Singular extensions and triangulated categories (original) (raw)

Triangulated Subcategories of Extensions and Triangles of Recollements

2016

In a triangulated category T with a pair of triangulated subcategories X and Y, one may consider the subcategory of extensions X * Y. We give conditions for X * Y to be triangulated and use them to provide tools for constructing stable t-structures. In particular, we show how to construct so-called triangles of recollements, that is, triples of stable t-structures of the form (X, Y), (Y, Z), (Z, X). We easily recover some triangles of recollements known from the literature.

Universal Coefficient Theorem in Triangulated Categories

Algebras and Representation Theory, 2008

We consider a homology theory h : T → A on a triangulated category T with values in a graded abelian category A. If the functor h reflects isomorphisms, is full and is such that for any object x in A there is an object X in T with an isomorphism between h(X) and x, we prove that A is a hereditary abelian category, all idempotents in T split and the kernel of h is a square zero ideal which as a bifunctor on T is isomorphic to Ext 1 A (h(−)[1], h(−)).

Linked exact triples of triangulated categories and a calculus of t-structures

2006

We introduce a new formalism of exact triples of triangulated categories arranged in certain types of diagrams. We prove that these arrangements are well-behaved relative to the process of gluing and ungluing t-structures defined on the indicated categories and we connect our constructs s to• a problem (from number theory) involving derived categories. We also briefly address a possible connection with a result of R. Thomason.

From recollements of abelian categories to recollements of triangulated categories

arXiv: Rings and Algebras, 2020

In this paper, we first provide an explicit procedure to glue complete hereditary cotorsion pairs along the recollement (mathcalA,mathcalC,mathcalB)(\mathcal{A},\mathcal{C},\mathcal{B})(mathcalA,mathcalC,mathcalB) of abelian categories with enough projective and injective objects. As a consequence, we investigate how to establish recollements of triangulated categories from recollements of abelian categories by using the theory of exact model structures. Finally, we give applications to contraderived categories, projective stable derived categories and stable categories of Gorenstein injective modules over an upper triangular matrix ring.

Locally finite triangulated categories

Journal of Algebra, 2005

A k-linear triangulated category A is called locally finite provided X∈ind A dim k Hom A (X, Y ) < ∞ for any indecomposable object Y in A. It has Auslander-Reiten triangles. In this paper, we show that if a (connected) triangulated category has Auslander-Reiten triangles and contains loops, then its Auslander-Reiten quiver is of the formL n :

Ideal cotorsion theories in triangulated categories

2015

We study ideal cotorsion pairs associated to almost exact structures in extension closed subcategories of triangulated categories. This approach allows us to extend the recent ideal approximations theory developed by Fu, Herzog et al. for exact categories in the above mentioned context. In the last part of the paper we apply the theory in order to study projective classes (in particular localization or smashing subcategories) in compactly generated categories.

Uniqueness of triangulated categories with a heart of dimension 0 or 1

arXiv: Category Theory, 2020

In this article, we prove that every triangulated category with a heart (of a bounded t-structure) of homological dimension at most 1 is uniquely determined up to exact equivalence. The last section describes how to obtain a t-structure from a semiorthogonal decomposition with compatible t-structures. This will provide an associated heart of homological dimension at most 1 for any full strong exceptional sequence E1,E2E_1 , E_2E1,E2 with finite dimensional hom(E1,E2)\hom(E_1, E_2)hom(E1,E2). As a consequence, a triangulated category admitting such an exceptional sequence is determined by dimhom(E1,E2)\dim \hom (E_1, E_2)dimhom(E_1,E_2).