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

For any d≥ 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.