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Extending a Whitney class to K zero

PDF p. 11: Chern power series for virtual classes

Statement

Assume the Axiom of Choice. Let $H$ be an additively oriented bigraded theory with the projective bundle contract on a category $C$ of schemes. For each $X$ in $C$ , suppose that every finite locally free $O_{X}$ -module $E$ is assigned a series $c_{t} (E) = 1 + \sum_{i \geq 1} c_{i} (E) t^{i}, c_{i} (E) \in H^{2 i, i} (X),$ depending only on its isomorphism class. Assume that these series are natural under morphisms in $C$ and satisfy the Whitney identity $c_{t} (E) = c_{t} (E^{'}) c_{t} (E^{''})$ for every short exact sequence $0 \to E^{'} \to E \to E^{''} \to 0$ of finite locally free modules.

Write $K_{v b}^{0} (X)$ for the abelian group generated by the isomorphism classes $[E]$ of finite locally free modules, with the relations $[E] = [E^{'}] + [E^{''}]$ for all such exact sequences. Then there is a unique group homomorphism $c_{t} : K_{v b}^{0} (X) ⟶ {1 + \sum_{i \geq 1} a_{i} t^{i} : a_{i} \in H^{2 i, i} (X)},$ where the target group uses multiplication of formal series, taking $[E]$ to $c_{t} (E)$ . Every element can be written $[E] - [F]$ , and its image is $c_{t} (E) c_{t} (F)^{- 1}$ . The construction is natural in $X$ . When $X$ is locally Noetherian, $K_{v b}^{0} (X)$ is the vector-bundle Grothendieck group $K^{0} (X)$ of the cited library definition. The extension is conditional on the stated exact-sequence Whitney identity; the individual coefficients are not asserted to be additive. In particular, the second coefficient obeys the cross-term formula $c_{2} (x + y) = c_{2} (x) + c_{1} (x) c_{1} (y) + c_{2} (y)$ .

A finite locally free module is an $O_{X}$ -module locally isomorphic to a finite direct sum of copies of $O_{X}$ . Naturality of the given series means $f^{*} c_{i} (E) = c_{i} (f^{*} E)$ for every morphism $f : Y \to X$ in $C$ . A short exact sequence here is an exact sequence of sheaves whose three terms are finite locally free. Formal-series multiplication uses the finite coefficient sums $(a b)_{n} = \sum_{j = 0}^{n} a_{j} b_{n - j}$ .

Statement references

Bigraded cohomology and the projective bundle contract

Grothendieck groups of coherent sheaves and of vector bundles on a scheme

The Axiom of Choice

Facts & Assumptions

Given: The category, theory, natural bundle series and exact-sequence Whitney identity in the statement, together with the Axiom of Choice.

Goal: Construct the unique natural extension, establish the difference formula and the stated coefficient identity, and identify the locally Noetherian case.

[F1]

The theory has associative unital bigraded rings, multiplication of bidegree $(0, 0)$ , graded commutativity in the first degree, and degree-preserving contravariant unital ring pullbacks. These are part of the given theory's contract (Bigraded cohomology and the projective bundle contract).

[F2]

Under Choice, the published vector-bundle Grothendieck group for a locally Noetherian scheme has exactly the generator-and-exact-sequence presentation stated above (Grothendieck groups of coherent sheaves and of vector bundles on a scheme, The Axiom of Choice).

[A1]

The assigned bundle series are invariant under isomorphism, natural, and multiplicative in every short exact sequence, as stated in the hypotheses.

Proof

technique · Construct the target group coefficient by coefficient and descend a map through the presentation

1.1F1construct

Put $R_{X} = ⨁_{i \geq 0} H^{2 i, i} (X)$ . By [F1], this is a commutative graded ring: the first degree of each homogeneous summand is even. The displayed target consists of series with constant coefficient $1$ and coefficient of $t^{i}$ in $H^{2 i, i} (X)$ . Finite coefficient sums show that it is closed under associative, commutative multiplication and contains $1$ . For $a (t) = 1 + \sum_{i \geq 1} a_{i} t^{i}$ , define $b_{0} = 1$ and recursively $b_{n} = - \sum_{j = 1}^{n} a_{j} b_{n - j}$ . Each $b_{n}$ has bidegree $(2 n, n)$ , and the coefficient of $t^{n}$ in $a (t) b (t)$ is zero for $n > 0$ . Thus $b (t)$ is an inverse in the displayed set, which is therefore an abelian group.

1.2constructalgebra

The isomorphism classes of finite locally free modules form a set: choose local trivializations on open subsets, and encode a bundle by a set-indexed open cover, finite ranks, and invertible transition matrices satisfying the overlap identities. The opens, covers and matrices form sets, and the usual gluing construction produces the represented sheaves. Take the free abelian group on these classes, realized as the finitely supported integer-valued functions on that set, and quotient by the subgroup generated by the exact-sequence relations. This gives the group in the statement. The split exact sequence $0 \to E \to E \oplus F \to F \to 0$ gives $[E \oplus F] = [E] + [F]$ , and the zero exact sequence gives $[0] = 0$ . Consequently, collecting positive and negative terms in any finite expression writes every element as $[E] - [F]$ , using the zero bundle for an empty sum.

2.1A1step 1.1step 1.2

By [A1] and step 1.1, sending the generator $[E]$ to $c_{t} (E)$ defines a homomorphism from the free abelian group to the target. For every defining exact-sequence relation its image is $c_{t} (E) c_{t} (E^{'})^{- 1} c_{t} (E^{''})^{- 1} = 1$ by the Whitney identity. It therefore descends to the quotient. Its value on $[E] - [F]$ is $c_{t} (E) c_{t} (F)^{- 1}$ . Since the classes $[E]$ generate the quotient, these prescribed values also make the extension unique.

3.1F1F2A1step 2.1

For $f : Y \to X$ , pullback sends finite locally free modules to finite locally free modules and preserves the exact sequences used in the presentation: locally the quotient term is free, so the sequence splits, and a split exact sequence remains exact after pullback. Thus $[E] \mapsto [f^{*} E]$ defines a homomorphism on the presented groups. Degree-preserving ring pullback acts coefficientwise on the target series and respects multiplication and inverses. The assumed naturality gives $f^{*} c_{t} (E) = c_{t} (f^{*} E)$ on generators, hence on every virtual class by step 2.1. For locally Noetherian $X$ , [F2] identifies this same presentation with the published $K^{0} (X)$ ; no locally Noetherian hypothesis is used in the preceding construction.

4.1step 1.1step 2.1algebra∎

Comparing the coefficient of $t^{2}$ in $c_{t} (x + y) = c_{t} (x) c_{t} (y)$ gives $c_{2} (x + y) = c_{2} (x) + c_{1} (x) c_{1} (y) + c_{2} (y)$ . Thus coefficientwise additivity does not follow from the group homomorphism: for example, in the formal series group over $Z [u]$ , the homomorphism $n \mapsto (1 + u t)^{n}$ has second coefficient zero at $1$ and $u^{2}$ at $2$ . This formal example illustrates the cross term; it makes no additional assertion about a particular cohomology theory. The construction and all claimed properties follow.

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