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Very specific question. We work over $\mathbb{C}$, although really just want alg. closed of char. 0.

Suppose that $G$ is an algebraic group and $V$ is a finite-dimensional $G$-module, meaning that we have a comodule map $$ a:V\to\mathbb{C}[G]\otimes V. $$ Let $v\in V$ be a non-zero vector; then we may choose a basis $v_1=v, v_2,\dots,v_n$ of $V$, and write $$ a(v_1)=\sum\limits_if_i\otimes v_i. $$ Now it is clear that the ideal $(f_1-1,f_2,\dots,f_n)$ cuts out exactly the subgroup of $G$ given by the stabilizer of $v$ in $G$.

My question is as follows: how could I go about computing the cohomology of the Koszul complex defined by the elements $f_1-1,f_2,\dots,f_n$ in $\mathbb{C}[G]$?

Edit: I should add that I have an idea of what the answer 'should' be: let $\mathfrak{g}$ denote the Lie algebra of $G$, and write $H$ for the stabilizer of $v$ in $G$. Write $W\subseteq V$ for the subspace $\mathfrak{g}\cdot v$. Then this is a representation of $H$, so $V/W$ in particular is also a representation of $H$. I expect that the cohomology of the Koszul complex is isomorphic to the ring: $$ \mathbb{C}[H]\otimes \bigwedge(V/W) $$ Here I'm viewing the Koszul complex as an operator on the supercommutative superalgebra $\mathbb{C}[G]\otimes\bigwedge V$, where $f_i$ 'corresponds to' the basis vector $v_i\in V$.

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See arXiv:1411.7994, Proposition 1.28 for a description of the cohomology dg-algebra of a Koszul complex in the case when the zero locus has codimension smaller than the number of equations.

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  • $\begingroup$ Thank you this is great! $\endgroup$ Commented Jan 23, 2022 at 10:51

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