696 research outputs found
Good tilting modules and recollements of derived module categories
Let be an infinitely generated tilting module of projective dimension at
most one over an arbitrary associative ring , and let be the
endomorphism ring of . In this paper, we prove that if is good then
there exists a ring , a homological ring epimorphism B\ra C and a
recollement among the (unbounded) derived module categories \D{C} of ,
\D{B} of , and \D{A} of . In particular, the kernel of the total left
derived functor is triangle equivalent to the derived
module category \D{C}. Conversely, if the functor
admits a fully faithful left adjoint functor, then is a good tilting
module. We apply our result to tilting modules arising from ring epimorphisms,
and can then describe the rings as coproducts of two relevant rings.
Further, in case of commutative rings, we can weaken the condition of being
tilting modules, strengthen the rings as tensor products of two commutative
rings, and get similar recollements. Consequently, we can produce examples
(from commutative algebra and -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
Derived equivalences for mirror-reflective algebras
We show that the construction of mirror-reflective algebras inherits derived
equivalences of gendo-symmetric algebras. More precisely, suppose A and B are
gendo-symmetric algebras with both Ae and Bf faithful projective-injective left
ideals generated by idempotents e in A and f in B, respectively. If A and B are
derived equivalent, then the mirror-reflective algebras of (A,e) and (B,f) are
derived equivalent.Comment: 15 page
- β¦