LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pending

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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Exact-sequence counts equal endpoint automorphism counts times subobject counts

Statement

Let $A$ be a small finitary abelian category and let $M, N, R \in A$ . Let $P_{M, N}^{R}$ be the number of endpoint-labelled short exact sequences $0 \to N \to R \to M \to 0$ , and let $a_{X} = ∣ Aut (X) ∣$ . Then $P_{M, N}^{R} = a_{M} a_{N} g_{M, N}^{R},$ where $g_{M, N}^{R}$ counts subobjects $N ↪ R$ with quotient isomorphic to $M$ as in The untwisted Hall algebra of a finitary abelian category.

Facts & Assumptions

Given: A finitary abelian category and objects $M, N, R$ .

[F1]

$g_{M, N}^{R}$ counts subobject classes represented by monomorphisms $N ↪ R$ whose cokernel is isomorphic to $M$ (The untwisted Hall algebra of a finitary abelian category).

[F2]

Hom sets are finite, and a short exact sequence has a monomorphism and epimorphism with the indicated kernel and cokernel (Small finitary abelian categories for Hall multiplication, Exact sequence and short exact sequence in an abelian category).

Proof

technique · direct

1.1F1F2construct

Fix a subobject class counted by [F1], represented by $i : N \to R$ . Its quotient is a quotient object isomorphic to $M$ . The number of isomorphisms $N \to im (i)$ is $a_{N}$ , and the number of isomorphisms $R / im (i) \to M$ is $a_{M}$ . Each pair of isomorphisms determines exactly one endpoint-labelled short exact sequence with this subobject; every such sequence arises from that pair.

2.1step 1.1F1algebra

Distinct subobject classes give disjoint sets of labelled sequences, because the image of the first map recovers the subobject class. Therefore each counted subobject contributes exactly $a_{N} a_{M}$ sequences, and summing over the $g_{M, N}^{R}$ classes proves the formula.

3.1F2step 2.1∎

The total set of labelled sequences is a subset of $Hom (N, R) \times Hom (R, M)$ and hence finite by [F2]. The count $P_{M, N}^{R}$ is therefore a well-defined natural number.

Depends on

Used by

The extended Hall algebra and composition subalgebra Definition
The Green Hall completion and coproduct Definition
A one-vertex Green coproduct and twisted Hall square Example
Green coproduct coefficients are finite Lemma
The Green Hall coproduct is coassociative Lemma
The Green Hall pairing is nondegenerate and adjoint to product and coproduct Proposition
Green’s coproduct is an algebra map into the twisted tensor completion Theorem

Dependency tree · two levels

16 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

Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2 (standard reference, not scraped)
Claus Michael Ringel, The Hall Algebra Approach to Quantum Groups, E.L.A.M. Lectures (1993) (standard reference, not scraped)