ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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koszul euler characteristic empty sequence

Example

Assume AC. For any finite-length module $M$ over a commutative Noetherian local ring $(R, m)$ , the empty sequence satisfies $χ (K (\emptyset; M)) = e_{0} (0, M) = ℓ_{R} (M)$ . For the concrete instance $M = R / m$ , both numbers are $1$ . For $M = 0$ , both are $0$ .

Facts & Assumptions

Given: AC, a commutative Noetherian local ring $(R, m)$ , a finite-length $R$ -module $M$ , and the empty sequence. The displayed special instances are $M = R / m$ and $M = 0$ .

[A1]

We assume The Axiom of Choice for the bridge theorem cited below.

[F1]

The empty sequence has complex $M [0]$ , ideal zero, and constant polynomial: koszul euler characteristic and degree indexed multiplicity.

[F2]

The coefficient indexed by the sequence length equals the Koszul Euler characteristic: degree-r Hilbert–Samuel coefficient as Koszul Euler characteristic.

Verification

technique · direct

1.1

The empty Koszul complex has $K_{0} = M$ , all other terms zero and all differentials zero. Thus $H_{0} = M$ and $H_{i} = 0$ for $i \neq 0$ , giving $χ (K) = ℓ_{R} (M)$ .

F1given

2.1

For every $n \geq 0$ , $0^{n + 1} M = 0$ , so $M / 0^{n + 1} M = M$ and $P (T) = ℓ_{R} (M)$ . Hence $e_{0} = 0! [T^{0}] P = ℓ_{R} (M)$ . A composition series also makes $M$ finitely generated: take a lift of one nonzero generator of each simple factor; induction through the finite series shows these finitely many lifts generate $M$ . The module-relative hypothesis is exactly finite length of $M$ , so the bridge theorem with $r = 0$ applies under AC and agrees with this direct calculation.

A1F1F2step 1.1

3.1

In particular take $M = k = R / m$ . Its only submodules are zero and $k$ , since any nonzero vector spans this one-dimensional $k$ -space; its length is $1$ . We obtain $H_{0} = k$ , $P = 1$ and $χ = e_{0} = 1$ . With $M = 0$ the empty series has length zero and the same calculation gives $P = χ = e_{0} = 0$ .

F1step 1.1step 2.1algebra∎

Remarks

This is a locally calculated design example, not a named example attributed to a source. The empty-sequence convention is consistent with Hochster printed p.166 (the zero-generator case) and with Stacks 43.15.4 at index zero. The general bridge is only a consistency check here; the homology and polynomial were calculated directly.

Depends on

Used by

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Dependency tree · two levels

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Sources

Stacks Project, 43.15.4–6; local proof with stated module-relative and coefficient conventions (standard reference, not scraped)
Hochster, Math 615 Winter 2012, pp.104–108: Euler characteristics and the multiplicity theorem (standard reference, not scraped)