Geodesic
Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves
Journal of the European Mathematical Society, Tome 14 (2012) no. 2, pp. 521-540

Voir la notice de l'article provenant de la source EMS Press

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 DefF​ is Ext1(F,F), and obstructions are contained in Ext2(F,F). The main tool is the identification of the deformation functor associated with the Thom–Whitney DGLA of a semicosimplicial DGLA gΔ, whose cohomology is concentrated in nonnegative degrees, with a noncommutative \v{C}ech cohomology-type functor Hsc1​(expgΔ).
DOI : 10.4171/jems/310
Classification : 13-XX, 17-XX, 18-XX, 00-XX
Keywords: Differential graded Lie algebras, functors of Artin rings 
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     author = {Domenico Fiorenza and Donatella Iacono and Elena Martinengo},
     title = {Differential graded {Lie} algebras controlling infinitesimal deformations of coherent sheaves},
     journal = {Journal of the European Mathematical Society},
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     publisher = {mathdoc},
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Domenico Fiorenza; Donatella Iacono; Elena Martinengo. Differential graded Lie algebras controlling infinitesimal deformations of coherent sheaves. Journal of the European Mathematical Society, Tome 14 (2012) no. 2, pp. 521-540. doi: 10.4171/jems/310

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