Attached atoms in Grothendieck categories

Let 𝒜 be a Grothendieck category. In this paper we define attached atoms of nonzero objects of 𝒜 as a dual of the associated atoms of nonzero objects introduced by Kanda [2 Kanda, R. (2012). Classifying Serre subcategories via atom spectrum. Adv. Math. 231(3–4):1572–1588. [Google Scholar]]. If R is a (non)commutative ring, then we show that this new notion can be corresponded to the attached prime ideals of nonzero modules introduced by Macdonald [5 Macdonald, I. G. (1973). Secondary representation of modules over a commutative ring. Sympos. Math. 11:23–43. [Google Scholar]] and Annin [1 Annin, S. (2008). Attached primes over noncommutative rings. J. Pure Appl. Algebra 212:510–521.[Crossref], [Web of Science ®] , [Google Scholar]].     

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.     

Grothendieck Groups and Tilting Objects

Let C be a connected Noetherian hereditary Abelian category with a Serre functor over an algebraically closed field k, with finite-dimensional homomorphism and extension spaces. Using the classification of such categories from our 1999 preprint, we prove that if C has some object of infinite length, then the Grothendieck group of C is finitely generated if and only if C has a tilting object.     

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.     

Models for homotopy categories of injectives and Gorenstein injectives

A natural generalization of locally noetherian and locally coherent categories leads us to define locally type FP_∞ categories. They include not just all categories of modules over a ring, but also the category of sheaves over any concentrated scheme. In this setting we generalize and study the absolutely clean objects recently introduced in [5 Bravo, D., Gillespie, J., Hovey, M. The stable module category of a general ring (arXiv:1405.5768). [Google Scholar]]. We show that 𝒟(𝒜𝒞), the derived category of absolutely clean objects, is always compactly generated and that it is embedded in K(Inj), the chain homotopy category of injectives, as a full subcategory containing the DG-injectives. Assuming the ground category 𝒢 has a set of generators satisfying a certain vanishing property, we also show that there is a recollement relating 𝒟(𝒜𝒞) to the (also compactly generated) derived category 𝒟(𝒢). Finally, we generalize the Gorenstein AC-injectives of [5 Bravo, D., Gillespie, J., Hovey, M. The stable module category of a general ring (arXiv:1405.5768). [Google Scholar]], showing that they are the fibrant objects of a cofibrantly generated model structure on 𝒢.     

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.     

左n-角范畴的稳定化及Grothendieck群

本文证明了左n-角范畴的稳定化范畴具有n-角结构,并证明了当n为奇数时,左n-角范畴的Grothendieck群与其稳定化范畴的Grothendieck群同构.     

Grothendieck Representations of Categories and Canonical Noncommutative Topologies

From a representation of a category in terms of a class of Grothendieck categories, a canonical topologisation of the objects follows. The noncommutative topology recently introduced in noncommutative geometry is an example of this. Basic results of sheaf theory are derived in this setting.     

Group completion in the K-theory and Grothendieck–Witt theory of proto-exact categories

We study the algebraic K-theory and Grothendieck–Witt theory of proto-exact categories, with a particular focus on classes of examples of ${\mathbb{F}}_1$-linear nature. Our main results are analogues of theorems of Quillen and Schlichting, relating the K-theory or Grothendieck–Witt theory spaces of proto-exact categories defined using the (hermitian) Q-construction and group completion.     

Quantitative Grothendieck property

A Banach space X is Grothendieck if the weak and the weak^? convergence of sequences in the dual space X^?$X^{?}$ coincide. The space ?^∞${?}^{\infty }$ is a classical example of a Grothendieck space due to Grothendieck. We introduce a quantitative version of the Grothendieck property, we prove a quantitative version of the above-mentioned Grothendieck?s result and we construct a Grothendieck space which is not quantitatively Grothendieck. We also establish the quantitative Grothendieck property of L^∞(μ)$L^{\infty }(\mu )$ for a σ-finite measure μ.     

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.     

Morita Duality for Grothendieck Categories with Applications to Coalgebras

Finiteness conditions of reflexive objects of a Morita duality of Grothendieck category is studied. It is observed that the relation between coproduct and product is a fundamental fact. We show that for coalgebras with a duality, the class of reflexive objects coincides with the class of quasi-finite comodules.     

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.     

Abelian categories, almost split sequences, and comodules

The following are equivalent for a skeletally small abelian Hom-finite category over a field with enough injectives and each simple object being an epimorphic image of a projective object of finite length.

(a) Each indecomposable injective has a simple subobject.

(b) The category is equivalent to the category of socle-finitely copresented right comodules over a right semiperfect and right cocoherent coalgebra such that each simple right comodule is socle-finitely copresented.

(c) The category has left almost split sequences.

     

Grothendieck Institutions

We extend indexed categories, fibred categories, and Grothendieck constructions to institutions. We show that the 2-category of institutions admits Grothendieck constructions (in a general 2-categorical sense) and that any split fibred institution is equivalent to a Grothendieck institution of an indexed institution. We use Grothendieck institutions as the underlying mathematical structure for the semantics of multi-paradigm (heterogenous) algebraic specification. We recuperate the so-called extra theory morphisms as ordinary theory morphisms in a Grothendieck institution. We investigate the basic mathematical properties of Grothendieck institutions, such as theory colimits, liberality (free constructions), exactness (model amalgamation), and inclusion systems by globalisation from the local level of the indexed institution to the level of the Grothendieck institution.     

Coarse Geometry via Grothendieck Topologies

We show that the coarse cohomology, defined by J. ROE for complete metric spaces, coincides with the sheaf cohomology of the constant sheaf IR for a certain Grothendieck topology. Therefore many properties of coarse cohomology are easy consequences of general principles.     

Presentations of Grothendieck Constructions

We will give quiver presentations of the Grothendieck constructions of functors from a small category to the 2-category of 𝕜-categories for a commutative ring 𝕜.     

A measure-theoretic Grothendieck inequality

In this note we develop a notion of integration with respect to a bimeasure μ that allows integration of functions in the projective tensor product , where ν₁ and ν₂ are Grothendieck measures for μ. This integral, which agrees with the standard notion of integration with respect to a bimeasure, allows us to integrate inner products and provides a generalization of the Grothendieck inequality to a measure-theoretic setting.     

On Grothendieck Subspaces

The modulus of an order bounded functional on a Riesz space is the sum of a pair of Riesz homomorphisms if and only if the kernel of this functional is a Grothendieck subspace of the ambient Riesz space. An operator version of this fact is given.Original Russian Text Copyright © 2005 Kutateladze S. S.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 620–624, May–June, 2005.     

Grothendieck topologies and deformation Theory II

Starting from a sheaf of associative algebras over a scheme we show thatits deformation theory is described by cohomologies of a canonical object,called the cotangent complex, in the derived category of sheaves ofbi-modules over this sheaf of algebras. The passage from deformations tocohomology is based on considering a site which is naturally constructed outof our sheaf of algebras. It turns out that on the one hand, cohomology ofcertain sheaves on this site control deformations, and on the other hand,they can be rewritten in terms of the category of sheaves of bi-modules.