Gorenstein injective filtrations over Cohen-Macaulay rings with dualizing modules
Abstract
Over a noetherian ring, it is a classic result of Matlis that injective modules admit direct sum decompositions into injective hulls of quotients by prime ideals. We show that over a Cohen-Macaulay ring admitting a dualizing module, Gorenstein injective modules admit similar filtrations. We also investigate Tor-modules of Gorenstein injective modules over such rings. This extends work of Enochs and Huang over Gorenstein rings. Furthermore, we give examples showing the following: (1) the class of Gorenstein injective -modules need not be closed under tensor products, even when is local and artinian; (2) the class of Gorenstein injective -modules need not be closed under torsion products, even when is a local, complete hypersurface; and (3) the filtrations given in our main theorem do not yield direct sum decompositions, even when is a local, complete hypersurface.
Keywords
Cite
@article{arxiv.1412.2654,
title = {Gorenstein injective filtrations over Cohen-Macaulay rings with dualizing modules},
author = {Aaron J. Feickert and Sean Sather-Wagstaff},
journal= {arXiv preprint arXiv:1412.2654},
year = {2015}
}
Comments
20 pages. v.3 is substantially different from v.2: Lemmas 3.12 and 3.13 from v.2 are incorrect as stated, so have been removed; Theorem A has been revised back to version from v.2, and a counterexample to the previous version is given in v.3; Theorem B is corrected, with counterexample given to result stated in v.1; a counterexample to a question of Enochs and Huang is also given