Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves
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 $\Ext^2({\mathcal F},{\mathcal F})$. \par 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 $H^1_{ m sc}(\exp {\mathfrak g}^\Delta)$.
Anno di pubblicazione
2012
ISSN
1435-9855
ISBN
Non Disponibile
Numero di citazioni Wos
5
Ultimo Aggiornamento Citazioni
Non Disponibile
Numero di citazioni Scopus
5
Ultimo Aggiornamento Citazioni
Non Disponibile
Settori ERC
Non Disponibile
Codici ASJC
Non Disponibile
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