Gel′fand-Fuchs cohomology in algebraic geometry and factorization algebras

Hennion, Benjamin; Kapranov, Mikhail

doi:10.1090/jams/1001

Gel′fand-Fuchs cohomology in algebraic geometry and factorization algebras
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by Benjamin Hennion and Mikhail Kapranov;

J. Amer. Math. Soc. 36 (2023), 311-396

DOI: https://doi.org/10.1090/jams/1001

Published electronically: April 6, 2022

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Abstract:

Let $X$ be a smooth affine variety over a field $\mathbf k$ of characteristic $0$ and $T(X)$ be the Lie algebra of regular vector fields on $X$. We compute the Lie algebra cohomology of $T(X)$ with coefficients in $\mathbf k$. The answer is given in topological terms relative to any embedding $\mathbf k\subset \mathbb {C}$ and is analogous to the classical Gel′fand-Fuchs computation for smooth vector fields on a $C^\infty$-manifold. Unlike the $C^\infty$-case, our setup is purely algebraic: no topology on $T(X)$ is present. The proof is based on the techniques of factorization algebras, both in algebro-geometric and topological contexts.

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