Local Commutative Algebra and Hochschild Cohomology Through the Lens of Koszul Duality

This thesis splits into two halves, the connecting theme being Koszul duality. The first part concerns local commutative algebra. Koszul duality here manifests in the homotopy Lie algebra. In the second part, which is joint work with Vincent G\'elinas, we study Hochschild cohomology and its characteristic action on the derived category. We begin by defining the homotopy Lie algebra $\pi^*(\phi)$ of a local homomorphism $\phi$ (or of a ring) in terms of minimal models, slightly generalising a classical theorem of Avramov. Then, starting with work of F\'{e}lix and Halperin, we introduce a notion of Lusternik-Schnirelmann category for local homomorphisms (and rings). In fact, to $\phi$ we associate a sequence $\cat_{0}(\phi)\geq \cat_1(\phi)\geq \cat_2(\phi)\geq \cdots$ each $\cat_i(\phi)$ being either a natural number or infinity. We prove that these numbers characterise weakly regular, complete intersection, and (generalised) Golod homomorphisms. We present examples which demonstrate how they can uncover interesting information about a homomorphism. We give methods for computing these numbers, and in particular prove a positive characteristic version of F\'{e}lix and Halperin's Mapping Theorem. A motivating interest in L.S. category is that finiteness of $\cat_2(\phi)$ implies the existence of certain six-term exact sequences of homotopy Lie algebras, following classical work of Avramov. We introduce a variation $\pic(\phi)$ of the homotopy Lie algebra which enjoys long exact sequences in all situations, and construct a comparison $\pic(\phi)\to \pi^*(\phi)$ which is often an isomorphism. This has various consequences; for instance, we use it to characterise quasi-complete intersection homomorphisms entirely in terms of the homotopy Lie algebra. In the second part of this thesis we introduce a notion of $A_\infty$ centre for minimal $A_\infty$ algebras. If $A$ is an augmented algebra over a field $k$ we show that the image of the natural homomorphism $\chi_k:\HH(A,A)\to {\rm Ext}^*_A(k,k)$ is exactly the $A_\infty$ centre of $A$, generalising a theorem of Buchweitz, Green, Snashall and Solberg from the case of a Koszul algebra. This is deduced as a consequence of a much wider enrichment of the entire characteristic action $\chi:\HH(A,A)\to {\sf Z}(D(A))$. We give a number of representation theoretic applications.Ph.D