Maximal Cohen-Macaulay modules over local toric rings

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$\mathstrut_p$).