On local rings without small Cohen-Macaulay algebras in mixed characteristic

TL;DR

This paper constructs examples of high-dimensional mixed characteristic local rings lacking small Cohen-Macaulay algebras but possessing some Cohen-Macaulay modules, and explores conditions under which graded domains admit such modules.

Contribution

It provides the first known examples of mixed characteristic local rings without small Cohen-Macaulay algebras, using deformation theory, and characterizes graded domains over Abelian varieties that admit Cohen-Macaulay modules.

Findings

Existence of mixed characteristic local rings without small Cohen-Macaulay algebras

Presence of Cohen-Macaulay modules in these rings

Graded normal domains over Abelian varieties admit Cohen-Macaulay modules

Abstract

For any $d\ge 4$, by deformation theory of schemes, we present examples of (complete or excellent) $d$-dimensional mixed characteristic normal local domains admitting no small Cohen-Macaulay algebra, but admitting instances of small (maximal) Cohen-Macaulay modules. It is also shown that a graded normal domain over a field whose Proj is an Abelian variety admits a graded small (maximal) Cohen-Macaulay module.