Karoubi-Villamayor and Quillen functors for triangular categories (original) (raw)

Torsion Pairs in Categories of Modules over a Preadditive Category

Bulletin of the Iranian Mathematical Society

It is a result of Gabriel that hereditary torsion pairs in categories of modules are in bijection with certain filters of ideals of the base ring, called Gabriel filters or Gabriel topologies. A result of Jans shows that this bijection restricts to a correspondence between (Gabriel filters that are uniquely determined by) idempotent ideals and TTF triples. Over the years, these classical results have been extended in several different directions. In this paper we present a detailed and self-contained exposition of an extension of the above bijective correspondences to additive functor categories over small preadditive categories. In this context, we also show how to deduce parametrizations of hereditary torsion theories of finite type, Abelian recollements by functor categories, and centrally splitting TTFs.

Module and Comodule Categories-A Survey

Proc. of the Mathematics Conference (Birzeit University …, 2000

The theory of modules over associative algebras and the theory of comodules for coassociative coalgebras were developed fairly independently during the last decades. In this survey we display an intimate connection between these areas by the notion of categories subgenerated by an object. After a review of the relevant techniques in categories of left modules, applications to the bimodule structure of algebras and comodule categories are sketched.

On the K-theory of mathbbZ\mathbb{Z}mathbbZ-categories

arXiv (Cornell University), 2022

We establish connections between the concepts of Noetherian, regular coherent, and regular n-coherent categories for Z-linear categories with finitely many objects and the corresponding notions for unital rings. These connections enable us to obtain a negative K-theory vanishing result, a fundamental theorem, and a homotopy invariance result for the K-theory of Z-linear categories.

R A ] 9 A pr 2 00 8 MONADS AND COMONADS IN MODULE CATEGORIES

2021

Let A be a ring and MA the category of A-modules. It is well known in module theory that for any A-bimodule B, B is an A-ring if and only if the functor − ⊗A B : MA → MA is a monad (or triple). Similarly, an A-bimodule C is an A-coring provided the functor − ⊗A C : MA → MA is a comonad (or cotriple). The related categories of modules (or algebras) of −⊗A B and comodules (or coalgebras) of − ⊗A C are well studied in the literature. On the other hand, the right adjoint endofunctors HomA(B,−) and HomA(C,−) are a comonad and a monad, respectively, but the corresponding (co)module categories did not find much attention so far. The category of HomA(B,−)-comodules is isomorphic to the category of B-modules, while the category of HomA(C,−)-modules (called Ccontramodules by Eilenberg and Moore) need not be equivalent to the category of C-comodules. The purpose of this paper is to investigate these categories and their relationships based on some observations of the categorical background. Th...

Matlis category equivalences for a ring epimorphism

Journal of Pure and Applied Algebra, 2020

Under mild assumptions, we construct the two Matlis additive category equivalences for an associative ring epimorphism u : R −→ U. Assuming that the ring epimorphism is homological of flat/projective dimension 1, we discuss the abelian categories of u-comodules and u-contramodules and construct the recollement of unbounded derived categories of R-modules, U-modules, and complexes of R-modules with u-co/contramodule cohomology. Further assumptions allow to describe the third category in the recollement as the unbounded derived category of the abelian categories of u-comodules and u-contramodules. For commutative rings, we also prove that any homological epimorphism of projective dimension 1 is flat. Injectivity of the map u is not required. Contents 19 7. Two Fully Faithful Triangulated Functors 21 8. Kronecker Quiver Example 23 References 29

Grothendieck ring of pretriangulated categories

2004

We consider the abelian group PTPTPT generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semi-orthogonal decompositions of corresponding triangulated categories. We introduce an operation of "multiplication" bullet\bulletbullet on the collection of DG categories which makes this abelian group into a commutative ring. A few applications are considered: representability of "standard" functors between derived categories of coherent sheaves on smooth projective varieties and a construction of an interesting motivic measure.

Categorical Results in the Theory of Two-Crossed Modules of Commutative Algebras

arXiv (Cornell University), 2011

In this paper we explore some categorical results of 2-crossed module of commutative algebras extending work of Porter in [18]. We also show that the forgetful functor from the category of 2-crossed modules to the category of k-algebras, taking {L, M, P, ∂2, ∂1} to the base algebra P , is fibred and cofibred considering the pullback (coinduced) and induced 2-crossed modules constructions, respectively. Also we consider free 2crossed modules as an application of induced 2-crossed modules. Categorical Results in the Theory of Two-Crossed Modules of Commutative Algebras 2 We end with an application which leads to link free 2-crossed modules with induced 2-crossed modules. Conventions Throughout this paper k will be a fixed commutative ring and R will be a kalgebra with identity. All algebras will be commutative and actions will be left and the right actions in some references will be rewritten by using left actions.