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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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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
Published electronically: April 6, 2022


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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Bibliographic Information
  • Benjamin Hennion
  • Affiliation: Université Paris-Saclay, Laboratoire de mathématiques d’Orsay, 91405 Orsay, France
  • MR Author ID: 1219478
  • Email:
  • Mikhail Kapranov
  • Affiliation: Kavli IPMU, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8583, Japan
  • MR Author ID: 200368
  • Email:
  • Received by editor(s): April 4, 2019
  • Received by editor(s) in revised form: September 30, 2021
  • Published electronically: April 6, 2022
  • Additional Notes: The research of the second author was supported by World Premier International Reseach Center (WPI Initiative), MEXT, Japan and by the IAS School of Mathematics.
  • © Copyright 2022 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 36 (2023), 311-396
  • MSC (2020): Primary 14F10, 17B56
  • DOI:
  • MathSciNet review: 4536901