3,506 research outputs found
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
Definable orthogonality classes in accessible categories are small
We lower substantially the strength of the assumptions needed for the
validity of certain results in category theory and homotopy theory which were
known to follow from Vopenka's principle. We prove that the necessary
large-cardinal hypotheses depend on the complexity of the formulas defining the
given classes, in the sense of the Levy hierarchy. For example, the statement
that, for a class S of morphisms in a locally presentable category C of
structures, the orthogonal class of objects is a small-orthogonality class
(hence reflective) is provable in ZFC if S is \Sigma_1, while it follows from
the existence of a proper class of supercompact cardinals if S is \Sigma_2, and
from the existence of a proper class of what we call C(n)-extendible cardinals
if S is \Sigma_{n+2} for n bigger than or equal to 1. These cardinals form a
new hierarchy, and we show that Vopenka's principle is equivalent to the
existence of C(n)-extendible cardinals for all n. As a consequence, we prove
that the existence of cohomological localizations of simplicial sets, a
long-standing open problem in algebraic topology, is implied by the existence
of arbitrarily large supercompact cardinals. This result follows from the fact
that cohomology equivalences are \Sigma_2. In contrast with this fact, homology
equivalences are \Sigma_1, from which it follows (as is well known) that the
existence of homological localizations is provable in ZFC.Comment: 38 pages; some results have been improved and former inaccuracies
have been correcte
On compactly generated torsion pairs and the classification of co-t-structures for commutative noetherian rings
We classify compactly generated co-t-structures on the derived category of a
commutative noetherian ring. In order to accomplish that, we develop a theory
for compactly generated Hom-orthogonal pairs (also known as torsion pairs in
the literature) in triangulated categories that resembles Bousfield
localization theory. Finally, we show that the category of perfect complexes
over a connected commutative noetherian ring admits only the trivial
co-t-structures and (de)suspensions of the canonical co-t-structure and use
this to describe all silting objects in the category.Comment: 34 pages. Version 2: minor corrections, references added and update
The strong global dimension of piecewise hereditary algebras
Let T be a tilting object in a triangulated category equivalent to the
bounded derived category of a hereditary abelian category with finite
dimensional homomorphism spaces and split idempotents. This text investigates
the strong global dimension, in the sense of Ringel, of the endomorphism
algebra of T. This invariant is expressed using the infimum of the lengths of
the sequences of tilting objects successively related by tilting mutations and
where the last term is T and the endomorphism algebra of the first term is
quasi-tilted. It is also expressed in terms of the hereditary abelian
generating subcategories of the triangulated category.Comment: Final published version. After refereeing, historical considerations
were added and the length of the article was reduced: Introduction and
Section 1 were reformulated; Subsection 2.1 was moved to Section 1 (with an
abridged proof); Subsection 3.2 was reformulated (with an abridged proof);
The proof in A.5 was rewritten (now shorter); And minor rewording was
processed throughout the articl
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