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A rank-two bundle with a trivial quotient and determinant

PDF p. 15: Second exact sequence on V

Statement

Let $V$ be a scheme and $Q$ a rank-two vector bundle on $V$ with $\det Q ≅ O_{V}$ . If $Q ↠ O_{V}$ is a surjective morphism, its kernel is isomorphic to $O_{V}$ , so there is a short exact sequence $0 \to O_{V} \to Q \to O_{V} \to 0$ .

Assume also the Axiom of Choice and that $V$ is locally Noetherian. Then $[Q] = 2 [O_{V}]$ in the vector-bundle Grothendieck group $K^{0} (V)$ , whether or not the sequence splits globally. If in addition $P$ is a vector bundle in a short exact sequence $0 \to Q \to P \to O_{V} \to 0$ , then $[P] = 3 [O_{V}]$ in $K^{0} (V)$ .

Here a vector bundle of rank $r$ is an $O_{V}$ -module locally isomorphic to $O_{V}^{r}$ ; a line bundle is a rank-one vector bundle. The determinant of a rank- $r$ bundle is its top exterior power $Λ^{r} E$ , and a trivial determinant means an isomorphism with $O_{V}$ . A short exact sequence is exact at each term, equivalently on stalks. Under the stated Choice and locally Noetherian hypotheses, $K^{0} (V)$ is the abelian group generated by isomorphism classes $[E]$ of vector bundles, with $[E] = [E^{'}] + [E^{''}]$ for every short exact sequence $0 \to E^{'} \to E \to E^{''} \to 0$ .

Facts & Assumptions

Given: A scheme $V$ , a rank-two vector bundle $Q$ , an isomorphism $\det Q ≅ O_{V}$ , and a surjection $Q \to O_{V}$ . For the group assertions, also the Axiom of Choice and locally Noetherianity of $V$ ; for the final assertion, the displayed short exact sequence involving $P$ .

Goal: Identify the kernel with $O_{V}$ and prove the stated group identities under their additional hypotheses.

[F1]

The kernel of a surjection of finite locally free sheaves is finite locally free, with rank equal to the difference of ranks; the surjection splits locally. Kernels of surjections of vector bundles

[F2]

A short exact sequence $0 \to E^{'} \to E \to E^{''} \to 0$ of vector bundles has a canonical isomorphism $\det E^{'} \otimes \det E^{''} ≅ \det E$ . Determinants and exact sequences

[F3]

The determinant of a rank-one bundle $L$ is $Λ^{1} L$ , canonically $L$ ; in particular $\det O_{V} ≅ O_{V}$ . Tensoring an $O_{V}$ -module with $O_{V}$ gives that module canonically. Exterior powers and determinant of an algebraic bundle

[F4]

Under Choice and locally Noetherianity, isomorphic vector bundles have the same generator in $K^{0} (V)$ , and every short exact sequence of vector bundles gives the additive relation in the statement. Grothendieck groups of coherent sheaves and of vector bundles on a scheme The Axiom of Choice

Proof

technique · direct application of the determinant and exact-sequence relations

1.1givenF1

Let $L$ be the kernel of the given surjection. By [F1], $L$ is a vector bundle of rank $2 - 1 = 1$ , and $0 \to L \to Q \to O_{V} \to 0$ is short exact.

2.1givenF2F3step 1.1

By [F2] applied to that sequence, $\det L \otimes \det O_{V} ≅ \det Q$ . By [F3] and the given trivialization of $\det Q$ , this yields an isomorphism $L ≅ O_{V}$ . Substitution in the sequence of step 1.1 gives the asserted short exact sequence.

3.1givenF4step 2.1

Under the additional Choice and locally Noetherian hypotheses, [F4] applied to that sequence gives $[Q] = [L] + [O_{V}] = 2 [O_{V}]$ . This uses the exact-sequence relation and does not require a global splitting.

4.1givenF4step 3.1∎

If the additional sequence $0 \to Q \to P \to O_{V} \to 0$ is given, [F4] gives $[P] = [Q] + [O_{V}] = 3 [O_{V}]$ . Thus all conclusions hold under their stated hypotheses.

Depends on

Used by · 1

Curve-and-point support reduction

Dependency tree · 3 levels

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