29 research outputs found
Recollements of Module Categories
We establish a correspondence between recollements of abelian categories up
to equivalence and certain TTF-triples. For a module category we show,
moreover, a correspondence with idempotent ideals, recovering a theorem of
Jans. Furthermore, we show that a recollement whose terms are module categories
is equivalent to one induced by an idempotent element, thus answering a
question by Kuhn.Comment: Comments are welcom
Realisation functors in tilting theory
Derived equivalences and t-structures are closely related. We use realisation functors associated to t-structures in triangulated categories to establish a derived Morita theory for abelian categories with a projective generator or an injective cogenerator. For this purpose we develop a theory of (non-compact, or large) tilting and cotilting objects that generalises the preceding notions in the literature. Within the scope of derived Morita theory for rings we show that, under some assumptions, the realisation functor is a derived tensor product. This fact allows us to approach a problem by Rickard on the shape of derived equivalences. Finally, we apply the techniques of this new derived Morita theory to show that a recollement of derived categories is a derived version of a recollement of abelian categories if and only if there are tilting or cotilting t-structures glueing to a tilting or a cotilting t-structure. As a further application, we answer a question by Xi on a standard form for recollements of derived module categories for finite dimensional hereditary algebras
Criteria for flatness and injectivity
Let be a commutative Noetherian ring. We give criteria for flatness of
-modules in terms of associated primes and torsion-freeness of certain
tensor products. This allows us to develop a criterion for regularity if
has characteristic , or more generally if it has a locally contracting
endomorphism. Dualizing, we give criteria for injectivity of -modules in
terms of coassociated primes and (h-)divisibility of certain \Hom-modules.
Along the way, we develop tools to achieve such a dual result. These include a
careful analysis of the notions of divisibility and h-divisibility (including a
localization result), a theorem on coassociated primes across a \Hom-module
base change, and a local criterion for injectivity.Comment: 19 page
Discrete derived categories I: homomorphisms, autoequivalences and t-structures
Discrete derived categories were studied initially by Vossieck (J Algebra 243:168–176, 2001) and later by Bobiński et al. (Cent Eur J Math 2:19–49, 2004). In this article, we describe the homomorphism hammocks and autoequivalences on these categories. We classify silting objects and bounded t-structures