Gorenstein categories and Tate cohomology on projective schemes

We study Gorenstein categories. We show that such a category has Tate cohomological functors and Avramov-Martsinkovsky exact sequences connecting the Gorenstein relative, the absolute and the Tate cohomological functors. We show that such a category has what Hovey calls an injective model structure and also a projective model structure in case the category has enough projectives. As examples we show that if X is a locally Gorenstein projective scheme then the category Qco(X) of quasi-coherent sheaves on $X$ is such a category and so has these features.

💡 Research Summary

The paper introduces the notion of a Gorenstein category, a complete abelian category in which every object admits both a complete projective resolution and a complete injective resolution, and where the classes of Gorenstein‑projective and Gorenstein‑injective objects are sufficiently large. By abstracting the classical Gorenstein homological algebra from module categories to arbitrary abelian categories, the authors set up a framework that simultaneously accommodates relative (Gorenstein) and absolute homological functors.

The first major construction is a Tate cohomology functor (\widehat{\operatorname{Ext}}^{*}_{\mathcal{A}}(-,-)). For an object (A) one chooses a complete projective resolution (P^{\bullet}) and extends it infinitely in both directions to obtain a totally acyclic complex (\mathbf{T}(P^{\bullet})). The Tate groups are defined as the cohomology of (\operatorname{Hom}(\mathbf{T}(P^{\bullet}),B)). This definition mirrors the classical Tate cohomology for Gorenstein rings but works in any Gorenstein category.

A central result is the Avramov‑Martsinkovsky exact sequence linking three cohomology theories: \