# PhD Defense Malte Sander Leip

Title: On the Hochschild homology of hypersurfaces as a mixed complex

Abstract:

In this thesis, we describe Hochschild homology over k of quotients of polynomial algebras k[x1, . . . , xn]/f for certain polynomials f in n ≤ 2 variables, as an object of the ∞-category of mixed complexes Mixed, where k is a commutative ring in which 2 is invertible.

In 1992, the Buenos Aires Cyclic Homology Group [BACH] constructed, for any n and any commutative ring k, a quasiisomorphism between the standard Hochschild complex over k of k[x1, . . . , xn]/f and a quite small chain complex, under the assumption that f is monic with respect to a chosen monomial order. This result was improved upon by Larsen in 1995 [Lar95] by taking the mixed structure into account as well, though only considering polynomials f in n = 2 variables that are monic with respect to one of the variables. Assuming a conjectural description of Hochschild homology of polynomial rings, we
extend these previous results by constructing, for a large subset of the polynomials f considered in [BACH], a strict mixed structure on the chain complex described in [BACH] and showing that it represents the Hochschild homology over k of k[x1, . . . , xn]/f as an object in the ∞-category of mixed complexes. We also verify the conjecture in some cases, leading to unconditional results for n ≤ 2 variables, as long as 2 is invertible in k.

The results of this thesis do not rely on the two aforementioned prior results, but instead, use the modern approach to Hochschild homology based on ∞-categorical methods.

Along the way, to be able to state and prove our result in this setting, we prove some results that may be of independent interest.

## Assessment Committee:

Chair, Professor Søren Galatius, Department of Mathematical Sciences, University of Copenhagen

Professor, Michael Larsen, Indiana University

Professor, Thomas Nikolaus, Universität Münster