Henrik Holm

Gorenstein derived functors

Over any associative ring R it is standard to derive Hom R (-,-) using projective resolutions in the first variable, or injective resolutions in the second variable, and doing this, one obtains Ext n R(-,-) in both cases. We examine the situation where projective and injective modules are replaced by Gorenstein projective and Gorenstein injective ones, respectively. Furthermore, we derive the tensor product - ⊗ R - using Gorenstein flat modules.

Cotorsion pairs induced by duality pairs

We introduce the notion of a duality pair and demonstrate how the left half of such a pair is “often” covering and preenveloping. As an application, we generalize a result by Enochs et al. on Auslander and Bass classes, and we prove that the class of Gorenstein injective modules—introduced by Enochs and Jenda—is covering when the ground ring has a dualizing complex.

Rings with finite Gorenstein injective dimension

In this paper we prove that for any associative ring R, and for any left R-module M with finite projective dimension, the Gorenstein injective dimension Gid R M equals the usual injective dimension id R M. In particular, if Gid R R is finite, then also id R R is finite, and thus R is Gorenstein (provided that R is commutative and Noetherian).

Beyond totally reflexive modules and back

Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory’s connections with relative homological algebra and with studies of local ring homomorphisms. It ends close to the starting point: with a characterization of Gorenstein rings in terms of total acyclicity of complexes.

Construction of totally reflexive modules from an exact pair of zero divisors

Let A be a local ring that admits an exact pair x,y of zero divisors as defined by Henriques and Sega. Assuming that this pair is orthogonal and that there exists a regular element on the A-module A/(x,y), we explicitly construct an infinite family of non-isomorphic indecomposable A-modules whose minimal free resolutions are periodic of period 2, and which are totally reflexive. In this setting, our construction provides an answer to a question by Christensen, Piepmeyer, Striuli, and Takahashi. Furthermore, we compute the module of homomorphisms between any two given modules from the infinite family mentioned above.

Relative Ext groups, resolutions, and Schanuel classes

Given a precovering (also called contravariantly finite) cl ass F there are three natural approaches to a homological dimension with respect to F: One based on Ext functors relative to F, one based on F-resolutions, and one based on Schanuel classes relative to F. In general these approaches do not give the same result. In this paper we study relations between the three approaches above, and we give necessary and sufficient conditions for them to agree.