LemmaStatement: AI-generatedProof: AI-generated

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.

Length multiplicity for one proper Cartier face

PDF p. 4: Smooth comparison with Bloch cycles

Statement

Let $(A, m)$ be a noetherian commutative local ring, $t \in m$ a nonzerodivisor on $A$ , and $I$ a prime ideal with $t \notin I$ and $\sqrt{I + (t)} = m$ . Put $M = A / I$ . For the free resolution $Q = (0 \to A \overset{t}{\to} A \to 0)$ of $A / (t)$ , the homology of $Q \otimes_{A} M$ is $A / (I, t)$ in degree zero and zero in positive degrees. The nonzero module $A / (I, t)$ has finite length. Hence the alternating sum of these homology lengths is $ℓ_{A} (A / (I, t))$ . This length is unchanged by an isomorphism of local rings carrying the ideals to their counterparts, or by replacing $t$ by a unit multiple.

Facts & Assumptions

Given: The ring, element and prime ideal in the Statement.

Goal: Compute the specified-resolution Tor groups and their length Euler characteristic.

[F1]

Specified-resolution Tor is the homology of the chosen projective resolution tensored with the other module. Tor from a projective resolution of the right module

[F2]

Length is the number of simple factors in a composition series, independent of the series. Composition series and length of a module

Proof

1.1givenF1algebra

Since multiplication by $t$ is injective on $A$ , $Q$ is a free resolution of $A / (t)$ . Tensoring with $M$ gives $0 \to M \overset{t}{\to} M \to 0$ . The quotient $M$ is a domain and the image of $t$ is nonzero, so the map is injective and its cokernel is $A / (I, t)$ . This proves the homology computation.

1.2givenconstructalgebra

Let $J = I + (t)$ and choose generators $x_{1}, \dots, x_{s}$ of $m$ . Since $\sqrt{J} = m$ , choose positive $e_{a}$ with $x_{a}^{e_{a}} \in J$ . Every monomial of total degree $N = 1 + \sum_{a} (e_{a} - 1)$ contains some $x_{a}^{e_{a}}$ , so $m^{N} \subset J$ ; if $s = 0$ , take $N = 1$ .

2.1F2step 1.1step 1.2construct

Filter $A / J$ by the images of the powers of $m$ through $m^{N}$ . Each factor is finitely generated and killed by $m$ , hence is a finite-dimensional vector space over $A / m$ . Refining by basis flags gives a finite composition series. Also $J \subset m$ , so $A / J \neq 0$ . The Euler characteristic is therefore the stated length.

3.1F2step 2.1algebra∎

A local-ring isomorphism carries composition series to composition series, and a unit multiple of $t$ generates the same ideal. The length is consequently unchanged, completing the proof.

Depends on

Used by · 1

Admissible face restrictions and their cycle identities

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.