Tangent Lie algebra of derived Artin stacks

Benjamin Hennion

doi:10.1515/crelle-2015-0065

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Tangent Lie algebra of derived Artin stacks

Benjamin Hennion

IMAG, Université de Montpellier, Place Eugène Bataillon, 34095 Montpellier Cedex, France
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Published Online: 2015-12-10 | DOI: https://doi.org/10.1515/crelle-2015-0065

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Abstract

Since the work of Mikhail Kapranov in [Compos. Math. 115 (1999), no. 1, 71–113], it is known that the shifted tangent complex $𝕋_{X}$ [-1] of a smooth algebraic variety X is endowed with a weak Lie structure. Moreover, any complex of quasi-coherent sheaves E on X is endowed with a weak Lie action of this tangent Lie algebra. This Lie action is given by the Atiyah class of E. We will generalise this result to (finite enough) derived Artin stacks, without any smoothness assumption. This in particular applies to singular schemes. This work uses tools of both derived algebraic geometry and $\infty$ -category theory.

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About the article

Received: 2014-04-02

Revised: 2015-07-20

Published Online: 2015-12-10

Published in Print: 2018-08-01

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018, Issue 741, Pages 1–45, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2015-0065.

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