Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves

Details Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves

2009-04-08

arxiv.org/abs/0904.1301v3

Mathematics

Quantum Algebra

Several typos corrected. 22 pages, uses xy-pic

Scientific paper

We use the Thom-Whitney construction to show that infinitesimal deformations of a coherent sheaf F are controlled by the differential graded Lie algebra of global sections of an acyclic resolution of the sheaf End(E), where E is any locally free resolution of F. In particular, one recovers the well known fact that the tangent space to deformations of F is Ext^1(F,F), and obstructions are contained in Ext^2(F,F). The main tool is the identification of the deformation functor associated with the Thom-Whitney DGLA of a semicosimplicial DGLA whose cohomology is concentrated in nonnegative degrees with a noncommutative Cech cohomology-type functor.

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