Cohomology and Obstructions: Commutative Algebras
نویسندگان
چکیده
Associated with each of the classical cohomology theories in algebra has been a theory relating H (H as classically numbered) to obstructions to non-singular extensions and H with coefficients in a “center” to the non-singular extension theory (see [Eilenberg & MacLane (1947), Hochschild (1947), Hochschild (1954), MacLane (1958), Shukla (1961), Harrison (1962)]). In this paper we carry out the entire process using triple cohomology. Because of the special constructions which arise, we do not know how to do this in any generality. Here we restrict attention to the category of commutative (associative) algebras. It will be clear how to make the theory work for groups, associative algebras and Lie algebras. My student, Grace Orzech, is studying more general situations at present. I would like to thank her for her careful reading of the first draft of this paper. The triple cohomology is described at length elsewhere in this volume [Barr & Beck (1969)]. We use the adjoint pair
منابع مشابه
e–Quillen cohomology of commutative S-algebras
We de ne topological Andr e–Quillen cohomology of commutative S-algebras and construct a spectral sequence that calculates the Andr e–Quillen cohomology of commutative S-algebras over HFp. c © 1999 Elsevier Science B.V. All rights reserved.
متن کاملInfinitesimal deformation quantization of complex analytic spaces
For the physical aspects of deformation quantization we refer to the expository and survey papers [4] and [10]. Our objective is to initiate a version of global theory of quantization deformation in the category of complex analytic spaces in the same lines as the theory of (commutative) deformation. The goal of inifinitesimal theory is to do few steps towards construction of a star-product in t...
متن کاملMassey Products and Deformations
It is common knowledge that the construction of one-parameter deformations of various algebraic structures, like associative algebras or Lie algebras, involves certain conditions on cohomology classes, and that these conditions are usually expressed in terms of Massey products, or rather Massey powers. The cohomology classes considered are those of certain differential graded Lie algebras (DGLA...
متن کاملTriple Cohomology of Lie–Rinehart Algebras and the Canonical Class of Associative Algebras
We introduce a bicomplex which computes the triple cohomology of Lie– Rinehart algebras. We prove that the triple cohomology is isomorphic to the Rinehart cohomology [13] provided the Lie–Rinehart algebra is projective over the corresponding commutative algebra. As an application we construct a canonical class in the third dimensional cohomology corresponding to an associative algebra.
متن کاملOn some Hochschild cohomology classes of fusion algebras
The obstructions for an arbitrary fusion algebra to be a fusion algebra of some semisimple monoidal category are constructed. Those obstructions lie in groups which are closely related to the Hochschild cohomology of fusion algebras with coefficients in the K-theory of the ground (algebraically closed) field. The special attention is devoted to the case of fusion algebra of invariants of finite...
متن کامل