Preadditive Category Research Articles

SOME REMARKS ON CATEGORIES OF MODULES MODULO MORPHISMS WITH ESSENTIAL KERNEL OR SUPERFLUOUS IMAGE

For an ideal <TEX>$\mathcal{I}$</TEX> of a preadditive category <TEX>$\mathcal{A}$</TEX>, we study when the canonical functor <TEX>$\mathcal{C}:\mathcal{A}{\rightarrow}\mathcal{A}/\mathcal{I}$</TEX> is local. We prove that there exists a largest full subcategory <TEX>$\mathcal{C}$</TEX> of <TEX>$\mathcal{A}$</TEX>, for which the canonical functor <TEX>$\mathcal{C}:\mathcal{C}{\rightarrow}\mathcal{C}/\mathcal{I}$</TEX> is local. Under this condition, the functor <TEX>$\mathcal{C}$</TEX>, turns out to be a weak equivalence between <TEX>$\mathcal{C}$</TEX>, and <TEX>$\mathcal{C}/\mathcal{I}$</TEX>. If <TEX>$\mathcal{A}$</TEX> is additive (with splitting idempotents), then <TEX>$\mathcal{C}$</TEX> is additive (with splitting idempotents). The category <TEX>$\mathcal{C}$</TEX> is ample in several cases, such as the case when <TEX>$\mathcal{A}$</TEX>=Mod-R and <TEX>$\mathcal{I}$</TEX> is the ideal <TEX>${\Delta}$</TEX> of all morphisms with essential kernel. In this case, the category <TEX>$\mathcal{C}$</TEX> contains, for instance, the full subcategory <TEX>$\mathcal{F}$</TEX> of Mod-R whose objects are all the continuous modules. The advantage in passing from the category <TEX>$\mathcal{F}$</TEX> to the category <TEX>$\mathcal{F}/\mathcal{I}$</TEX> lies in the fact that, although the two categories <TEX>$\mathcal{F}$</TEX> and <TEX>$\mathcal{F}/\mathcal{I}$</TEX> are weakly equivalent, every endomorphism has a kernel and a cokernel in <TEX>$\mathcal{F}/{\Delta}$</TEX>, which is not true in <TEX>$\mathcal{F}$</TEX>. In the final section, we extend our theory from the case of one ideal<TEX>$\mathcal{I}$</TEX> to the case of <TEX>$n$</TEX> ideals <TEX>$\mathcal{I}_$</TEX>, <TEX>${\ldots}$</TEX>, <TEX>$\mathca{l}_n$</TEX>.

On Some Noteworthy Pairs of Ideals in Mod-R

An additive functor \(F \colon {\mathcal A}\to{\mathcal B}\) between preadditive categories \(\mathcal A\) and \(\mathcal B\) is said to be a local functor if, for every morphism \(f\colon A\to A'\) in \(\mathcal A\), F(f) isomorphism in \(\mathcal B\) implies f isomorphism in \(\mathcal A\). We show that there exist several pairs \((\mathcal I_1,\mathcal I_2)\) of ideals of \(\mathcal A\) for which the canonical functor \(\mathcal A\to\mathcal A/\mathcal I_1\times \mathcal A/\mathcal I_2\) is a local functor. In most of our examples, the category \(\mathcal A\) is a full subcategory of the category Mod -R of all right modules over a ring R. These pairs of ideals arise in a surprisingly natural way and enjoy several properties. Ideals are kernels of functors, and most of our examples of ideals are kernels of important and well studied functors. E.g., (1) the kernel Δ of the canonical functor P of Mod -R into its spectral category Spec(Mod -R), so that Δ is the ideal of all morphisms with an essential kernel; (2) the kernel Σ of the dual functor F of P, so that Σ is the ideal of all morphisms with a superfluous image; (3) the kernels Δ(1) and Σ(1) of the first derived functors P(1) and F(1) of P and F, respectively; (4) the kernels of suitable functors Hom and ⊗ and their first derived functors \({\rm Ext}^1_R\) and \({\rm Tor}^R_1\).

Tilting Theory and Functor Categories II. Generalized Tilting

In this paper we continue the project of generalizing tilting theory to the category of contravariant functors \(\mathrm{Mod}(\mathcal{C})\), from a skeletally small preadditive category \(\mathcal{C}\) to the category of abelian groups, initiated in [15]. We introduce the notion of a generalized tilting category \(\mathcal{T}\), and we concentrate here on extending Happel’s theorem to \(\mathrm{Mod}(\mathcal{C})\); more specifically, we prove that there is an equivalence of triangulated categories \(\mathcal{D}^{b}( \mathrm{Mod}(\mathcal{C}))\cong \mathcal{D}^{b}(\mathrm{Mod}(\mathcal{T}))\). We then add some restrictions on our category \(\mathcal{C}\), in order to obtain a version of Happel’s theorem for the categories of finitely presented functors. We end the paper proving that some of the theorems for artin algebras, relating tilting with contravariantly finite categories proved in Auslander and Reiten (Adv Math 12(3):306–366, 1974; Adv Math 86(1):111–151, 1991), can be extended to the category of finitely presented functors \(\mathrm{mod}(\mathcal{C})\), with \(\mathcal{C}\) a dualizing variety.

MAXIMAL IDEALS IN PREADDITIVE CATEGORIES AND SEMILOCAL CATEGORIES

The first aim of this article is to study maximal ideals of a preadditive category [Formula: see text]. Maximal ideals, which do not exist in general for arbitrary preadditive categories, are associated to a maximal ideal of the endomorphism ring of an object and always exist when the category is semilocal. If [Formula: see text] is additive and semilocal, any skeleton [Formula: see text] of [Formula: see text] is a Krull monoid and we are able to characterize the essential valuations of [Formula: see text] and provide some natural divisor homomorphisms and divisor theories of [Formula: see text].

One-sided Grothendieck quotients

We introduce one-sided thick subcategories \({\mathcal{C}}\) of an arbitrary preadditive category \({\mathcal{A}}\) and define a quotient category \({\mathcal{A}}/ {\mathcal{C}}\) . When \({\mathcal{A}}\) is abelian, this concept specializes to Grothendieck’s quotient for two-sided thick \({\mathcal{C}}\) . We determine the left noetherian rings for which the injective modules form a left thick subcategory. We exhibit a class of one-sided thick subcategories in categories of coherent functors which are ubiquitous in representation theory.

Subdirect Products of Preadditive Categories and Weak Equivalences

We study the notions of coproduct and subdirect product of preadditive categories and prove a Birkhoff type theorem showing that every skeletally small preadditive category is a subdirect product of subdirectly irreducible, skeletally small, preadditive categories. Moreover, we show that every direct-sum decomposition of the monoid \(V({\cal A})\) of the isomorphism classes of objects of \(\cal A\) is weakly induced by a coproduct decomposition of the preadditive category \(\cal A\).

When ∏ is Isomorphic to ⊕

For a category 𝒞, we investigate the problem of when the coproduct ⊕ and the product functor ∏ from 𝒞 I to 𝒞 are isomorphic for a fixed set I, or, equivalently, when the two functors are Frobenius functors. We show that for an Ab category 𝒞 this happens if and only if the set I is finite (and even in a much general case, if there is a morphism in 𝒞 that is invertible with respect to addition). However, we show that ⊕ and ∏ are always isomorphic on a suitable subcategory of 𝒞 I which is isomorphic to 𝒞 I but is not a full subcategory. If 𝒞 is only a preadditive category, then we give an example that shows that the two functors can be isomorphic for infinite sets I. For the module category case, we provide a different proof to display an interesting connection to the notion of Frobenius corings.

A generalization of the Gabriel–Popescu theorem

In this paper we give necessary and sufficient conditions for an additive functor u : u→ C , from a small pre-additive category u to a Grothendieck category C , to realize C as a localization of the category of presheaves on u . This is a generalization of the Gabriel–Popescu theorem, which considers the case where u is fully faithful.

Products in categories of fractions and universal inversion of homomorphisms

Main result. If finite direct products exist in a category and the class of morphisms is such that the category of fractions and the canonical functor preserves these products. Using this theorem analogues of the theory of matrix localization of rings are constructed for arbitrary varieties of universal algebras and for preadditive categories.

Spectral categories and varieties of preadditive categories

In this paper we reinterpret the main results about the spectral categoric; by making use of the theory of sheaves and introducing the notion of ‘variety of preadditive categories’. This approach allows us to visualize the structure of the spectral categories better, to explain the different decompositions of a spectral category (discrete and continuous part [4], type I, II and III [14]) and to give an incisive interpretation to the dimension theory for the objects of a spectral category [7]. Moreover, we give an explicit description of the Grothendieck groups df the dense subcategories of a spectral category. Spectral categories, that is abelian categories with exact direct limits in which every exact sequence splits, naturally arise in the study of injective modules (or, more generally, in the study of the injective objects of any Grothendieck category) and their Krull-Remak-Schmidt-Gabriel decompositions [ 121. They were introduced by Gabriel and Oberst [4], who discovered that any spectral category is the product of a discrete spectral category and a continuous one. Later, on the lines of Kaplansky’s theory of types for A W*-algebras [8], Roos [ 141 discovered that every spectral category can be decomposed into a product of categories of three distinct types (type I, II and III). Finally Goodearl and Boyle [7] constructed a complete and beautiful dimension theory for the objects of a spectral category. The directly finite case of that theory is partially based on ideas of Von Neumann and Loomis. We study the spectral categories by means of the varieties of preadditive categories (their definition is given in Section 1). Essentially a variety of preadditive categories is for a preadditive category [ 121 what a ringed space is for a ring. Ringed spaces have been extensively used by Dauns and Hofmann [l] and Pierce [ 1 l] in the study of Von Neumann (bi)regular rings. Here we study the spectral categories by means of varieties of categories. Given any abelian category %, we construct an associated variety of preadditive categories having the spectrum W[ ‘c’] of the Boolean algebra of all idempotents of the center of (z’ as a basis and suitable quotient

The Dominion of Isbell

A well-known characterization of epimorphisms in the category of rings with identity is imitated to give a similar characterization of epimorphisms in the category of small pre-additive categories. From this one deduces Isbell’s “Zigzag Theorem” concerning epimorphisms in Cat.

The dominion of Isbell

A well-known characterization of epimorphisms in the category of rings with identity is imitated to give a similar characterization of epimorphisms in the category of small pre-additive categories. From this one deduces Isbell’s “Zigzag Theorem” concerning epimorphisms in Cat.