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 $T$. 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-Hölder theorem fails even for stratifications by derived module categories, and also answers negatively an open problem by Angeleri-Hügel, König and Liu.