The goal of this dissertation is to explicitly construct a rigid dualizing complex for the algebra kx₁, . . . , 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 kx₁, . . . , xₙ. This serves as a proof of concept that these methods are generalizable, with hopeful applications to pro-p Iwahori Hecke algebras.
Justin Daniel Lawrence (Thu,) studied this question.