# Manchester Algebra Seminar - Matt Booth

Dates: | 16 February 2021 |
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Times: | 13:00 - 14:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, Current University students |

Speaker: Matt Booth

Title: Mac Lane cohomology and topological Hochschild cohomology

Abstract: The deformation theory of an algebra over a field is controlled by its Hochschild cohomology, and in particular square-zero extensions of A are bijection with HH^2(A). For general rings, Hochschild cohomology does not capture enough information, and one must use a cohomology theory called Mac Lane cohomology instead, which records nonlinear data.

I'll begin the talk by describing the above picture, and then report on ongoing joint work with Dmitry Kaledin and Wendy Lowen where we try to generalise this situation to abelian categories, which one can view as many-object rings. We start by showing that for rings, Mac Lane cohomology agrees with topological Hochschild cohomology, an invariant constructed using techniques from algebraic topology - the analogous statement for homology has been known since the 90s. This allows us to define the Mac Lane cohomology of an abelian category.

In particular, we get a definition for the Mac Lane cohomology of an algebraic variety, and using some powerful invariance techniques for THH we're able to do some quite explicit computations.

Time: 1pm Place: Zoom