35 research outputs found
Tilting modules over an algebra by Igusa, SmalĂž and Todorov
AbstractThe finiteness of the little finitistic dimension of an artin algebra R is known to be equivalent to the existence of a tilting R-module T such that {T}â„=(P<â)â„ where P<â is the category of all finitely presented R-modules of finite projective dimension. Moreover, T can be taken finitely generated if and only if P<â is contravariantly finite.In this paper, we describe explicitly the structure of T for the IST-algebra, a finite-dimensional algebra with P<â not contravariantly finite. We also characterize the indecomposable modules in P<â, and all tilting classes over this algebra
-Structures on stable derivators and Grothendieck hearts
We prove that given any strong, stable derivator and a -structure on its
base triangulated category , the -structure canonically lifts to all
the (coherent) diagram categories and each incoherent diagram in the heart
uniquely lifts to a coherent one. We use this to show that the -structure
being compactly generated implies that the coaisle is closed under directed
homotopy colimit which in turns implies that the heart is an (Ab.) Abelian
category. If, moreover, is a well generated algebraic or topological
triangulated category, then the heart of any accessibly embedded (in
particular, compactly generated) -structure has a generator. As a
consequence, it follows that the heart of any compactly generated -structure
of a well generated algebraic or topological triangulated category is a
Grothendieck category.Comment: 47 page
-Structures with Grothendieck hearts via functor categories
We study when the heart of a t-structure in a triangulated category
with coproducts is AB5 or a Grothendieck category. If
satisfies Brown representability, a t-structure has an AB5 heart
with an injective cogenerator and coproduct-preserving associated homological
functor if, and only if, the coaisle has a pure-injective t-cogenerating
object. If is standard well generated, such a heart is
automatically a Grothendieck category. For compactly generated t-structures (in
any ambient triangulated category with coproducts), we prove that the heart is
a locally finitely presented Grothendieck category.
We use functor categories and the proofs rely on two main ingredients.
Firstly, we express the heart of any t-structure in any triangulated category
as a Serre quotient of the category of finitely presented additive functors for
suitable choices of subcategories of the aisle or the co-aisle that we,
respectively, call t-generating or t-cogenerating subcategories. Secondly, we
study coproduct-preserving homological functors from to complete
AB5 abelian categories with injective cogenerators and classify them, up to a
so-called computational equivalence, in terms of pure-injective objects in
. This allows us to show that any standard well-generated
triangulated category possesses a universal such
coproduct-preserving homological functor and to develop a purity theory in such
triangulated categories.Comment: 56 pages; version 2: new sections 8.3 (a classification of
t-structures with definable co-aisles in terms of suspended ideals of the
compacts) and 8.4 (existence of right adjacent co-t-structures) adde
Deconstructibility and the Hill lemma in Grothendieck categories
A full subcategory of a Grothendieck category is called deconstructible if it
consists of all transfinite extensions of some set of objects. This concept
provides a handy framework for structure theory and construction of
approximations for subcategories of Grothendieck categories. It also allows to
construct model structures and t-structures on categories of complexes over a
Grothendieck category. In this paper we aim to establish fundamental results on
deconstructible classes and outline how to apply these in the areas mentioned
above. This is related to recent work of Gillespie, Enochs, Estrada, Guil
Asensio, Murfet, Neeman, Prest, Trlifaj and others.Comment: 20 pages; version 2: minor changes, misprints corrected, references
update
On exact categories and applications to triangulated adjoints and model structures
We show that Quillen's small object argument works for exact categories under
very mild conditions. This has immediate applications to cotorsion pairs and
their relation to the existence of certain triangulated adjoint functors and
model structures. In particular, the interplay of different exact structures on
the category of complexes of quasi-coherent sheaves leads to a streamlined and
generalized version of recent results obtained by Estrada, Gillespie, Guil
Asensio, Hovey, J{\o}rgensen, Neeman, Murfet, Prest, Trlifaj and possibly
others.Comment: 38 pages; version 2: major revision, more explanation added at
several places, reference list updated and extended, misprints correcte