Good tilting modules and recollements of derived module categories

Let $T$ be an infinitely generated tilting module of projective dimension at most one over an arbitrary associative ring $A$, and let $B$ be the endomorphism ring of $T$. In this paper, we prove that if $T$ is good then there exists a ring $C$, a homological ring epimorphism B\ra C and a recollement among the (unbounded) derived module categories \D{C} of $C$, \D{B} of $B$, and \D{A} of $A$. In particular, the kernel of the total left derived functor $T\otimes_B^{\mathbb L}-$ is triangle equivalent to the derived module category \D{C}. Conversely, if the functor $T\otimes_B^{\mathbb L}-$ admits a fully faithful left adjoint functor, then $T$ is a good tilting module. We apply our result to tilting modules arising from ring epimorphisms, and can then describe the rings $C$ as coproducts of two relevant rings. Further, in case of commutative rings, we can weaken the condition of being tilting modules, strengthen the rings $C$ as tensor products of two commutative rings, and get similar recollements. Consequently, we can produce examples (from commutative algebra and $p$-adic number theory, or Kronecker algebra) to show that two different stratifications of the derived module category of a ring by derived module categories of rings may have completely different derived composition factors (even up to ordering and up to derived equivalence),or different lengths. This shows that the Jordan-H\"older theorem fails even for stratifications by derived module categories, and also answers negatively an open problem by Angeleri-H\"ugel, K\"onig and Liu

On the Quasi-Heredity of Birman–Wenzl Algebras

AbstractIn this paper we consider the Birman–Wenzl algebras over an arbitrary field and prove that they are cellular in the sense of Graham and Lehrer. Furthermore, we determine for which parameters the Birman–Wenzl algebras are quasi-hereditary. So the general theory of cellular algebras and quasi-hereditary algebras applies to Birman–Wenzl algebras. As a consequence, we can determine all irreducible representations of the Birman–Wenzl algebras by linear algebra methods. We prove also that the new Hecke algebras induced from Birman–Wenzl algebras are Frobenius over a field (but not always cellular)