An Introduction to Noncommutative Deformations of Modules
نویسنده
چکیده
Let k be an algebraically closed (commutative) field, let A be an associative k-algebra, and let M = {M1, . . . , Mp} be a finite family of left A-modules. We study the simultaneous formal deformations of this family, described by the noncommutative deformation functor DefM : ap → Sets introduced in Laudal [8]. In particular, we prove that this deformation functor has a pro-representing hull, and describe how to calculate this hull using the cohomology groups ExtnA(Mi, Mj) and their matric Massey products.
منابع مشابه
Computing Noncommutative Deformations of Presheaves and Sheaves of Modules
We describe a noncommutative deformation theory for presheaves and sheaves of modules that generalizes the commutative deformation theory of these global algebraic structures, and the noncommutative deformation theory of modules over algebras due to Laudal. In the first part of the paper, we describe a noncommutative deformation functor for presheaves of modules on a small category, and an obst...
متن کاملComputing Noncommutative Global Deformations of D-modules
Let (X,D) be a D-scheme in the sense of Beilinson and Bernstein, given by an algebraic variety X and a morphism OX → D of sheaves of rings on X. We consider noncommutative deformations of quasi-coherent sheaves of left D-modules on X, and show how to compute their pro-representing hulls. As an application, we compute the noncommutative deformations of the left DX -module OX when X is any ellipt...
متن کاملCategories of holomorphic line bundles on higher dimensional noncommutative complex tori
We construct explicitly noncommutative deformations of categories of holomorphic line bundles over higher dimensional tori. Our basic tools are Heisenberg modules over noncommutative tori and complex/holomorphic structures on them introduced by A. Schwarz. We obtain differential graded (DG) categories as full subcategories of curved DG categories of Heisenberg modules over the complex noncommut...
متن کاملThe Generalized Burnside Theorem in Noncommutative Deformation Theory
Let A be an associative algebra over a field k, and let M be a finite family of right A-modules. Study of the noncommutative deformation functor DefM : ar → Sets of the family M leads to the construction of the algebra O(M) of observables and the Generalized Burnside Theorem, due to Laudal [2]. In this paper, we give an overview of aspects of noncommutative deformations closely connected to the...
متن کاملNoncommutative Deformations of Sheaves and Presheaves of Modules
We work over an algebraically closed field k, and consider the noncommutative deformation functor DefF of a finite family F of presheaves of modules defined over a presheaf of k-algebras A on a small category c. We develop an obstruction theory for DefF , with certain global Hochschild cohomology groups as the natural cohomology. In particular, we show how to calculate the prorepresenting hull ...
متن کامل