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Finite projective complements

PDF p. 12: Critical rank representative construction

Statement

Let $A$ be a commutative ring with unit. Every finite projective $A$ -module $P$ admits a finite projective $A$ -module $Q$ and an integer $N \geq 0$ such that $P \oplus Q ≅ A^{N}$ .

Assume additionally the Axiom of Choice and that $X = Spec A$ is locally noetherian. Every class in the vector-bundle Grothendieck group $K^{0} (X)$ is of the form $[E] - [O_{X}^{N}]$ for a vector bundle $E$ and an integer $N \geq 0$ . Its rank at each point $x \in X$ is ${rank}_{x} E - N$ ; this is a locally constant integer-valued function, hence is constant on each connected component.

A finite projective module is a finitely generated module $P$ such that every homomorphism from $P$ to the target of a surjective module homomorphism lifts through that surjection. A vector bundle is a finite locally free $O_{X}$ -module: locally it is isomorphic to a finite direct sum of copies of $O_{X}$ . Its rank at a point is the number of copies in a local trivialization containing that point. A locally noetherian scheme is one covered by spectra of noetherian rings. The group $K^{0} (X)$ is generated by isomorphism classes $[E]$ of vector bundles, with $[E] = [E^{'}] + [E^{''}]$ for each short exact sequence $0 \to E^{'} \to E \to E^{''} \to 0$ of vector bundles.

Facts & Assumptions

Given: A commutative ring $A$ with unit and a finite projective $A$ -module $P$ . For the assertions about $K^{0}$ , also assume the Axiom of Choice and that $X = Spec A$ is locally noetherian.

Goal: Construct a finite projective complement to $P$ in a finite free module, then prove the representative and rank assertions.

[A1]

The Axiom of Choice is used only to invoke the bundle and Grothendieck-group suppliers [F1]–[F2]; steps 1.1–3.1 use finite choices only. The Axiom of Choice

[F1]

Under the Axiom of Choice, associated sheaf and global sections give an equivalence between finite projective $A$ -modules and vector bundles on $X$ , preserving finite direct sums and ranks. In particular $\tilde{A^{N}} ≅ O_{X}^{N}$ . Affine bundles and finite projective modules

[F2]

For a locally noetherian scheme under the Axiom of Choice, $K^{0} (X)$ has the generators and exact-sequence relations stated above. Grothendieck groups of coherent sheaves and of vector bundles on a scheme

Proof

technique · Split a finite free presentation and use complements to replace negative bundle classes by negative free classes

1.1givenconstruct

Choose finitely many generators $p_{1}, \dots, p_{N}$ of $P$ and let $q : A^{N} \to P$ send the standard basis vectors to them. This map is surjective. By the lifting property of $P$ , the identity map of $P$ lifts to a homomorphism $s : P \to A^{N}$ with $q s = {id}_{P}$ . This construction uses only the given finite generating family, including the empty family when $P = 0$ .

2.1step 1.1construct

Put $Q = \ker q$ . The map $P \oplus Q \to A^{N}$ , $(p, z) \mapsto s (p) + z$ , is an isomorphism: its inverse sends $a$ to $(q (a), a - s q (a))$ . The projection $e = {id}_{A^{N}} - s q$ has image $Q$ , so $Q$ is generated by the finitely many elements $e (e_{i})$ , where $e_{i}$ are the standard basis vectors.

3.1step 2.1construct

The module $A^{N}$ has the lifting property: lift the images of its finitely many basis vectors separately through any given surjection. A direct summand of a module with this property also has it: compose a map from the summand with the projection from the ambient module, lift, and restrict the lift to the summand. By step 2.1, $Q$ is such a summand of $A^{N}$ and is finitely generated. Thus $Q$ is finite projective, proving the first assertion.

4.1F2assume-hypstep 3.1

Now impose the additional hypotheses. Every element of $K^{0} (X)$ is a finite integer linear combination of bundle classes by [F2]. Collecting positive and negative terms using the relation for the split exact sequence $0 \to E^{'} \to E^{'} \oplus E^{''} \to E^{''} \to 0$ , write it as $[B] - [C]$ for vector bundles $B, C$ ; the zero bundle handles an empty sum.

5.1F1F2step 3.1step 4.1

By [F1], $C ≅ \tilde{P}$ for a finite projective module $P$ . Apply steps 1.1–3.1 to obtain $P \oplus Q ≅ A^{N}$ . With $D = \tilde{Q}$ , [F1] gives $C \oplus D ≅ O_{X}^{N}$ . Therefore the split exact-sequence relation gives $[B] - [C] = [B \oplus D] - [C \oplus D] = [B \oplus D] - [O_{X}^{N}] .$ Taking $E = B \oplus D$ proves the representative assertion.

6.1F2step 5.1construct

For a point $x$ , rank is additive in every short exact sequence of vector bundles: after passing to the stalk and tensoring with the residue field, the sequence becomes a short exact sequence of finite-dimensional vector spaces, since the quotient stalk is free; dimensions are additive. Thus assigning a bundle its pointwise rank respects the relations in [F2] and defines a homomorphism from $K^{0} (X)$ to integer-valued functions on $X$ . Local triviality makes every such rank function locally constant. Consequently the representative in step 5.1 has rank ${rank}_{x} E - N$ at $x$ . A locally constant function to the discrete space $Z$ is constant on each connected component, since otherwise its distinct nonempty fibers would separate that component.

7.1step 3.1step 5.1step 6.1∎

Steps 3.1, 5.1 and 6.1 establish all the assertions under their stated hypotheses.

Depends on

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