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Title Secondary Hochschild and Cyclic (Co)homologies
Publication Date
Degree PhD
Discipline/Department Mathematics
Degree Level doctoral
University/Publisher Bowling Green State University
Abstract Hochschild cohomology was originally introduced in 1945. Much more recently in 2013 a generalization of this theory, the secondary Hochschild cohomology, was brought to light. In this dissertation we provide the details behind the simplicial structure for the chain complexes associated to the (secondary) Hochschild (co)homology. For this we introduce the notion of simplicial algebras and simplicial modules. The key results are two lemmas (3.4.1 and 3.4.2) that can be thought of as analogues of the Tor and Ext functors in the context of simplicial modules. It was a pleasant surprise that the higher order Hochschild homology over the 2-sphere can also be described using simplicial structures. We study some other related concepts like the secondary Hochschild and cyclic homologies associated to the triple (A,B,ε), as well as some of their properties.
Subjects/Keywords Mathematics; homological algebra; deformation theory; associative rings and algebras; Hochschild cohomology; cyclic cohomology
Contributors Staic, Mihai D. (Advisor)
Language en
Rights unrestricted ; This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws.
Country of Publication us
Format application/pdf
Record ID oai:etd.ohiolink.edu:bgsu1489422065908758
Repository ohiolink
Date Indexed 2020-10-19
Grantor Bowling Green State University

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…x5B;28], [31], and [42]. Definition 2.5.1. ([12]) We say that an associative ring A is a k-algebra if there exists a morphism of rings ϕ : k −→ A such that ϕ(k) ⊆ Z(A). In this section we…

…long exact sequence for the cohomology, and both have proved useful for computations. In 2013 Staic introduced the secondary Hochschild cohomology of the triple (A, B, ε) with coefficients in M in [39]. Here A is an associative k…

…Conventions For this dissertation we fix k to be a field. Furthermore, all tensor products are over k unless otherwise stated (that is, ⊗ = ⊗k ). We fix the following: A is an associative k-algebra and M is an A-bimodule. We fix B to be a commutative…

…fix A to be an associative k-algebra and M an A-bimodule. Define C n (A, M ) = Homk (A⊗n , M ) and dn : C n (A, M ) −→ C n+1 (A, M ) determined by dn (f )(a0 ⊗ a1 ⊗ · · · ⊗ an ) = a0 f (…

…x5B;21] in order to study extensions of associative algebras over fields. Almost twenty years later, Gerstenhaber made a connection between Hochschild cohomology and deformation theory. His result is below. Theorem 2.5.7. ([16])…

…i) The multiplication law mt is associative mod t2 if and only if c1 is a 2-cocycle. That is, c1 ∈ Z 2 (A, A). (ii) Suppose that mt is associative mod tn . Then mt can be extended to be associative mod tn+1 if and only if c1…

…the secondary Hochschild cohomology in [39] which describes deformations that admit a B-algebra structure. We give here a short overview. Let A be an associative k-algebra, B a commutative k-algebra, ε : B −→ A a morphism of k-algebras such…

…We consider a family of products M = {mα,t }α∈B where mα,t (a ⊗ b) = ε(α)ab + c1 (a ⊗ b ⊗ α)t + c2 (a ⊗ b ⊗ α)t2 + c3 (a ⊗ b ⊗ α)t3 + · · · Then (i) The family M is associative mod…