Abstract:

This thesis introduces a framework for defining new classes of rings analogous to various well-studied combinatorial algebras over a field. Given a discrete valuation ring (V, t), the polynomial ring V[x1, ... , xn] is zn-graded by multidegree in the variables, and up to multi-plication by a unit the homogeneous elements are of the form tm0x7{1'1... x;;in. We call these elements t-monomials. Likewise, the Laurent polynomial ring L[xf1, ... , x;1] over the frac- tion field L of V is a zn-graded V-algebra whose homogeneous elements are t-monomials, up to a unit in V. We define t-Stanley-Reisner rings, t-semigroup rings, and t-toric face rings by replacing monomials with t-monomials in the definitions of Stanley-Reisner rings, semigroup rings, and toric face rings, respectively.

Evaluation at t induces an algebraically well-behaved bijection between t-monomials and monomials in one extra variable; we exploit this bijection to prove results about our new ring classes by comparison to the original versions. This technique is particularly useful to study open questions which are known for equicharacteristic. As a case study, we demonstrate that the Q-Sequence Conjecture of Hochster and Zhang (proven in equicharacteristic by Hochster and Huneke) holds for all three of these new ring classes.