Auslander and Buchweitz have proved that every finitely generated module
over a Cohen–Macaulay (CM) ring with a dualizing module admits a
so-called maximal CM approximation. In terms of relative homological
algebra, this means that every finitely generated module has a special
maximal CM precover. In this paper, we prove the existence of special maximal
CM preenvelopes and, in the case where the ground ring is henselian, of
maximal CM envelopes. We also characterize the rings over which every
finitely generated module has a maximal CM envelope with the unique
lifting property. Finally, we show that cosyzygies with respect to the class
of maximal CM modules must eventually be maximal CM, and we compute
some examples.
Keywords
cosyzygy, envelope, maximal Cohen-Macaulay module, special
preenvelope, unique lifting property
