Good tilting modules and recollements of derived module categories

Abstract

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. We prove that if T is good, then there exists a ring C, a homological ring epimorphism B→C and a recollement among the (unbounded) derived module categories 𝒟C of C, 𝒟B of B and 𝒟A of A. In particular, the kernel of the total left-derived functor T⊗B𝕃- is triangle equivalent to the derived module category 𝒟C. Conversely, if T⊗B𝕃- admits a fully faithful left adjoint functor, then T is good. Moreover, if T arises from an injective ring epimorphism, then C is isomorphic to the coproduct of two relevant rings. In the case of commutative rings, the ring C can be strengthened as the tensor product of two commutative rings. Consequently, we produce a large variety of examples (from Dedekind domains 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–Hölder theorem fails even for stratifications by derived module categories.