Reduction of exact structures

Abstract

Examples of exact categories in representation theory are given by the category of ∆−filtered modules over quasi-hereditary algebras, but also by various categories related to matrix problems, such as poset representations or representations of bocses. Motivated by the matrix problem background, we study in this article the reduction of exact structures, and consider the poset (Ex(A), ⊂) of all exact structures on a fixed additive category A. This poset turns out to be a complete lattice, and under suitable conditions results of Enomoto's imply that it is boolean.We initiate in this article a detailed study of exact structures E by generalizing notions from abelian categories such as the length of an object relative to E and the quiver of an exact category (A, E). We investigate the Gabriel-Roiter measure for (A, E), and further study how these notions change when the exact structure varies. category of finite type, using Auslander algebra, or [INP18] where the more general concept of extriangulated structures is studied.While every exact category (A, E) can be embedded into a module category, notions like length or simple object cannot be borrowed from such an embedding. The first goal of this paper is to give an intrinsic definition relative to the class of morphisms in E, thus, in section 3, we call an object E−simple if it does not admit proper monomorphisms that belong to the class E. And we say that X is anThis change of definition requires to work out a number of notions and results that are granted in abelian categories, such as the notion of simple objects, artinian and finite objects or the length of an object. It turns out that in general not all the desired properties can be guaranteed. We also define, in section 3, the notion of the quiver of an exact category Q(A, E).The motivation for studying reductions of exact structures stems from the matrix reduction technique. The method of matrix reduction has been applied successfully by the Kiev school to solve various important problems in representation theory, like the Brauer-Thrall conjectures, or to show the tame-wild dichotomy. While the basic technique is elementary, the formalism of matrix reductions is somewhat complicated. Various models have been proposed to formalise matrix reductions: poset representations or bimodule problems cover only some cases. For the general case, one needs to study bocs representations, as introduced by Roiter in [Ro79], or iterated quotients of bimodule problems as in [Brü].This result allows to show that the length function l E of a finite essentially small exact category (A, E) is a measure for the poset ObjA in the sense of Gabriel [Gab]. We further show that most of the work of Krause [Kr11] on the Gabriel-Roiter measure for abelian length categories can be generalized to the context of exact categories: For the partially ordered set (indA, ⊂ E ) equipped with the length function l E , we define the Gabriel-Roiter measure as a morphism of partially ordered sets which refines the length function l E , see Theorem 7...

On the lattices of exact and weakly exact structures

“…Lattice of exact structures revisited. We know by [BHLR,Theorem 5.3] that the class of exact structures on an additive category Ex(A) forms a lattice. In order to study the properties of this lattice, we show that it is isomorphic to the lattice of closed additive sub-bifunctors of Ext 1 A (−, −) defined in Section 4.…”

“…Theorem 7.11. [BHLR,5.1,5.3,5.4] Let A be an additive category. The poset Ex(A) of exact structures E on A forms a bounded complete lattice (Ex(A), ⊆, ∧, ∨ E ) under the following operations:…”

“…As discussed in Proposition 4.9, for any additive category A there is a bijection between exact structures and closed additive sub-bifunctors of E max . We already know that the exact structures form a lattice [BHLR,Theorem 5.3]. In this section we define a lattice structure on the class CBiFun(A) of closed additive sub-bifunctors of E max .…”

Exact structures and degeneration of Hall algebras

“…This lattice can be identified with the lattice of the subsets of the set of all the indecomposable objects that are not projective with respect to the maximal exact structure on A. Brüstle, Hassoun, Langford and Roy investigated relations between changes of exact structures to smaller ones (i.e. having strictly less conflations) and some matrix reduction problems, see [16] for the details (see also [26]). This inspired the name of the reduction of exact structures for this procedure.…”

Examples and non-examples of integral categories and the admissible intersection property

“…In studying lengths of objects in exact categories, Brüstle, Hassoun, Langford and Roy [3,Exam. 6.9] showed that an analogue of the classic Jordan-Hölder property can fail for an arbitrary exact category; see also Enomoto [8].…”