LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Koszul Differential Coordinate Formula

Statement

For $I = (i_{1} < \dots < i_{p})$ , $d (e_{I} \otimes m) = \sum_{j = 1}^{p} (- 1)^{j - 1} e_{I ∖ i_{j}} \otimes x_{i_{j}} m$ .

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Complex Of A Sequence With Coefficients, Exterior Algebra Basis Monomials.

Proof

technique · direct

1.1

Apply the graded Leibniz rule to the ordered wedge and use $d (e_{i_{j}}) = x_{i_{j}}$ .

givenalgebra

2.1

Passing the differential through $j - 1$ degree-one factors gives $(- 1)^{j - 1}$ , yielding the formula.

step 1.1algebra∎

Depends on

Used by

Koszul Complex One And Two Elements Example
Koszul D Square Sign Check Three Elements Example
Koszul Differential Is Well Defined And Squares To Zero Lemma
Koszul Differential Square Pairwise Cancellation Lemma
Koszul Generator Matrix Chain Map Lemma
Koszul Resolution Minimality Maximal Ideal Sequence Lemma
Basic Koszul Homology Theorem

Dependency tree · two levels

5 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

The Stacks Project, Koszul complexes and regular sequences (standard reference, not scraped)