Symmetric Quotients of Polynomial Rings and Stanley--Reisner Rings

Abstract

This thesis studies the fixed quotient of a commutative ring with group action, which is naturally a module over the ring of invariants. The perspective of graded syzygies is used to analyze the module structure. Chapter 4 concerns fixed quotients of polynomial rings, focusing on a stability pattern that emerges as the number of indeterminates increases. Chapter 5 studies a deformation of the polynomial ring, working towards a proof that the syzygies of the deformation's fixed quotient bound those of the polynomial ring.