Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-23 (gpt-6-sol)
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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Completion reflects depth

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let (R,m) be Noetherian local and M finite. Then depth⁡R^(M⊗RR^)=depth⁡R(M).

Facts & Assumptions

Given: The Axiom of Choice; the residue fields of R and R^ agree, and completion is faithfully flat.

Proof

technique · direct
1.1given

For each i, flat base change for a free resolution of R/m whose terms are finite free gives Ext⁡Ri(R/m,M)⊗RR^≅Ext⁡R^i(R^/mR^,M^).

2.1step 1.1algebra∎

Faithful flatness says the left group vanishes exactly when its tensor product does. Thus the first nonzero Ext degree is unchanged; the Ext characterization of depth proves the equality, including the +∞ case.

Depends on

Used by

Dependency tree · two levels

14 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