Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06
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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 Complex Concatenation Tensor Isomorphism

Statement

For finite sequences x,y, the graded tensor-product identification gives a signed chain isomorphism K(x,y;M)≅K(x;R)⊗RK(y;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 Multiplication Koszul Sign Rule.

Proof

technique · direct
1.1

The exterior algebra of the direct sum of the two based free modules is the graded tensor product of their exterior algebras.

givenalgebra
2.1

Its total differential d⊗1+(−1)deg⁡1⊗d agrees termwise with the Koszul deletion differential.

step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

6 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