Mathematics Ph.D. Dissertations

Title

Secondary Hochschild and Cyclic (Co)homologies

Date of Award

2017

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.)

Department

Mathematics

First Advisor

Mihai D. Staic (Advisor)

Second Advisor

John Laird (Other)

Third Advisor

Xiangdong Xie (Committee Member)

Fourth Advisor

Juan Bes (Committee Member)

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.

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