Edgar E. Enochs

Relative homological coalgebras

We study classes of relative injective and projective comodules and extend well-known results about projective comodules given in [7]. The existence of covers and envelopes by these classes of comodules is also studied and used to characterize the projective dimension of a coalgebra. We also compare this homological coalgebra with the very intensively studied homological algebra of the dual algebra (see [5]).

Injective and flat covers, envelopes and resolvents

Using the dual of a categorical definition of an injective envelope, injective covers can be defined. For a ringR, every leftR-module is shown to have an injective cover if and only ifR is left noetherian. Flat envelopes are defined and shown to exist for all modules over a regular local ring of dimension 2. Using injective covers, minimal injective resolvents can be defined.

All Modules Have Flat Covers

In this paper we give two different proofs that the flat cover conjecture is true: that is, every module has a flat cover. The two proofs are of completely different nature, and, we hope, will have different applications. The first of the two proofs (due to the third author) is essentially an application of the work of P. Eklof and J. Trlifaj (work which is more set-theoretic). The second proof (due to the first two authors) is more direct, and has a model-theoretic flavour.

Foxby duality and Gorenstein injective and projective modules

In 1966, Auslander introduced the notion of the G-dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of G-dimensions. It seemed appropriate to call the modules with G-dimension 0 Gorenstein projective, so the basic problem was to define Gorenstein injective modules. These were defined in Math. Z. 220 (1995), 611–633 and were shown to have properties predicted by Auslander’s results. The way we define Gorenstein injective modules can be dualized, and so we can define Gorenstein projective modules (i.e. modules of G-dimension 0) whether the modules are finitely generated or not. The investigation of these modules and also Gorenstein flat modules was continued by Enochs, Jenda, Xu and Torrecillas. However, to get good results it was necessary to take the base ring Gorenstein. H.-B. Foxby introduced a duality between two full subcategories in the category of modules over a local CohenMacaulay ring admitting a dualizing module. He proved that the finitely generated modules in one category are precisely those of finite G-dimension. We extend this result to modules which are not necessarily finitely generated and also prove the dual result, i.e. we characterize the modules in the other class defined by Foxby. The basic result of this paper is that the two classes involved in Foxby’s duality coincide with the classes of those modules having finite Gorenstein projective and those having finite Gorenstein injective dimensions. We note that this duality then allows us to extend many of our results to the original Auslander setting.

Torsion free covering modules

Let A be an integral domain and K its field of fractions. An Amodule E is said to be torsion free if ax = 0 for azA, x EE implies a = 0 or x = 0. We will say that a submodule E1 of an A -module E is pure in E if axE,=acEC\E1 for all aGA. Then if E is torsion free, a submodule E1 of E is pure in E if and only if E/E, is torsion free. Clearly the union of a chain of pure submodules of a module is still a pure submodule and if E2CE1, are submodules of E such that E2 is pure in E1 and E1/E2 pure in E/E2 then El is pure in E. It is well known that for any A-module E there exists a torsion free A -module E1 and an epimorphism p: E-*E1 such that if 4 is any linear mapping from E into a torsion free module F then there is a unique linear mapping f: El-*F such that f o p =4, i.e., the diagram

Gorenstein injective and projective complexes

In this article we extend the notion of Gorenstein injective and projective modules to that of complexes and characterize such complexes. We prove that over an n-Gorenstein ring every complex has a Gorenstein injective envelope and we show that every such envelope is a quasi-isomorphim.. When the ring is commutative, local and Gorenstein, Auslander announced that every finitely generated R-module has a finitely generated Gorenstein projective cover. We show that every bounded above complex having all terms finitely generated over such a ring has a Gorenstein projective cover and we show that these covers are quasi-isomorphisms.

DUALIZING MODULES AND n-PERFECT RINGS

In this article we extend the results about Gorenstein modules and Foxby duality to a non-commutative setting. This is done in §3 of the paper, where we characterize the Auslander and Bass classes which arise whenever we have a dualizing module associated with a pair of rings. In this situation it is known that flat modules have finite projective dimension. Since this property of a ring is of interest in its own right, we devote §2 of the paper to a consideration of such rings. Finally, in the paper’s final section, we consider a natural generalization of the notions of Gorenstein modules which arises when we are in the situation of §3, i.e. when we have a dualizing module. AMS 2000 Mathematics subject classification: Primary 16D20

Flat Covers in the Category of Quasi-coherent Sheaves Over the Projective Line

Abstract In this paper we prove the existence of a flat cover and a cotorsion envelope for any quasi-coherent sheaf over the projective line , where R is any commutative ring. We first prove a general result that guarantees the existence of ℱ-covers and ℱ⊥-envelopes in the general setting of a Grothendieck category (not necessarily with enough projectives) provided that the class ℱ satisfies some “standard” conditions. This will generalize some results of the earlier work. [Aldrich, S. T., Enochs, E., García Rozas, J. R., Oyonarte, L. (2001). Covers and envelopes in Grothendieck categories. Flat covers of complexes with applications. J. Algebra 243:615–630].

A Homotopy of Quiver Morphisms with Applications to Representations

It is shown that a morphism of quivers having a certain path lifting property has a decomposition that mimics the decomposition of maps of topological spaces into homotopy equivalences composed with fibrations. Such a decomposition enables one to describe the right adjoint of the restriction of the represen- tation functor along a morphism of quivers having this path lifting property. These right adjoint functors are used to construct injective representations of quivers. As an application, the injective representations of the cyclic quivers are classified when the base ring is left noetherian. In particular, the indecomposable in- jective representations are described in terms of the injective indecomposable R-modules and the injective indecomposable R(x; x 1 )-modules. Let Q be a quiver and R a ring. In this paper, we shall study the category (Q, R-Mod) of representations of Q by left R-modules. As in the work of Riedtmann (7) and Bongartz and Gabriel (1), we will be interested in representations induced by morphisms of quivers. More precisely, we shall refine an argument of Jensen (4) to construct, using adjoint pairs of functors, injective objects of (Q, R-Mod) with specific features. Our first result is rem- iniscent of the decomposition theorem (8, Theorem II.8.9) for maps of topological spaces which asserts that every continuous function is a homotopy equivalence composed with a fibration. It relates the following two properties of quiver morphisms:

Projective Representations of Quivers

In the first part of this article, we describe the projective representations in the category of representations by modules of a quiver which does not contain any cycles and the quiver A ∞ as a subquiver, that is, the so-called rooted quivers. As a consequence of this, we show when the category of representations by modules of a quiver admits projective covers. In the second part, we develop a technique involving matrix computations for the quiver A ∞, which will allow us to characterize the projective representations of A ∞. This will improve some previous results and make more accurate the statement made in Benson (1991). We think this technique can be applied in many other general situations to provide information about the decomposition of a projective module.