Exact Category of Modules of Constant Jordan Type

AbstractWe exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.

Download Full-text

Pseudo-atoms, atoms and a Jordan type decomposition in effect algebras

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2008.03.003 ◽

2008 ◽

Vol 344 (1) ◽

pp. 238-252 ◽

Cited By ~ 7

Author(s):

Mona Khare ◽

Akhilesh Kumar Singh

Keyword(s):

Effect Algebras ◽

Jordan Type ◽

Type Decomposition

Download Full-text

Extension of matrix pencil reduction to abelian categories

Journal of Algebra and Its Applications ◽

10.1142/s0219498818500627 ◽

2018 ◽

Vol 17 (04) ◽

pp. 1850062

Author(s):

Olivier Verdier

Keyword(s):

Normal Form ◽

Vector Space ◽

Decomposition Theorem ◽

Abelian Category ◽

Jordan Canonical Form ◽

Main Technique ◽

Abelian Categories ◽

Category Of Modules ◽

Space Case ◽

And Control

Matrix pencils, or pairs of matrices, are used in a variety of applications. By the Kronecker decomposition theorem, they admit a normal form. This normal form consists of four parts, one part based on the Jordan canonical form, one part made of nilpotent matrices, and two other dual parts, which we call the observation and control part. The goal of this paper is to show that large portions of that decomposition are still valid for pairs of morphisms of modules or abelian groups, and more generally in any abelian category. In the vector space case, we recover the full Kronecker decomposition theorem. The main technique is that of reduction, which extends readily to the abelian category case. Reductions naturally arise in two flavors, which are dual to each other. There are a number of properties of those reductions which extend remarkably from the vector space case to abelian categories. First, both types of reduction commute. Second, at each step of the reduction, one can compute three sequences of invariant spaces (objects in the category), which generalize the Kronecker decomposition into nilpotent, observation and control blocks. These sequences indicate whether the system is reduced in one direction or the other. In the category of modules, there is also a relation between these sequences and the resolvent set of the pair of morphisms, which generalizes the regular pencil theorem. We also indicate how this allows to define invariant subspaces in the vector space case, and study the notion of strangeness as an example.

Download Full-text

Homological Dimensions Relative to Special Subcategories

Algebra Colloquium ◽

10.1142/s1005386721000122 ◽

2021 ◽

Vol 28 (01) ◽

pp. 131-142

Author(s):

Weiling Song ◽

Tiwei Zhao ◽

Zhaoyong Huang

Keyword(s):

Abelian Category ◽

Projective Dimension ◽

Homological Dimensions ◽

Category Of Modules ◽

The Right

Let [Formula: see text] be an abelian category, [Formula: see text] an additive, full and self-orthogonal subcategory of [Formula: see text] closed under direct summands, [Formula: see text] the right Gorenstein subcategory of [Formula: see text] relative to [Formula: see text], and [Formula: see text] the left orthogonal class of [Formula: see text]. For an object [Formula: see text] in [Formula: see text], we prove that if [Formula: see text] is in the right 1-orthogonal class of [Formula: see text], then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical; if the [Formula: see text]-projective dimension of [Formula: see text] is finite, then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical. We also prove that the supremum of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension and that of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension coincide. Then we apply these results to the category of modules.

Download Full-text

PROPERLY STRATIFIED ALGEBRAS AND TILTING

Proceedings of the London Mathematical Society ◽

10.1017/s0024611505015431 ◽

2005 ◽

Vol 92 (1) ◽

pp. 29-61 ◽

Cited By ~ 8

Author(s):

ANDERS FRISK ◽

VOLODYMYR MAZORCHUK

Keyword(s):

Projective Dimension ◽

Finitistic Dimension ◽

Tilting Module ◽

Tilting Modules ◽

Category Of Modules ◽

Cotilting Modules ◽

Contravariantly Finite ◽

Properly Stratified Algebras ◽

Stratified Algebra ◽

Generalized Tilting Module

We study the properties of tilting modules in the context of properly stratified algebras. In particular, we answer the question of when the Ringel dual of a properly stratified algebra is properly stratified itself, and show that the class of properly stratified algebras for which the characteristic tilting and cotilting modules coincide is closed under taking the Ringel dual. Studying stratified algebras whose Ringel dual is properly stratified, we discover a new Ringel-type duality for such algebras, which we call the two-step duality. This duality arises from the existence of a new (generalized) tilting module for stratified algebras with properly stratified Ringel dual. We show that this new tilting module has a lot of interesting properties; for instance, its projective dimension equals the projectively defined finitistic dimension of the original algebra, it guarantees that the category of modules of finite projective dimension is contravariantly finite, and, finally, it allows one to compute the finitistic dimension of the original algebra in terms of the projective dimension of the characteristic tilting module.

Download Full-text