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Products of Green coproducts converge coefficientwise

Statement

For any $M, N$ , the product $Δ^{'} ([M]) \cdot Δ^{'} ([N])$ in the twisted tensor multiplication of The Green Hall completion and coproduct has a finite coefficient sum at every fixed basis pair $[R] \otimes [S]$ . Thus the product is defined in the coefficientwise completion. The claim does not extend to arbitrary pairs of completed tensors.

Facts & Assumptions

Given: Objects $M, N, R, S$ in a finitary abelian category.

[F1]

Every nonzero Hall coefficient is witnessed by a short exact sequence with the stated quotient and subobject, and Hom sets are finite (Small finitary abelian categories for Hall multiplication).

[F2]

A product coefficient can be nonzero only for exact sequences $0 \to L_{1} \to M \to K_{1} \to 0$ , $0 \to L_{2} \to N \to K_{2} \to 0$ , $0 \to K_{2} \to R \to K_{1} \to 0$ , and $0 \to L_{2} \to S \to L_{1} \to 0$ (The Green Hall completion and coproduct).

Proof

technique · direct

1.1F1F2construct

Fix $S, R$ . A nonzero contribution to their coefficient determines the four exact sequences in [F2]. Compose $S ↠ L_{1}$ with the monomorphism $L_{1} ↪ M$ ; its image is $L_{1}$ because the first map is epic and the second is monic. Thus the isomorphism class $[L_{1}]$ occurs as the image of a morphism $S \to M$ . There are finitely many such morphisms by [F1], hence finitely many possible $[L_{1}]$ .

2.1F1F2step 1.1construct

Compose $N ↠ K_{2}$ with $K_{2} ↪ R$ . Its image is $K_{2}$ , and its kernel is $L_{2}$ , so finite $Hom (N, R)$ gives only finitely many possible pairs $(K_{2}, L_{2})$ . The finite set $Hom (S, M)$ similarly gives only finitely many pairs $(L_{1}, K_{1})$ from the image and cokernel of $S ↠ L_{1} ↪ M$ . Hence only finitely many quadruples $(K_{1}, L_{1}, K_{2}, L_{2})$ contribute.

3.1step 2.1algebra∎

Each quadruple contributes one scalar, finite Hall coefficients and a finite sum by step 2.1. Therefore the coefficient of $[R] \otimes [S]$ is a finite sum, for every output pair.

Depends on

Used by

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15 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)
Matthew B. Young, Degenerate versions of Green’s theorem for Hall modules, arXiv:1810.08238v2 (standard reference, not scraped)