TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Support of an arbitrary direct sum is the union of the supports

Statement

For any family $(M_{i})_{i \in I}$ of left $R$ -modules, ${Supp}_{R} ⁣ (⨁_{i \in I} M_{i}) = ⋃_{i \in I} {Supp}_{R} (M_{i}) .$

Facts & Assumptions

Given: A commutative ring $R$ and a family $(M_{i})_{i \in I}$ of left $R$ -modules.

[L1]

A prime ideal $p$ lies in the support of a module exactly when the localisation at $p$ is nonzero (Support of a module).

[L2]

Localisation commutes with arbitrary direct sums (Localisation commutes with quotient modules and arbitrary direct sums).

Proof

technique · direct

1.1L2

Fix a prime ideal $p$ . By [L2], ${(⨁_{i} M_{i})}_{p} ≅ ⨁_{i} (M_{i})_{p}$ . This direct sum is nonzero exactly when at least one summand $(M_{i})_{p}$ is nonzero.

2.1L1step 1.1

By [L1], step 1.1 says exactly that $p$ lies in the support of $⨁_{i} M_{i}$ if and only if it lies in the support of some $M_{i}$ .

3.1step 2.1∎

Since this holds for every prime ideal $p$ , the support of the direct sum is the union of the supports.

Depends on

Used by

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Section 13 (standard reference, not scraped)
The Stacks Project, Section 10.40: Support (standard reference, not scraped)