On Differential Torsion Theories and Rings with Several Objects

Abstract

Let ${\mathcal{R}}$ be a small preadditive category, viewed as a “ring with several objects.” A right${\mathcal{R}}$-module is an additive functor from ${\mathcal{R}}^{\text{op}}$ to the category $Ab$ of abelian groups. We show that every hereditary torsion theory on the category $({\mathcal{R}}^{\text{op}},Ab)$ of right ${\mathcal{R}}$-modules must be differential.

$$(A,\delta )$$(A,δ)-modules, Hochschild homology and higher derivations

“…In other words, the objects that we call (A, δ)-modules in Definition 2.1 are those that Tanaka [17, Definition 3.1] called "pre-(A, δ)-modules" (see also Bland [6]). A similar notion, in the context of "rings with several objects" has been studied in [3].…”

Fredholm modules over categories, Connes periodicity and classes in cyclic cohomology

“…In this paper, we study Fredholm modules over linear categories, along with their Chern characters taking values in cyclic cohomology. Our idea is to have a counterpart of the algebraic notion of modules over a category, a subject which has been highly developed in the literature (see, for instance, [7], [17], [35], [36], [44], [45]). A small preadditive category is treated as a ring with several objects, following an idea first advanced by Mitchell [39].…”

Cycles over DGH-semicategories and pairings in categorical Hopf-cyclic cohomology

“…A small Hopf-module category may be treated as a "Hopf module algebra with several objects," in the same way as a small preadditive category plays the role of a "ring with several objects" in the sense of Mitchell [38]. In fact, the replacement of rings by small preadditive categories has been widely studied in the literature (see, for instance, [6], [16], [33], [35], [43], [44]). Further, cyclic modules associated to Hopf-module categories have been studied by Kaygun and Khalkhali [26], while the Hochschild-Mitchell cohomology of a Hopf-comodule category has been studied by Herscovich and Solotar [21].…”