Hans-Bjørn Foxby

Homological dimensions of unbounded complexes

Abstract This paper explores various notions of projective, injective, and flat dimensions, arising from recent constructions of resolutions of unbounded complexes, proposed by N. Spaltenstein and by S. Halperin with the authors. The different versions of each dimension are compared to each other, and also to the classical concepts, whenever these may be defined. Cohomological characterizations of the dimensions are provided in terms of vanishing of appropriate derived functors. The behavior of the dimensions under change of rings is investigated.

Bass series of local ring homomorphisms of finite flat dimension

Nontrivial relations between Bass numbers of local commutative rings are established in case there exists a local homomorphism φ: R → S which makes S into an R-module of finite flat dimension. In particular, it is shown that an inequality μ R i+depth R ≤ μ s i+depth S holds for all i ∈ Z.This is a consequence of an equality involving the Bass series I R M (t) = Σ i ∈ z μ R i (M)t i of a complex M of R-modules which has bounded above and finite type homology and the Bass series of the complex of S-modules M⊗ R S, where ⊗ denotes the derived tensor product. It is prove that there is an equa lity of formal Laurent series I s M⊗RS (t) = I R M (t)I F(φ) (t), where F(φ) is the fiber of φ considered as a homomorphism of commutative differential graded rings

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.

Gorenstein local homomorphisms

A Noetherian local ring is the algebraic version of a ring of germs of functions defined in neighborhoods of some point of an algebraic (or analytic) variety. Accordingly, local rings are naturally classified by the complexity of the singularity they describe, with the simplest class consisting of the regular rings, which correspond to nonsingular points. On the singular side a natural boundary is provided by the Cohen-Macaulay rings: beyond them pathological (that is, geometrically unpredictable) behavior becomes a common phenomenon. During the last three decades much of the work in commutative algebra has concentrated on rings whose singularities interpolate between these two extremes. One of the most important developments early in that period was the discovery of the intermediate class of Gorenstein rings by Bass and Grothendieck. These authors demonstrated that Gorenstein rings provide a perfect framework for the investigation of duality phenomena, and this is the main reason behind their ubiquitous appearance in commutative algebra and algebraic geometry. They also noted that among the Gorenstein singularities one finds all local complete intersections, which describe points of transversal intersection of hypersurfaces. The purpose of this note is to introduce some of the results of [2], where a relative theory of Gorenstein singularities is systematically developed. There are several aspects to our approach. First, it gives a unified treatment of hitherto unrelated relative Gorenstein notions, such as that of flat homomorphisms whose fibres are Gorenstein rings (Grothendieck), and surjective homomorphisms whose kernels are generated by regular sequences, or more generally, are Gorenstein ideals (Buchsbaum and Eisenbud). Next, it contains the theory of Gorenstein rings as the absolute case, that