Koszul Duality and Deformation Theory

Guan, Ai and Lazarev, Andrey (2021) Koszul Duality and Deformation Theory. PhD thesis, Lancaster University.

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Abstract

This thesis answers various questions related to Koszul duality and deformation theory. We begin by giving a general treatment of deformation theory from the point of view of homotopical algebra following Hinich, Manetti and Pridham. In particular, we show that any deformation functor in characteristic zero is controlled by a certain differential graded Lie algebra defined up to homotopy, and also formulate a noncommutative analogue of this result valid in any characteristic.  In the next part of this thesis, we introduce a notion of left homotopy for Maurer-Cartan elements in L∞-algebras and A∞-algebras, and show that it corresponds to gauge equivalence in the differential graded case. From this we deduce a short formula for gauge equivalence, and provide an entirely homotopical proof to Schlessinger-Stasheff’s theorem. As an application, we answer a question of T. Voronov, proving a non-abelian Poincaré lemma for differential forms taking values in an L∞-algebra.  In the final part of this thesis, we generalize previous formulations of Koszul duality for associative algebras by Keller-Lefèvre and Positselski. For any dg algebra A we construct a model category structure on dg A-modules such that the corresponding homotopy category is compactly generated by dg A-modules that are finitely generated and free over A (disregarding the differential). We prove that this model category is Quillen equivalent to the category of comodules over a certain, possibly nonconilpotent differential graded coalgebra, a so-called extended bar construction of A.