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Determinants and exact sequences

PDF p. 15: Trivial determinants of P and Q

Statement

Let $X$ be a scheme and let $0 \to E^{'} \overset{i}{\to} E \overset{p}{\to} E^{''} \to 0$ be a short exact sequence of vector bundles, meaning finite locally free $O_{X}$ -modules. There is a canonical isomorphism of line bundles

$\det E^{'} \otimes \det E^{''} \tilde{\to} \det E .$

It is natural for isomorphisms of short exact sequences and compatible with pullback along scheme morphisms. Ranks may be locally constant rather than constant: the determinant is defined on each open rank locus by the corresponding top exterior power.

Assume the Axiom of Choice and suppose $X$ is locally Noetherian. Then determinant induces a group homomorphism $K^{0} (X) \to Pic (X)$ from the vector-bundle Grothendieck group to the Picard group, given by $[E] - [F] \mapsto [\det E \otimes (\det F)^{- 1}]$ . Here $(\det F)^{- 1}$ is the tensor-inverse line bundle, and Picard group addition is tensor product. This assertion concerns $K^{0}$ , not the coherent-sheaf group $K_{0}$ .

Statement references

Exterior powers and determinant of an algebraic bundle

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

Facts & Assumptions

Given: A scheme $X$ and a short exact sequence of vector bundles as in the statement; for the Grothendieck-group assertion, also assume the Axiom of Choice and that $X$ is locally Noetherian.

Goal: Construct the canonical determinant isomorphism, prove its functorial properties, and show that determinant respects the defining relations of $K^{0} (X)$ .

[F1]

The surjection $E \to E^{''}$ splits locally. On a splitting neighbourhood, $E ≅ E^{'} \oplus E^{''}$ ; ranks add there. Kernels of surjections of vector bundles

[F2]

Exterior powers and determinant of a constant-rank bundle are defined by sheafified exterior algebra. For a free module with an ordered basis, increasing wedges form a basis of each exterior power; in particular the top wedge is a line and wedges above the rank vanish. Exterior powers commute naturally with pullback. The Picard group consists of line-bundle isomorphism classes under tensor product. Exterior powers and determinant of an algebraic bundle

[F3]

Under the stated choice and locally Noetherian hypotheses, $K^{0} (X)$ is generated by isomorphism classes of vector bundles with the relations $[E] = [E^{'}] + [E^{''}]$ for short exact sequences. Grothendieck groups of coherent sheaves and of vector bundles on a scheme

Proof

technique · direct; construct the map locally by wedging lifts and use the exact-sequence presentation

1.1givenF2construct

The rank of a vector bundle is locally constant because it is constant on each local trivialization. Its rank loci are disjoint open subsets covering $X$ . Define its determinant separately on these loci by [F2]; the pieces glue uniquely across the disjoint cover. The empty wedge defines the determinant of a rank-zero bundle as $O$ .

1.2F1F2construct

Work on an open set where the sequence splits and $E^{'}, E^{''}$ are free of ranks $a, b$ . For local sections $e_{1}^{'}, \dots, e_{a}^{'}$ and $e_{1}^{''}, \dots, e_{b}^{''}$ , choose lifts ${\tilde{e}}_{j}^{''}$ of the latter and send $(e_{1}^{'} \land \dots \land e_{a}^{'}) \otimes (e_{1}^{''} \land \dots \land e_{b}^{''})$ to $i (e_{1}^{'}) \land \dots \land i (e_{a}^{'}) \land {\tilde{e}}_{1}^{''} \land \dots \land {\tilde{e}}_{b}^{''}$ . This formula is multilinear and alternating in each block, so it induces a map from the tensor product of the two top exterior powers. If a lift changes, its difference is a section of $E^{'}$ ; every new term has $a + 1$ factors from $E^{'}$ and vanishes by [F2]. Thus the map is independent of the chosen lifts, including when either rank is zero.

2.1F1F2step 1.2construct

With bases of $E^{'}$ and $E^{''}$ and lifts supplied by a splitting, the concatenated vectors form a basis of $E$ . The map of step 1.2 sends the generator formed from the two top wedges to the top wedge of that basis, so it is an isomorphism. Its lift-independent formula agrees on overlaps; hence the local maps glue to the asserted global isomorphism. The formula is preserved by isomorphisms of short exact sequences. Pulling it back gives the same wedge formula for the pulled-back sequence, which remains locally split; the exterior-power pullback isomorphisms therefore identify the two determinant maps.

3.1F2F3step 2.1construct

Isomorphic bundles have isomorphic determinants by functoriality. Step 2.1 shows that every exact-sequence relation in [F3] maps to the identity relation in $Pic (X)$ . Consequently the assignment $[E] \mapsto [\det E]$ descends from the free abelian group on bundle classes to a homomorphism on $K^{0} (X)$ .

4.1F2F3step 3.1algebra∎

Every element of $K^{0} (X)$ is a difference $[E] - [F]$ : collect positive and negative terms into finite direct sums, using the split exact-sequence relations, and use the zero bundle for an empty sum. The homomorphism of step 3.1 therefore has the displayed formula, with inverse in the Picard group represented by the tensor-inverse line bundle. This completes the proof.

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