Localization on certain Grothendieck categories

Using categorical techniques we obtain some results on localization and colocalization theory in Grothendieck categories with a set of small projective generators. In particular, we give a sufficient condition for such category to be semiartinian. For semiartinian Grothendieck categories where every simple object has a projective cover, we obtain that every localizing subcategory is a TTF-class. In addition, some applications to semiperfect categories are obtained.     

Homological Finiteness Conditions for Groups,Monoids, and Algebras

Let ? be an additive category and 𝒞 a full subcategory with split idempotents, and closed under isomorphic images and finite direct sums. We give conditions on ? and 𝒞 implying that ? embeds into an abelian category, so that the objects of 𝒞 turn into injective objects. This construction generalizes the embedding of exactly definable categories into locally coherent categories, while the dual construction generalizes the embedding of finitely accessible categories into Grothendieck categories with a family of finitely generated projective generators. As applications, we characterize exactly definable categories through intrinsic properties and study those locally coherent categories whose fp-injective objects form a Grothendieck category.     

Relative Regular Objects in Categories

We define the concept of a regular object with respect to another object in an arbitrary category. We present basic properties of regular objects and we study this concept in the special cases of abelian categories and locally finitely generated Grothendieck categories. Applications are given for categories of comodules over a coalgebra and for categories of graded modules, and a link to the theory of generalized inverses of matrices is presented. Some of the techniques we use are new, since dealing with arbitrary categories allows us to pass to the dual category.      

Are all localizing subcategories of stable homotopy categories coreflective?

We prove that, in a triangulated category with combinatorial models, every localizing subcategory is coreflective and every colocalizing subcategory is reflective if a certain large-cardinal axiom (Vopěnka?s principle) is assumed true. It follows that, under the same assumptions, orthogonality sets up a bijective correspondence between localizing subcategories and colocalizing subcategories. The existence of such a bijection was left as an open problem by Hovey, Palmieri and Strickland in their axiomatic study of stable homotopy categories and also by Neeman in the context of well-generated triangulated categories.     

The Ziegler Spectrum of a Locally Coherent Grothendieck Category

The general theory of locally coherent Grothendieck categoriesis presented. To each locally coherent Grothendieck categoryC a topological space, the Ziegler spectrum of C, is associated.It is proved that the open subsets of the Ziegler spectrum ofC are in bijective correspondence with the Serre subcategoriesof coh C the subcategory of coherent objects of C. This is aNullstellensatz for locally coherent Grothendieck categories.If R is a ring, there is a canonical locally coherent Grothendieckcategory RC (respectively, CR) used for the study of left (respectively,right) R-modules. This category contains the category of R-modulesand its Ziegler spectrum is quasi-compact, a property used toconstruct large (not finitely generated) indecomposable modulesover an artin algebra. Two kinds of examples of locally coherentGrothendieck categories are given: the abstract category theoreticexamples arising from torsion and localization and the examplesthat arise from particular modules over the ring R. The dualitybetween coh-(RC) and coh-CR is shown to give an isomorphismbetween the topologies of the left and right Ziegler spectraof a ring R. The Nullstellensatz is used to give a proof ofthe result of Crawley-Boevey that every character : K₀(coh-C) Z is uniquely expressible as a Z-linear combination of irreduciblecharacters. 1991 Mathematics Subject Classification: 16D90,18E15.     

Two criteria for locally noetherian Grothendieck categories

We give two criteria for a Grothendieck category to be locally noetherian. The first of these is in terms of direct sum decompositions into uniform components. The second is related to the existence of finitely injective objects that cannot be expressed as a direct sum of injectives.     

A recollement construction of Gorenstein derived categories

We first give an equivalence between the derived category of a locally finitely presented category and the derived category of contravariant functors from its finitely presented subcategory to the category of abelian groups, in the spirit of Krause’s work [Math. Ann., 2012, 353: 765–781]. Then we provide a criterion for the existence of recollement of derived categories of functor categories, which shows that the recollement structure may be induced by a proper morphism defined in finitely presented subcategories. This criterion is then used to construct a recollement of derived category of Gorenstein injective modules over CM-finite 2-Gorenstein artin algebras.     

Hereditary noetherian categories of positive Euler characteristic

We develop the fundamentals of hereditary noetherian categories with Serre duality over an arbitrary field k, where the category of coherent sheaves over a smooth projective curve over k serves as the prime example and others are coming from the representation theory of finite dimensional algebras. The proper way to view such a category is to think of coherent sheaves on a possibly non-commutative smooth projective curve. We define for each such category notions like function field and Euler characteristic, determine its Auslander-Reiten components and study stable and semistable bundles for an appropriate notion of degree. We provide a complete classification of hereditary noetherian categories for the case of positive Euler characteristic by relating these to finite dimensional representations of (locally bounded) hereditary k-algebras whose underlying valued quiver admits a positive additive function. Dedicated to Otto Kerner on the occasion of his 60th birthday     

Cleft Comodule Categories

We define crossed product categories and we show that they are equivalent with cleft comodule categories. We also prove that a comodule category is cleft if and only if it is Hopf–Galois and has a normal basis. As an application we show that the category of Hopf modules over a cleft linear category and the category of modules over the coinvariant subcategory are equivalent.     

Gorenstein Homological Aspects of Monomorphism Categories via Morita Rings

In this paper we construct Gorenstein-projective modules over Morita rings with zero bimodule homomorphisms and we provide sufficient conditions for such rings to be Gorenstein Artin algebras. This is the first part of our work which is strongly connected with monomorphism categories. In the second part, we investigate monomorphisms where the domain has finite projective dimension. In particular, we show that the latter category is a Gorenstein subcategory of the monomorphism category over a Gorenstein algebra. Finally, we consider the category of coherent functors over the stable category of this Gorenstein subcategory and show that it carries a structure of a Gorenstein abelian category.     

Stable Projective Homotopy Theory of Modules,Tails, and Koszul Duality

A contravariant functor is constructed from the stable projective homotopy theory of finitely generated graded modules over a finite-dimensional algebra to the derived category of its Yoneda algebra modulo finite complexes of modules of finite length. If the algebra is Koszul with a noetherian Yoneda algebra, then the constructed functor is a duality between triangulated categories. If the algebra is self-injective, then stable homotopy theory specializes trivially to stable module theory. In particular, for an exterior algebra the constructed duality specializes to (a contravariant analog of) the Bernstein–Gelfand–Gelfand correspondence.     

Induced and higher-dimensional stable independence

We provide several crucial technical extensions of the theory of stable independence notions in accessible categories. In particular, we describe circumstances under which a stable independence notion can be transferred from a subcategory to a category as a whole, and examine a number of applications to categories of groups and modules, extending results of [16]. We prove, too, that under the hypotheses of [11], a stable independence notion immediately yields higher-dimensional independence as in [26].     

Auslander-Reiten Triangles, Ziegler Spectra and Gorenstein Rings

We investigate (existence of) Auslander—Reiten triangles in a triangulated category in connection with torsion pairs, existence of Serre functors, representability of homological functors and realizability of injective modules. We also develop an Auslander—Reiten theory in a compactly generated triangulated category and we study the connections with the naturally associated Ziegler spectrum. Our analysis is based on the relative homological theory of purity and Brown's Representability Theorem. Our main interest lies in the structure of Auslander—Reiten triangles in the full subcategory of compact objects. We also study the connections and the interplay between Auslander—Reiten theory, pure-semisimplicity and the finite type property, Grothendieck groups, and we give applications to derived categories of Gorenstein rings.     

Biset Functors as Module Mackey Functors and its Relation to Derivators

In this article, we will show that the category of biset functors can be regarded as a reflective monoidal subcategory of the category of Mackey functors on the 2-category of finite groupoids. This reflective subcategory is equivalent to the category of modules over the Burnside functor. As a consequence of the reflectivity, we can associate a biset functor to any derivator on the 2-category of finite categories.     

One-Point Extensions of t-Koszul Algebras

We introduce the category of t-fold modules which is a full subcategory of graded modules over a graded algebra. We show that this subcategory and hence the subcategory of t-Koszul modules are both closed under extensions and cokernels of monomorphisms. We study the one-point extension algebras, and a necessary and sufficient condition for such an algebra to be t-Koszul is given. We also consider the conditions such that the category of t-Koszul modules and the category of quadratic modules coincide.     

Desingularization of quasi-excellent schemes in characteristic zero

Grothendieck proved in EGA IV that if any integral scheme of finite type over a locally noetherian scheme X admits a desingularization, then X is quasi-excellent, and conjectured that the converse is probably true. We prove this conjecture for noetherian schemes of characteristic zero. Namely, starting with the resolution of singularities for algebraic varieties of characteristic zero, we prove the resolution of singularities for noetherian quasi-excellent Q-schemes.     

The Exchange Property in Grothendieck Categories: Applications

We introduce the concept of an object with the (finite) exchange property in an arbitrary Grothendieck category, and we present the basic properties of such an object. Applications are given for categories of graded modules and for categories of comodules over a coalgebra. Among other results, it is proved that an arbitrary coalgebra 𝒞 over a field has the finite exchange property.     

Stable model categories are categories of modules

A stable model category is a setting for homotopy theory where the suspension functor is invertible. The prototypical examples are the category of spectra in the sense of stable homotopy theory and the category of unbounded chain complexes of modules over a ring. In this paper we develop methods for deciding when two stable model categories represent ‘the same homotopy theory’. We show that stable model categories with a single compact generator are equivalent to modules over a ring spectrum. More generally stable model categories with a set of generators are characterized as modules over a ‘ring spectrum with several objects’, i.e., as spectrum valued diagram categories. We also prove a Morita theorem which shows how equivalences between module categories over ring spectra can be realized by smashing with a pair of bimodules. Finally, we characterize stable model categories which represent the derived category of a ring. This is a slight generalization of Rickard's work on derived equivalent rings. We also include a proof of the model category equivalence of modules over the Eilenberg-Mac Lane spectrum HR and (unbounded) chain complexes of R-modules for a ring R.     

Cluster-tilted algebras are Gorenstein and stably Calabi-Yau

We prove that in a 2-Calabi-Yau triangulated category, each cluster tilting subcategory is Gorenstein with all its finitely generated projectives of injective dimension at most one. We show that the stable category of its Cohen-Macaulay modules is 3-Calabi-Yau. We deduce in particular that cluster-tilted algebras are Gorenstein of dimension at most one, and hereditary if they are of finite global dimension. Our results also apply to the stable (!) endomorphism rings of maximal rigid modules of [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press]. In addition, we prove a general result about relative 3-Calabi-Yau duality over non-stable endomorphism rings. This strengthens and generalizes the Ext-group symmetries obtained in [Christof Geiß, Bernard Leclerc, Jan Schröer, Rigid modules over preprojective algebras, arXiv: math.RT/0503324, Invent. Math., in press] for simple modules. Finally, we generalize the results on relative Calabi-Yau duality from 2-Calabi-Yau to d-Calabi-Yau categories. We show how to produce many examples of d-cluster tilted algebras.