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Serre free-summand theorem for modules

PDF p. 1: Serre splitting theorem

Statement

Let $A$ be a commutative unital ring whose maximal spectrum $Ω$ , with its subspace topology, is noetherian of finite dimension $e$ . Let $M$ be a finitely presented $A$ -module. Suppose that for every maximal ideal $m$ , the localization $M_{m}$ has a free direct summand of rank greater than $e$ . Then $M ≅ A \oplus M^{'}$ for some $A$ -module $M^{'}$ . The dimension of a topological space is the supremum of the lengths of strict chains of irreducible closed subsets.

External input

This is Theorem 15.129.5 of the cited Stacks Project section. It is imported as an established result; its underlying dimension-avoidance argument is not reconstructed in this item.

Facts & Assumptions

Given: A ring $A$ , maximal spectrum $Ω$ , integer $e$ , and module $M$ satisfying the hypotheses in the statement.

Goal: Obtain a free rank-one direct summand of $M$ .

[F1]

The cited external theorem asserts exactly this conclusion under the displayed hypotheses.

Proof

1.1F1given

The given noetherianity, finite dimension, finite presentation, and local free-summand bounds are precisely the hypotheses of the imported theorem.

2.1F1step 1.1∎

Applying that theorem yields an $A$ -module $M^{'}$ and an isomorphism $M ≅ A \oplus M^{'}$ . This is an application of the external input, not a local reconstruction of its proof.

Depends on

The Stacks Project, More on Algebra, Section 15.129: Splitting off a free module · Theorem 15.129.5 (tag 0GVC), statement and displayed proof

Used by · 1

Serre splitting above the dimension

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

The Stacks Project, More on Algebra, Section 15.129: Splitting off a free module · Theorem 15.129.5 (tag 0GVC), statement and displayed proof

If the maximal spectrum of a ring is noetherian of finite dimension e and a finitely presented module has, at every maximal ideal, a free localized direct summand of rank greater than e, then the module is isomorphic to the ring plus a complementary module.

Hypotheses: Noetherian finite-dimensional maximal spectrum; finite presentation; the stated localized free-summand rank bound at every maximal ideal.

Reading limits: The theorem statement and its short displayed proof were read. The proof invokes Proposition 15.129.4 and other lemmas; their proofs were not reconstructed here. The original Serre paper was not read.

Sources

The Stacks Project, More on Algebra, Section 15.129: Splitting off a free module · Theorem 15.129.5 (tag 0GVC), statement and displayed proof