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The Green Hall pairing is nondegenerate and adjoint to product and coproduct

Statement

On the direct-sum Hall space define $([M], [N])_{G} = \frac{δ_{[M], [N]}}{a_{M}} .$ This bilinear form is nondegenerate over $Q (ν)$ and satisfies $([M] ⋆ [N], [R])_{G} = ([M] \otimes [N], Δ^{'} ([R]))_{G \otimes G},$ where the tensor pairing is the product of the two Green pairings.

Facts & Assumptions

Given: A finite-field hereditary Hall category and the Hall basis.

[F1]

$a_{M}$ is a finite positive integer and the Green coproduct has coefficient $ν^{χ_{0} (M, N)} P_{M, N}^{R} / a_{R}$ (The Green Hall completion and coproduct).

[F2]

$P_{M, N}^{R} = a_{M} a_{N} g_{M, N}^{R}$ (Exact-sequence counts equal endpoint automorphism counts times subobject counts).

Proof

technique · evaluate on basis vectors

1.1F1algebra

In the Hall basis, the pairing matrix is diagonal with entries $1 / a_{M}$ . Each is nonzero in $Q (ν)$ by [F1], so the pairing is nondegenerate: if $x = \sum_{M} c_{M} [M]$ pairs to zero with every $[M]$ , then $c_{M} / a_{M} = 0$ for every coefficient, hence every $c_{M} = 0$ .

2.1F1F2algebra∎

The left side of the adjointness identity is $ν^{χ_{0} (M, N)} g_{M, N}^{R} / a_{R}$ . The right side is $ν^{χ_{0} (M, N)} P_{M, N}^{R} / (a_{R} a_{M} a_{N})$ , which is the same scalar by [F2]. Thus the identity holds on basis vectors and extends bilinearly.

Depends on

Used by

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Sources

Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2 (standard reference, not scraped)
Claus Michael Ringel, Hall algebras and quantum groups, Inventiones Mathematicae 101 (1990), 583–591 (standard reference, not scraped)