We examine some recent work of Phillip Griffith on étale covers and fibered products from the point of view of tight closure theory. While it is known that cyclic covers of Gorenstein rings with rational singularities are Cohen-Macaulay, we show this is not true in general in the absence of the Gorenstein hypothesis. Specifically, we show that the canonical cover of a Q-Gorenstein ring with rat...

We describe a duality phenomenon for cohomology theories with the character of Gorenstein rings.

We introduce the notion of relative singularity category with respect to any self-orthogonal subcategory ω of an abelian category. We introduce the Frobenius category of ω-Cohen-Macaulay objects, and under some reasonable conditions, we show that the stable category of ω-Cohen-Macaulay objects is triangle-equivalent to the relative singularity category. As applications, we relate the stable cat...

We compute the Gorenstein weak dimension of a coherent power series rings over a commutative rings and we show that, in general, Gwdim (R) ≤ 1 does not imply that R is an arithmetical ring.