it turns out, we get a well-defined group homomorphism by taking

f_{∗}[F] = ∑_{i≥0}(−1)^{i}[R^{i}f_{∗}(F)].

Proposition 5.4. Let f : X → Y be a proper morphism. Then for all x ∈ G_{0}(X) and all y ∈ K_{0}(Y), one has the projection formula

f_{∗}(x.f^{∗}y) = f_{∗}(x).y ∈ G_{0}(Y).

Proof: Let E be a vector bundle on Y and let F be a coherent sheaf on X. It is enough to show that there are natural isomorphisms

R^{i}f_{∗}(F ⊗ f^{∗}E) ≈ R^{i}f_{∗}(F) ⊗ E

for all i. Fixing E and viewing both sides as functors of F, we can use the fact that tensoring with a locally free sheaf preserves exact sequences to see that both sides are delta functors. When F is an injective sheaf, both sides vanish for i > 0. So, it is enough to verify the isomorphisms when i = 0. But this is just the projection formula of Lem. 4.11.

Intersection Theory

Suppose Y and Z are irreducible closed subschemes of codimension r and s, respectively, on X. Assume further that Y and Z are in “general position”. Then the irreducible components of the intersection Y ∩Z all have codimension equal to the sum r + s. (As a special case, two curves on a surface intersect in finitely many points.) This suggests that we can define an intersection pairing CH^{r}(X) × CH^{s}(X) → CH^{r+s}(X).

Theorem 5.5. There exists a unique intersection pairing on the class of irreducible nonsingular varieties over a field k satisfying the following properties:

(CH1) The pairing makes CH(X) = ⨁_{r=0}^{dim(X)} CH^{r}(X) into a graded ring with iden-

tity equal to the class of X itself. (CH2) For any morphism f : X → Y, there is a natural ring homomorphism

f^{∗} : CH(Y) → CH(X). (Thus, CH is a contravariant functor from the category of nonsingular varieties to the category of rings.)

◯^{c}Coombes 1996 projection formula intersection multiplicity generic point effective Cartier divisor divisor, Cartier Chow’s Moving Lemma algebraic cycle cycle, algebraic

(CH3) For every proper morphism f : X → Y, there is a natural morphism f_{∗} : CH(X) → CH(Y) of graded groups (that shifts degrees). (Thus, CH can also be considered as a covariant functor from the category of varieties with proper morphisms to the category of graded groups.)

(CH4) If f : X → Y is proper, then for all x ∈ CH(X) and all y ∈ CH(Y), one has the projection formula

f_{∗}(x.f_{∗}y) = f_{∗}(x).y.

(CH5) If Y and Z are cycles on X and if Δ : X → X×X is the diagonal morphism,

then

Y.Z = Δ^{∗}(Y × Z).

(CH6) If Y and Z are cycles on X that meet properly (in that the codimension of the components of their intersection is equal to the sum of their codimensions), then there are local intersection multiplicities (that depend only on a neighborhood of the generic point of each component) such that

Y.Z = ∑ i(Y, Z; W_{i})W_{i}.

(CH7) If Y is a subvariety of X and if Z is an effective Cartier divisor on X meeting Y properly, then Y.Z is the cycle associated to the effective Cartier divisor Y ∩ Z on Y.

Proof: The two key ingredients of the classical proof are (i) a correct definition of the local intersection multiplicities and (ii) Chow’s Moving Lemma, which says that given two cycles Y and Z, you can find a cycle Z′ rationally equivalent to Z that meets Y properly. Fulton @@ 465 @@ gives a different treatment that avoids both these elements and holds more generally than the result stated here.

Remark 5.6. The maps f_{∗} and f^{∗} associated with a morphism f : X → Y are defined as follows. First, let Z be an irreducible closed subscheme of X. If dim(f(Z)) < dim(Z), we set f_{∗}(Z) = 0. Otherwise, we define

f_{∗}(Z) = [k(Z) : k(f(Z))]\overline{f(Z)},

which makes sense in this case because the extension of function fields is finite.

Next, let Z be an irreducible closed subscheme of Y. Let Γ_{f} denote the graph of f, considered as an algebraic cycle on the product X × Y. Let Z̃ = p_{Y}^{−1}(Z) = X × Z ⊂ X × Y also be viewed as a cycle on the product. Now define

f^{∗}(Z) = (p_{X})_{∗}(Γ_{f}.Z̃).

◯^{c}Coombes 1996 algebraic cycle cycle, algebraic

Chern Classes

On a strictly formal level, we see that the theory of algebraic cycles looks a lot like the K-theory of vector bundles. We also know that there is a direct connection between line bundles and divisors on X. The next step is to see that higher dimensional vector bundles are also related to algebraic cycles. First, we look at some additional properties of algebraic cycles.

Proposition 5.7. (The Exactness Property) Let Y be a nonsingular irreducible closed subscheme of X, with open complement U. Then there is an exact sequence

CH(Y)⟶^{i_{∗}}CH(X)⟶^{j^{∗}}CH(U) → 0.

Proof: Let s be the codimension of Y in X. You can separate the points of codimension r in X into two pieces: those in Y (of codimension r − s) and those in U. Now the result follows from a diagram chase when you write down the exact sequences defining the r^{th} grade pieces of the Chow groups.

Proposition 5.8. (The Homotopy Property) For every positive integer m, we have natural ring isomorphisms CH(X) → CH(X ×A^{m}), induced by pulling back along the projection map.

Proof: By induction, we can reduce to the case m = 1. Let π : X ×A^{1} → X be the projection. Any prime cycle of codimension r on the product is either completely contained in a fibre, or else it projects onto a cycle of the same dimension in X. From this observation, it suffices to check that cycles contained in fibres are trivial. But that is just the statement that Pic(A^{1}) = 0.

Corollary 5.9. For all m, one has CH(P^{m}) = Z[h]/h^{m+1}, where h is the class of a hyperplane.

Proof: We proceed by induction on m, with the result being trivial for m = 0, which is just a point. Take a hyperplane section and use the exactness property to write

CH(P^{m−1}) → CH(P^{m}) → CH(A^{m}) → 0.

By the homotopy property, one sees that CH(A^{m}) is isomorphic to Z (in codimension 0). The result follows once you check that no linear subvariety of P^{m} can be rationally equivalent to 0, since the degree of a rationally trivial cycle on projective space is 0.

Proposition 5.10. Let E be a locally free sheaf of rank m on X. Let P(E) be the associated projective bundle. (So, there is a morphism π : P(E) → X whose fibres are isomorphic to P^{r−1}.) Let ξ ∈ CH^{1}(P(E)) be the class of the divisor

◯^{c}Coombes 1996 projective bundle Chern class total Chern class Chern polynomial

corresponding to the line bundle O_{P(E)}(1). Then π^{∗} makes CH(P(E)) into a free module over CH(X) with basis 1, ξ, …, ξ^{r−1}.

Proof: See Grothendieck’s appendix to @@ 375 @@ or see Fulton @@ 465 @@.

Remark 5.11. This proposition generalizes the previous computation on projective space. It is important because it allows us to make the following definition.

Definition 5.12. Let E be a locally free sheaf of rank m on X, and let P(E) be the associated projective bundle. For each i = 0, …, r, we define the i^{th} Chern class c_{i}(E) ∈ CH^{i}(X) by the requirement that c_{0}(E) = 1 and

∑_{i=0}^{r}(−1)^{i}π^{∗}(c_{i}(E)).ξ^{r−i} = 0

in CH^{r}(P(E)).

Definition 5.13. Define the total Chern class

c(E) = c_{0}(E) + c_{1}(E) + ⋯ + c_{r}(E),

and the Chern polynomial

c_{t}(E) = c_{0}(E) + c_{1}(E)t + ⋯ + c_{r}(E)t^{r}.

Theorem 5.14. The Chern classes are characterized by the following properties:

(C1) If E = O_{X}(D) is the line bundle associated to a divisor D, then c_{t}(E) =

1 + Dt. (C2) If f : Y → X and E is locally free on X, then for all i,

c_{i}(f^{∗}E) = f^{∗}c_{i}(E).

(C3) If 0 → E → F → G → 0 is an exact sequence of locally free sheaves, then

c_{t}(F) = c_{t}(E)c_{t}(G).

Proof: We first show that these three properties determine the Chern classes. Observe that the pullback of E to P(E) is an extension of a line bundle by a bundle of rank r−1. So, one can show inductively that there exists a morphism π : Y → X such that π^{∗}(E) has a filtration whose quotients are line bundles L_{i} and such that π^{∗} : CH(X) → CH(Y) is injective. In this case, one computes the Chern classes on Y, where

c_{t}(E) = ∏ c_{t}(L_{i}) = ∏(1 + D_{i}t).

It only remains to verify that the three properties really do hold for the Chern classes.

◯^{c}Coombes 1996

(i) The projectivization of a line bundle is isomorphic to the original scheme. exponential Chern character Under the natural isomorphism π : P(O_{X}(D)) → X, one has π^{∗}(O_{X}(D)) = O_{P}(1), and so ξ = D ∈ CH^{1}(X). It follows that the first Chern class is D.

(ii) Follows immediately from the naturality of the construction of projective bundles.

(iii) As above, we can replace X by a scheme on which E, F, and G have compatible filtrations as a successive extension of line bundles. By induction on the rank, therefore, it suffices to prove the result when E = L is a line bundle. Let g : P(G) → X and f : P(F) → X be the canonical morphisms. There is a surjection

g^{∗}(F) → g^{∗}(G) → O_{G}(1)

of sheaves on P(G). This surjection naturally determines a morphism α : P(G) → P(F) such that α^{∗}(O_{F}(1)) = O_{G}(1). In particular, α^{∗} : CH(P(F)) → CH(P(G)) takes the canonical ring generator ξ_{F} to ξ_{G}. It now follows from the presentation of the Chow rings of projective bundles as modules over the Chow ring of the base that c_{t}(G) divides c_{t}(F), with the quotient being a linear polynomial. Writing the quotient in the form 1 + Dt, we see that D is determined by the equation c_{1}(F) = c_{1}(G) + D. Since E is a line bundle, it now follows that D = c_{1}(E).

Proposition 5.15. Let E and F be locally free sheaves on X. Formally, write

c_{t}(E) = ∏_{i=1}^{r}(1 + a_{i}t) c_{t}(F) = ∏_{i=1}^{s}(1 + b_{i}t).

Then one has

(i) c_{t}(E ⊗ F) = ∏_{i,j}(1 + (a_{i} + b_{j})t),

(ii) c_{t}(∧^{p}E) = ∏_{1≤i_{1}≤···≤i_{p}≤r}(1 + (a_{i_{1}} + ⋯ + a_{i_{p}})t), and

(iii) c_{t}(E^{∧}) = c_{−t}(E).

Proof: Left as an exercise.

The Riemann-Roch Theorem

Definition 5.16. Let E be a locally free sheaf on X, and write its Chern polynomial as above. Define the exponential Chern character to be

ch(E) = ∑_{i=1}^{r} e^{a_{i}}.

Remark 5.17. The computations on tensor products can be used to show that the exponential Chern character is a ring homomorphism

ch : K_{0}(X) → CH(X) ⊗ Q.

◯^{c}Coombes 1996 Euler characteristic zero cycle Euler characteristic degree Todd class

By property (C2), associated to every morphism f : X → Y, there is a commutative diagram

K_{0}(Y) ⟶^{ch} CH(Y) ⊗ Q
↓^{f^{∗}}  ↓^{f^{∗}}
K_{0}(X) ⟶^{ch} CH(X) ⊗ Q.

Remark 5.18. Now let’s consider the special case of the morphism f : X → Spec(k) for a nonsingular projective variety over a field k. We can easily see that ch : K_{0}(k) = Z → CH(k) ⊗ Q = Q is the homomorphism that takes a finite dimensional k-vector space to its dimension. The natural f^{∗} maps, being ring homomorphisms, are then completely uninteresting, since they simply take 1 to 1. Since f is proper, however, we also have f_{∗} maps. On the K-theory level, the map is given by

f_{∗}[E] = ∑_{i≥0}(−1)^{i}[R_{∗}^{i}E] = ∑_{i≥0}(−1)^{i}[H^{i}(X, E)],

and so ch(f_{∗}(E)) = χ(E) is just the Euler characteristic. On the level of algebraic cycles, f_{∗} only takes on a nonzero value at a zero cycle, where it maps zero cycles to their degrees. Thus, if we can understand how ch behaves compared to the two different f_{∗} maps, then we will get a formula that computes the Euler characteristic of a vector bundle in terms of the degree of some associated algebraic zero cycle.

Definition 5.19. Define the Todd class of E to be

td(E) = ∏_{i=1}^{r} \frac{a_{i}}{1 − e^{−a_{i}}} .

Remark 5.20. One can compute (tediously) that

ch(E) = r + c_{1}(E) + \frac{1}{2}(c_{1}^{2}(E) − 2c_{2}(E)) + ⋯

and

td(E) = 1 + \frac{1}{2}c_{1}(E) + \frac{1}{12}(c_{1}^{2}(E) + c_{2}(E)) + · · · .

Theorem 5.21. (Grothendieck-Hirzebruch-Riemann-Roch) Let E be a locally free sheaf on a nonsingular projective variety of dimension n. Then

χ(E) = deg[ch(E).td(T )]_{n}.

Proof: (Actually, this hardly qualifies as a sketch, but let’s proceed anyway. For full details, the K-theoretic purist can refer to @@ 375 @@, while those of a more strongly geometric bent might prefer @@ 465 @@.) The idea is to prove the result in a more general setting, showing that ch(f_{∗}(x)) = f_{∗}(ch(x).td(T_{f})), where T_{f} is the relative tangent bundle associated to a suffciently nice morphism between schemes. (A smooth projective morphism between quasiprojective nonsingular schemes would

◯^{c}Coombes 1996 Riemann-Roch for Curves Serre Duality Riemann-Roch for Surfaces Serre Duality Noether’s Formula

certainly qualify.) You do this by observing that every projective morphism X → Y factors as a closed immersion X → P_{Y}^{n} followed by the projection P_{Y}^{N} → Y. You can prove the result in these two cases and then check that it behaves correctly with respect to composition. For closed immersions, you use the fact that the morphism is finite and the higher direct images vanish; for projective space, you can compute everything explicitly.

Corollary 5.22. (Riemann-Roch for Curves) When X is a smooth projective curve of genus g, we get

χ(E) = deg[c_{1}(E) + \frac{r}{2}(−K)]

If E = O_{X}(D), this becomes

l(D) − l(K − D) = deg(D) + 1 − g.

Proof: The first part follows from the general Riemann-Roch Theorem, the earlier computations of the exponential Chern character and the Todd class, and the observation that the divisor associated to the tangent bundle is the negative of the canonical class. To get the formula for a line bundle associated to a divisor D, use Serre Duality to write

l(D) − l(K − D) = χ(O_{X}(D)) = deg(D) + (1/2) deg(−K) = deg(D) + 1 − g.

Corollary 5.23. (Riemann-Roch for Surfaces) Let D be a divisor on a smooth surface X over a field k. Then one has

l(D) − s(D) + l(K − D) = \frac{1}{2}D.(D − K) + \frac{1}{12}(K^{2} + c_{2}).

Proof: Once again, we use Serre Duality to write

χ(O_{X}(D)) = l(D) − s(D) + l(K − D).

Now the general Riemann-Roch Theorem yields

\begin{aligned}χ(O_{X}(D)) & = deg[(1 + D + \frac{1}{2}D^{2}).(1 − \frac{1}{2}K + \frac{1}{12}(K^{2} + c_{2})] \\ & = \frac{1}{2}D.(D − K) + \frac{1}{12}(K^{2} + c_{2}). \end{aligned}

Corollary 5.24. (Noether’s Formula) If X is a smooth projective surface over a field, then

1 + p_{a}(X) = \frac{1}{12}(K^{2} + c_{2}).

Hilbert polynomial Genus Formula arithmetic genus

Proof: Choose a very ample divisor D to embed X into projective space, and let p_{X}(z) ∈ Q[z] be the Hilbert polynomial associated to O_{X} under this embedding. Plugging z = 0 into the Hilbert polynomial gives

1 + p_{a}(X) = p_{X}(0) = χ(O_{X}) = \frac{1}{12}(K^{2} + c_{2}).

Remark 5.25. The proof of the Riemann-Roch Theorem for Surfaces given, for example, in Hartshorne @@ 98 @@ only uses the Riemann-Roch Theorem for Curves and the Genus Formula for a smooth curve on a surface. The proof for surfaces from this viewpoint requires us to write D as a difference of smooth curves, manipulate some exact sequences, and then apply the Genus Formula to the curves. As a consequence of Riemann-Roch, however, we recover the following more general version of the Genus Formula.

Proposition 5.26. (Genus Formula) Let C be any curve of arithmetic genus p_{a}(C) on a smooth projective surface X over a field k. Let K be the canonical divisor on X. Then

2p_{a}(C) − 2 = C.(C + K).

Proof: View C as an effective divisor on X. Then we have an exact sequence

0 → O_{X}(−C) → O_{X} → O_{C} → 0.

Use the additivity of the Euler characteristic and the Riemann-Roch Theorem for Surfaces to get

p_{a}(C) − 1 = χ(O_{C}) = χ(O_{X}) − χ(O_{X}(−C))

= (1 + p_{a}(X)) − (\frac{1}{2}(−C).(−C − K) + 1 + p_{a}(X))

= \frac{1}{2}C.(C + K).

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