We show that the deformation functor of an orthogonal (resp. symplectic) sheaf over a smooth projective scheme admits miniversal pro-family, identifying its space first-order deformations with first hypercohomology complex which is naturally constructed out sheaf. also provide obstruction theory these objects whose target second this complex.

1 The Cotangent Complex: General Theory 4 1.1 Stable Envelopes and Tangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.2 Relative Adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 The Tangent Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.4 The Relative Cotangent Complex...

This is an expository paper written in 1999 and published in Seminari di Geometria Algebrica 1998-1999, Scuola Normale Superiore (1999). Six years later some arguments used here appear quite naive and obsolete but, in view of the several citations that this paper has obtained in the meantime, I preferred don’t change the mathematical contents and to fix only some typos and minor mistakes. For a...

In this lecture 1 I would like to address the following question: given an associative algebra A 0 , what are the possible ways to deform it? Consideration of this question for concrete algebras often leads to interesting mathematical discoveries. I will discuss several approaches to this question, and examples of applying them. 1. Deformation theory 1.1. Formal deformations. The most general a...

A deformation U , of a graded K-algebra A is said to be of PBW type if grU is A. It has been shown for Koszul and N -Koszul algebras that the deformation is PBW if and only if the relations of U satisfy a Jacobi type condition. In particular, for these algebras the determination of the PBW property is a finite and explicitly determined linear algebra problem. We extend these results to an arbit...

We study an algebraic deformation problem which captures the data of the general deformation problem for a quantum vertex algebra. We derive a system of coupled equations which is the counterpart of the Maurer-Cartan equation on the usual Hochschild complex of an associative algebra. We show that this system of equations results from an action principle. This might be the starting point for a p...

Abstract In this continuation of [4], we investigate the deformations holomorphic Cartan geometries where underlying complex manifold is allowed to move. The space infinitesimal a flat geometry computed. We show that natural forgetful map, from principal bundle on topological manifold, an isomorphism.

Quantization, at least in some formulations, involves replacing some algebra of observables by a (more non-commutative) deformed algebra. In view of the fundamental role played by K-theory in non-commutative geometry and topology, it is of interest to ask to what extent K-theory remains “rigid” under this process. We show that some positive results can be obtained using ideas of Gabber, Gillet-...

For every local ring R in what follows, mR denotes the maximal ideal. Let Λ̃ be a complete, regular, local Noetherian ring, let E ⊂ m Λ̃ be an ideal, and denote the quotient by Λ. Thus, Λ is also a complete, local Noetherian ring. Denote the residue field Λ/mΛ by k. Denote by C = CΛ the category whose objects are Λ-algebras A such that (i) A is a local, Artin ring with mΛA ⊂ mA, and (ii) the indu...