Stratifying derived module categories

The concept of recollement is used to obtain a stratification of the derived module category of a ring which may be regarded as an analogue of a composition series for groups or modules. This analogy raises the problem whether a ‘derived’ Jordan Hölder theorem holds true; that is, are such stratifications unique up to ordering and equivalence? This is indeed the case for several classes of rings, including semi-simple rings, commutative Noetherian rings, group algebras of finite groups, and finite dimensional algebras which are piecewise hereditary.     

On free commutative regular rings

It has been observed that the category of all regular rings, when viewed as a full subcategory of the category of all rings, is not an algebraic variety. However if a regular ring is viewed as a ring equipped with a unary operation q such that xx^qx = x and 0000^q= 0, then the category of all rings with this added structure is indeed a variety but it is not, in any natural way, a subcategory of the category of all rings. In this article regular rings are viewed in this way and the free commutative regular rings are constructed. They are derived from the universal commutative regular ring associated with certain polynomial rings.     

Generators and Dimensions of Derived Categories of Modules

Several years ago, Bondal, Rouquier, and Van den Bergh introduced the notion of the dimension of a triangulated category, and Rouquier proved that the bounded derived category of coherent sheaves on a separated scheme of finite type over a perfect field has finite dimension. In this article, we study the dimension of the bounded derived category of finitely generated modules over a commutative Noetherian ring. The main result of this article asserts that it is finite over a complete local ring containing a field with perfect residue field. Our methods also give a ring-theoretic proof of the affine case of Rouquier's theorem.     

Invariants of t-structures and classification of nullity classes

We construct an invariant of t-structures on the derived category of a commutative noetherian ring. This invariant is complete when restricting to the category of complexes with finitely generated bounded homology, and also gives a classification of nullity classes with the same restriction. On the full derived category of Z we show that the class of distinct t-structures do not form a set.     

Even More Spectra: Tensor Triangular Comparison Maps via Graded Commutative 2-rings

We initiate the theory of graded commutative 2-rings, a categorification of graded commutative rings. The goal is to provide a systematic generalization of Paul Balmer’s comparison maps between the spectrum of tensor-triangulated categories and the Zariski spectra of their central rings. By applying our constructions, we compute the spectrum of the derived category of perfect complexes over any graded commutative ring, and we associate to every scheme with an ample family of line bundles an embedding into the spectrum of an associated graded commutative 2-ring.     

二面体群的Grothendieck环结构

二面体群的表示范畴为对称半单monoidal范畴,因而其Grothendieck环为有限多个元素生成的交换环.本文确定了该Grothendieck环的极小生成元,并且进一步证明了该Grothendieck环与某一多项式环的商环同构.     

Classifying subcategories of modules over a commutative noetherian ring

Let R be a quotient ring of a commutative coherent regular ringby a finitely generated ideal. Hovey gave a bijection betweenthe set of coherent subcategories of the category of finitelypresented R-modules and the set of thick subcategories of thederived category of perfect R-complexes. Using this bijection,he proved that every coherent subcategory of finitely presentedR-modules is a Serre subcategory. In this paper, it is provedthat this holds whenever R is a commutative noetherian ring.This paper also yields a module version of the bijection betweenthe set of localizing subcategories of the derived categoryof R-modules and the set of subsets of Spec R which was givenby Neeman.     

二面体群的Grothendieck环结构

二面体群的表示范畴为对称半单monoidal范畴,因而其Grothendieck环为有限多个元素生成的交换环.本文确定了该Grothendieck环的极小生成元,并且进一步证明了该Grothendieck环与某一多项式环的商环同构.     

Picard groups of derived categories

We investigate the group of isomorphism classes of invertible objects in the derived category of -modules for a commutative unital ringed Grothendieck topos with enough points. When the ring has connected prime ideal spectrum for all points p of we show that is naturally isomorphic to the Cartesian product of the Picard group of -modules and the additive group of continuous functions from the space of isomorphism classes of points of to the integers . Also, for a commutative unital ring R, the group is isomorphic to the Cartesian product of Pic(R) and the additive group of continuous functions from spec R to the integers .     

New notions in derived categories of local rings

In the derived category of a local commutative noetherian ring, we define irreducible chain complexes, atomic chain complexes, minimal atomic chain complexes and chain complexes having no mod m detectable homology. Also, we define nuclear chain complexes and core of chain complexes. After defining these notions, we establish the connection between them.     

Tensor Product of Massey Products

In this paper, we interpret Massey products in terms of realizations （twitsting cochains） of certain differential graded coalgebras with values in differential graded algebras. In the case where the target algebra is the cobar construction of a differential graded commutative Hopf algebra, we construct the tensor product of realizations and show that the tensor product is strictly associative, and commutative up to homotopy.     

定义在范畴 RMnl上的张量函数

In this paper, we define a tensor functor on the category of R-n modules, where R is a commutative ring with 1, and prove the following theorems:     

Projectivity of acts and morita equivalence of monoids

In this paper we shall consider a non-additive category of A-modules, that is, instead of a ring A we take a monoid A which acts on sets from the left. These objects will be called A-acts. We investigate indecomposable A-acts and generators and characterize projectives in this category. For a given monoid A we describe all monoids B such that the category of B-acts is equivalent to the category of A-acts. In particular we find that equivalence of these categories yields an isomorphism between the monoids A and B if A is a group or finite or commutative. This differs from the additive case where the categories of modules over a commutative field and its ring of nxn matrices are equivalent. Finally we give examples of non-isomorphic monoids A and B such that the corresponding categories are equivalent.     

Local coherence of hearts associated with Thomason filtrations

Any Thomason filtration of a commutative ring yields (at least) two t-structures in the derived category of the ring, one of which is compactly generated [19], [20]. We study the hearts of these two t-structures and prove that they coincide in case of a weakly bounded below filtration. Prompted by [39], in which it is proved that the heart of a compactly generated t-structure in a triangulated category with coproduct is a locally finitely presented Grothendieck category, we study the local coherence of the hearts associated with a weakly bounded below Thomason filtration, achieving a useful recursive characterisation in case of a finite length filtration. Low length cases involve hereditary torsion classes of finite type of the ring, and even their Happel–Reiten–Smalø hearts; in these cases, the relevant characterisations are given by few module-theoretic conditions.     

Irreducible maps of commutative rings

We give necessary conditions for a map to be irreducible (in the category of finitely generated, torsion free modules) over a non-local, commutative ring and sufficient conditions when the ring is Bass. In particular, we show that an irreducible map of ZG, where G is a square free abelian group, must be a monomorphism with a simple cokernel. We also show that local endomorphism rings are necessary and sufficient for the existence of almost split sequences over a commutative Bass ring and we explicitly describe the modules and the maps in those sequences. The results in this paper enable us to describe the Auslander-Reiten quiver of a non-local Bass ring in [8].     

The Category of Maximal Cohen–Macaulay Modules as a Ring with Several Objects

Over a commutative local Cohen–Macaulay ring, we view and study the category of maximal Cohen–Macaulay modules as a ring with several objects. We compute the global dimension of this category and thereby extend some results of Iyama and Leuschke.     

On Finite-Valued Cohomology Theories

In this paper, we describe cohomology theories constructed on the category of pairs of topological spaces and continuous maps and based on finite-valued cochains with coefficients in abelian group.     

Abstract Hilbert Schemes

In analogy with classical projective algebraic geometry, Hilbert functors can be defined for objects in any Abelian category. We study the moduli problem for such objects. Using Grothendieck's general framework. We show that with suitable hypotheses the Hilbert functor is representable by an algebraic space locally of finite type over the base field. For the category of the graded modules over a strongly Noetherian graded ring, the Hilbert functor of graded modules with a fixed Hilbert series is represented by a commutative projective scheme. For the projective scheme corresponding to a suitable noncommutative graded algebra, the Hilbert functor is represented by a countable union of commutative projective schemes.     

Skew category, Galois covering and smash product of a -category

In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew category and we show that these constructions agree with the usual ones for algebras. In the case of the smash product for an infinite group our construction specialized for a ring agrees with M. Beattie's construction of a ring with local units. We recover in a categorical generalized setting the Duality Theorems of M. Cohen and S. Montgomery (1984), and we provide a unification with the results on coverings of quivers and relations by E. Green (1983). We obtain a confirmation in a quiver and relations-free categorical setting that both constructions are mutual inverses, namely the quotient of a free action category and the smash product of a graded category. Finally we describe functorial relations between the representation theories of a category and of a Galois cover of it.