Maximal Cohen-Macaulay modules over local toric rings

TL;DR

This paper investigates the existence of maximal Cohen-Macaulay modules over local toric rings, providing new constructions and results in positive, zero, and mixed characteristic cases, advancing understanding of Hochster's conjectures.

Contribution

It introduces the concept of local toric rings, constructs families satisfying Hochster's small MCM conjecture in positive characteristic, and proves that purely toric rings satisfy Hochster's big MCM conjecture in mixed characteristic.

Findings

Constructed families of local toric rings satisfying Hochster's small MCM conjecture in positive characteristic.

Showed that in equal characteristic zero, certain extensions satisfy Hochster's small MCM conjecture if bounded multiplicity conditions hold.

Proved that purely toric local rings satisfy Hochster's big MCM conjecture in mixed characteristic.

Abstract

In analogy with the classical, affine toric rings, we define a local toric ring as the quotient of a regular local ring modulo an ideal generated by binomials in a regular system of parameters with unit coefficients; if the coefficients are just $\pm1$, we call the ring purely toric. We prove the following results on the existence of MCM's (=maximal Cohen-Macaulay modules): (EQUI$\mathstrut_p$) we construct certain families of local toric rings satisfying Hochster's small MCM conjecture in positive characteristic; (EQUI$\mathstrut_0$) provided Hochster's small MCM conjecture holds in positive characteristic with the additional condition that the multiplicity of the small MCM is bounded in terms of the parameter degree of the ring, then any local ring (not necessarily toric) in equal characteristic zero admits a formally etale extension satisfying Hochster's small MCM conjecture (this applies in particular to the families from (EQUI$\mathstrut_p$)); and (MIX), in mixed characteristic, we show that all purely toric local rings satisfy Hochster's big MCM conjecture, and so do those belonging to the families from (EQUI$\mathstrut_p$).