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Abstract

The goal of this dissertation is to explicitly construct a rigid dualizing complex for the algebra k[x₁, . . . , xₙ], with k a field, using the methods of [CO24]. Along the way, we give an overviewof the theory of derived categories with the needed results (§1), an overview of the background and methods of [CO24] (§2), and develop some associated tools using cubical homology (§3). In particular, analogous to how [CO24] constructs a rigid dualizing complex for a Hecke algebra by using an exact sequence arising from geometric properties of the associated Weyl group, we do so in this case using an exact sequence arising from a space of translations associated to k[x₁, . . . , xₙ]. This serves as a proof of concept that these methods are generalizable, with hopeful applications to pro-p Iwahori Hecke algebras.