# Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves

### Domenico Fiorenza

Università di Roma La Sapienza, Italy### Donatella Iacono

Bonn, Germany### Elena Martinengo

Università di Roma La Sapienza, Italy

## Abstract

We use the Thom-Whitney construction to show that infinitesimal deformations of a coherent sheaf ${\mathcal F}$ are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf $\Eps nd^*(\Eps^\cdot)$, where $\Eps^\cdot$ is any locally free resolution of ${\mathcal F}$. In particular, one recovers the well known fact that the tangent space to $\Def_{\mathcal F}$ is $\Ext^1({\mathcal F},{\mathcal F})$, and obstructions are contained in $\Ext^2({\mathcal F},{\mathcal F})$. The main tool is the identification of the deformation functor associated with the Thom-Whitney DGLA of a semicosimplicial DGLA ${\mathfrak g}^\Delta$, whose cohomology is concentrated in nonnegative degrees, with a noncommutative \v{C}ech cohomology-type functor $H^1_{\rm sc}(\exp {\mathfrak g}^\Delta)$.

## Cite this article

Domenico Fiorenza, Donatella Iacono, Elena Martinengo, Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves. J. Eur. Math. Soc. 14 (2012), no. 2, pp. 521–540

DOI 10.4171/JEMS/310