Semi-dualizing complexes and their Auslander categories

Let be a commutative Noetherian ring. We study -modules, and complexes of such, with excellent duality properties. While their common properties are strong enough to admit a rich theory, we count among them such, potentially, diverse objects as dualizing complexes for on one side, and on the other, the ring itself. In several ways, these two examples constitute the extremes, and their well-understood properties serve as guidelines for our study; however, also the employment, in recent studies of ring homomorphisms, of complexes ``lying between' these extremes is incentive.

     

Gorenstein injective modules and Auslander categories

In this paper, we study Gorenstein injective modules over a local Noetherian ring R. For an R-module M, we show that M is Gorenstein injective if and only if Hom _R (Ȓ,M) belongs to Auslander category B(Ȓ), M is cotorsion and Ext ⁱ _R (E,M) = 0 for all injective R-modules E and all i > 0. Received: 24 August 2006 Revised: 30 October 2006     

Cotorsion pairs generated by modules of bounded projective dimension

We apply the theory of cotorsion pairs to study closure properties of classes of modules with finite projective dimension with respect to direct limit operations and to filtrations. We also prove that if the ring is an order in an ℵ₀-noetherian ring Q of little finitistic dimension 0, then the cotorsion pair generated by the modules of projective dimension at most one is of finite type if and only if Q has big finitistic dimension 0. This applies, for example, to semiprime Goldie rings and to Cohen Macaulay noetherian commutative rings.     

Mutation pairs and quotient categories of Abelian categories

The notion of 𝒟-mutation pairs of subcategories in an abelian category is defined in this article. When (𝒵,𝒵) is a 𝒟-mutation pair in an abelian category 𝒜, the quotient category 𝒵∕𝒟 carries naturally a triangulated structure. Moreover, our result generalize the construction of the quotient triangulated category by Happel [10 Happel, D. (1988). Triangulated Categories in the Representation of Finite Dimensional Algebras. London Mathematical Society, LMN, Vol. 119. Cambridge: Cambridge University Press.[Crossref] , [Google Scholar], Theorem 2.6]. Finally, we find a one-to-one correspondence between cotorsion pairs in 𝒜 and cotorsion pairs in the quotient category 𝒵∕𝒟, and study homological finiteness of subcategories in a mutation pair.     

Cotorsion pairs, model category structures, and representation theory

We make a general study of Quillen model structures on abelian categories. We show that they are closely related to cotorsion pairs, which were introduced by Salce [Sal79] and have been much studied recently by Enochs and coauthors [EJ00]. This gives a method of constructing model structures on abelian categories, which we illustrate by building two model structures on the category of modules over a (possibly noncommutative) Gorenstein ring. The homotopy category of these model structures is a generalization of the stable module category much used in modular representation theory. This stable module category has also been studied by Benson [Ben97]. Received: 14 December 2000; in final form: 17 December 2001 / Published online: 5 September 2002     

n-angulated quotient categories induced by mutation pairs

Geiss, Keller and Oppermann (2013) introduced the notion of n-angulated category, which is a “higher dimensional” analogue of triangulated category, and showed that certain (n-2)-cluster tilting subcategories of triangulated categories give rise to n-angulated categories. We define mutation pairs in n-angulated categories and prove that given such a mutation pair, the corresponding quotient category carries a natural n-angulated structure. This result generalizes a theorem of Iyama-Yoshino (2008) for triangulated categories.     

Torsion pairs in recollements of abelian categories

For a recollement (A,B, C) of abelian categories, we show that torsion pairs in A and C can induce torsion pairs in B; and the converse holds true under certain conditions.     

Modular categories associated with unipotent groups

Let $G$ be a unipotent algebraic group over an algebraically closed field $\mathtt{k }$ of characteristic $p>0$ and let $l\ne p$ be another prime. Let $e$ be a minimal idempotent in $\mathcal{D }_G(G)$ , the $\overline{\mathbb{Q }}_l$ -linear triangulated braided monoidal category of $G$ -equivariant (for the conjugation action) $\overline{\mathbb{Q }}_l$ -complexes on $G$ under convolution (with compact support) of complexes. Then, by a construction due to Boyarchenko and Drinfeld, we can associate to $G$ and $e$ a modular category $\mathcal{M }_{G,e}$ . In this paper, we prove that the modular categories that arise in this way from unipotent groups are precisely those in the class $\mathfrak{C }_p^{\pm }$ .     

Recollements arising from cotorsion pairs on extriangulated categories

This paper is devoted to constructing some recollements of additive categories associated to concentric twin cotorsion pairs on an extriangulated category. As an application, this result generalizes the work by W. J. Chen, Z. K. Liu, and X. Y. Yang in a triangulated case [J. Algebra Appl., 2018, 17(5): 1–15]. Moreover, it highlights new phenomena when it applied to an exact category. Finally, we give some applications of our main results. In particular, we obtain Krause's recollement whose proofs are both elementary and very general.     

Auslander correspondence

We study Auslander correspondence from the viewpoint of higher-dimensional analogue of Auslander-Reiten theory [O. Iyama, Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories, Adv. Math. 210 (1) (2007) 22-50 (this issue)] on maximal orthogonal subcategories. We give homological characterizations of higher dimensional analogue of Auslander algebras in terms of global dimension, Auslander-type conditions and so on. Especially we give an answer to a question of M. Artin [M. Artin, Maximal orders of global dimension and Krull dimension two, Invent. Math. 84 (1) (1986) 195-222]. They are also closely related to Auslander's representation dimension of Artin algebras [M. Auslander, Representation dimension of Artin algebras, in: Lecture Notes, Queen Mary College, London, 1971] and Van den Bergh's non-commutative crepant resolutions of Gorenstein singularities [M. Van den Bergh, Non-commutative crepant resolutions, in: The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 749-770].     

Auslander systems

The authors generalize the dynamical system constructed by
J. Auslander in 1959, resulting in perhaps the simplest family of examples of minimal but not strictly ergodic systems. A characterization of unique ergodicity and mean-L-stability is given. The new systems are also shown to have zero topological entropy and fail to be weakly rigid. Some results on the set of idempotents in the enveloping semigroup are also achieved.

     

Boundary pairs associated with quadratic forms

We introduce a purely functional analytic framework for elliptic boundary value problems in a variational form. We define abstract Neumann and Dirichlet boundary conditions and a corresponding Dirichlet‐to‐Neumann operator, and develop a theory relating resolvents and spectra of these operators. We illustrate the theory by many examples including Jacobi operators, Laplacians on spaces with (non‐smooth) boundary, the Zaremba (mixed boundary conditions) problem and discrete Laplacians.     

Some pairs of adjoint functors involving track homotopy categories

K. A. Hardie and K. H. Kamps investigated the track homotopy categoryH _B over a fixed spaceB ([1]). They have introduced two pairs of adjoint functors:P _B N _B andm _* m ^*, whereP _B :H _B →H ^B , andm _*:H _A →H _B for a fixed mapm:A→B. We have introduced a split fibration of categoriesL:H _b →H _B and provedL J, J L in [2]. This paper first extendsP _B N _B to for any fixed mapb: . Moreover we also extend these results to obtain two pairs of adjoint functors involving track homotopy categoriesH _b andH ^b whereH ^b is the dual ofH _b . One of our results isN _b P _b . This differs fromP _B N _B . Supported by National Natural Science Foundation of China     

Tilting Cotorsion Pairs

Let R be a ring and T a 1-tilting right R-module. Then T isof countable type. Moreover, T is of finite type in the casewhere R is a Prüfer domain. 2000 Mathematics Subject Classification16D90, 16D30, 13G05 (primary), 03E75, 20K40, 16G99 (secondary).     

Left Cotorsion Rings

It is proved that if R is an associative ring that is cotorsionas a left module over itself, and J is the Jacobson radicalof R, then the quotient ring R/J is a left self-injective vonNeumann regular ring and idempotents lift modulo J. In particular,if R is indecomposable, then it is a local ring. 2000 MathematicsSubject Classification 16D50, 16D90, 16L30.     

几乎余挠模

In this paper, we introduce the concept of almost cotorsion modules. A module is called almost cotorsion if it is subisomorphic to its cotorsion envelope. Some characterizations of almost cotorsion modules are given. It is also proved that every module is a direct summand of an almost cotorsion module. As an application, perfect rings are characterized in terms of almost cotorsion modules.     

Cotorsion pair extensions

Assume that S is an almost excellent extension of R. Using functors HomR（S,-） and -×R S, we establish some connections between classes of modules lR and lS, cotorsion pairs （AR, BR） and （AS, BS）. If lS is a T-extension or （and） H-extension of lR, we show that lS is a （resp., monomorphic, epimorphic, special） preenveloping class if and only if so is lR. If （AS, BS） is a TH- extension of （AR, BR）, we obtain that （AS, BS） is complete （resp., of finite type, of cofinite type, hereditary, perfect, n-tilting） if and only if so is （AR, BR）.