Gorenstein AC-injective/projective modules

Jim Gillespie (Ramapo College, New Jersey)

Frank Adams 1,

The key results in the development of Gorenstein homological algebra seem to depend on the ring R being Noetherian. In joint work with Daniel Bravo and Mark Hovey, we introduced a way to extend many ideas from Gorenstein homological algebra to arbitrary rings. To do this we strengthen the definition of Gorenstein injective (resp. Gorenstein projective) modules to get what we call Gorenstein AC-injective (resp. Gorenstein AC-projective) modules. We show that, over any ring, they are the right half (resp. left half) of a complete hereditary cotorsion pair. These cotorsion pairs are so nice that they are equivalent to abelian model structures, on R-Mod, and so induce stable module categories. We also get existence of Gorenstein AC-injective preenvelopes and Gorenstein AC-projective precovers, which we will interpret for Noetherian and coherent rings. Time permitting we will also characterize the Gorenstein AC-injective/projective modules over the rings I call Ding-Chen rings.

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