Tangent Lie algebra of derived Artin stacks

Abstract

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 \mathbb{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.