Full Record

Author | Laubacher, Jacob C |

Title | Secondary Hochschild and Cyclic (Co)homologies |

URL | http://rave.ohiolink.edu/etdc/view?acc_num=bgsu1489422065908758 |

Publication Date | 2017 |

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 |

Sample Search Hits | Sample Images | Cited Works

…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…