LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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Idempotents lift through finite commutative algebra quotients

Statement

For a finite-dimensional commutative unital $k$ -algebra $A$ and any ideal $I$ , every idempotent of $A / I$ lifts to an idempotent of $A$ .

Facts & Assumptions

Given: The stated algebra, ideal, and ${\overset{ˉ}{e}}^{2} = \overset{ˉ}{e}$ in $A / I$ .

[F1]

The algebra and its quotient decompose into local factors, with zero quotient factors allowed. (Finite-dimensional commutative algebras decompose into local factors)

Proof

technique · direct

1.1

Use [F1] to write $A = \prod A_{i}$ and $A / I = \prod A_{i} / I_{i}$ . In a nonzero local quotient, $x$ and $1 - x$ cannot both be nonunits: their sum is $1$ and nonunits form its maximal ideal. For an idempotent $x$ , $x (1 - x) = 0$ , so if $x$ is a unit then $x = 1$ , and if $1 - x$ is a unit then $x = 0$ . Thus each nonzero quotient coordinate of $\overset{ˉ}{e}$ is $0$ or $1$ .

2.1

In each factor $A_{i}$ choose the same coordinate $0$ or $1$ , choosing $0$ for a zero quotient factor. Their finite tuple $e$ satisfies $e^{2} = e$ coordinatewise and maps to $\overset{ˉ}{e}$ . If $A = 0$ , the empty tuple is its sole idempotent and is already a lift.

F1step 1.1∎

Sources

Jacobsen, Block fusion systems and the center of the group ring, Lemma 2.32 and Theorem 2.33, pp.18–19; general-field lifting proved locally. Local argument and conventions as displayed above.

Depends on

Finite-dimensional commutative algebras decompose into local factors

Used by

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Sources

Jacobsen, Block fusion systems and the center of the group ring, Lemma 2.32 and Theorem 2.33, pp.18–19; general-field lifting proved locally (standard reference, not scraped)