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Higher Hochschild cohomology vanishes for linearly reductive groups

Statement

Let $k$ be a field and let $G$ be a linearly reductive affine algebraic group over $k$ (Affine schemes and their coordinate rings): every finite-dimensional rational representation of $G$ is a direct sum of simple representations, equivalently $H^{1} (G, V) = 0$ for every finite-dimensional representation $V$ (Linear reductivity is equivalent to vanishing of first Hochschild cohomology). Then $H^{n} (G, V) = 0 for all n \geq 1$ and for every rational representation $V$ of $G$ , where $H^{∙}$ is Hochschild cohomology (Hochschild cohomology of algebraic groups and the classification of Hochschild extensions).

Facts & Assumptions

Given: A field $k$ , a linearly reductive affine algebraic group $G$ over $k$ , a rational representation $V$ of $G$ , and an integer $n \geq 1$ .

[F1]

Cohomology is computed from the Hochschild complex $C^{∙} (G, M)$ ; a short exact sequence of rational $G$ -modules induces a long exact sequence in cohomology through degreewise tensor exactness. (Hochschild cohomology of algebraic groups and the classification of Hochschild extensions)

[F2]

Every rational representation is the filtered union of its finite-dimensional subrepresentations, and Hochschild cohomology commutes with filtered colimits of coefficient modules: $H^{n} (G, {\underset{\to}{l i m}}_{i} V_{i}) = {\underset{\to}{l i m}}_{i} H^{n} (G, V_{i})$ for directed systems of subrepresentations. (Milne, Algebraic Groups, Section 15(e); the colimit statement is the standard exactness of filtered colimits applied degreewise to the rational cochain complex $C^{n} (G, V) = V \otimes_{k} O (G)^{\otimes n}$ : tensor products commute with filtered colimits, which are exact over a field.)

[F3]

Milne's Lemma 15.14: every class $x \in H^{n} (G, V)$ for finite-dimensional $V$ and $n \geq 1$ dies in $H^{n} (G, W)$ for some finite-dimensional representation $W$ containing $V$ . (Milne, Algebraic Groups, Lemma 15.14; the vanishing input is Shapiro's lemma for the trivial subgroup and acyclicity of free comodules.)

[F4]

$H^{1} (G, V) = 0$ for every finite-dimensional representation $V$ of a linearly reductive $G$ . (Linear reductivity is equivalent to vanishing of first Hochschild cohomology)

Proof

Given: A field $k$ , a linearly reductive affine algebraic group $G$ over $k$ , a rational representation $V$ , and $n \geq 1$ .

1.1F2

By [F2] the representation $V$ is the filtered union of its finite-dimensional subrepresentations $V_{i}$ , and $H^{n} (G, V) = {\underset{\to}{l i m}}_{i} H^{n} (G, V_{i})$ . It therefore suffices to prove $H^{n} (G, W) = 0$ for every finite-dimensional representation $W$ and every $n \geq 1$ ; fix such a $W$ .

2.1F1F3F4step 1.1∎

I prove $H^{n} (G, W) = 0$ by induction on $n \geq 1$ . For $n = 1$ this is [F4]. For $n \geq 2$ , let $x \in H^{n} (G, W)$ ; by [F3] there is a finite-dimensional representation $U$ containing $W$ such that $x$ maps to zero in $H^{n} (G, U)$ . The short exact sequence $0 \to W \to U \to U / W \to 0$ of finite-dimensional representations gives, by [F1], the exact sequence $H^{n - 1} (G, U / W) \overset{δ}{\to} H^{n} (G, W) \to H^{n} (G, U)$ , so $x = δ (y)$ for some $y \in H^{n - 1} (G, U / W)$ . By the induction hypothesis and [step 1.1] applied to the finite-dimensional representation $U / W$ , the group $H^{n - 1} (G, U / W)$ vanishes; hence $y = 0$ and $x = 0$ . Therefore $H^{n} (G, W) = 0$ for all finite-dimensional $W$ and all $n \geq 1$ , and by [step 1.1] the same holds for every rational representation.

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Sources

J. S. Milne, Algebraic Groups (corrected 2022 printing, Cambridge University Press) (standard reference, not scraped)
J. S. Milne, Algebraic Groups (v2.00, 20 December 2015 author-hosted preliminary edition) (standard reference, not scraped)