Good tilting modules and recollements of derived module categories (Q2888910)

Let \(A\) be a ring with identity, and \(T\) a left \(A\)-module which may be infinitely generated. The module \(T\) is called a tilting module (of projective dimension at most \(1\)) provided thatNEWLINENEWLINE\((T1)\) \(T\) has projective dimension at most one,NEWLINENEWLINE\((T2)\) \(\text{Ext}^i_A(T,T^{({\alpha})})=0\) for each \(i\geq 1\) and each cardinal \(\alpha\), andNEWLINENEWLINE\((T3)\) there exists an exact sequence \(0\rightarrow A\rightarrow T_0\rightarrow T_1\rightarrow 0\) of left \(A\)-modules, where \(T_0\) and \(T_1\) are isomorphic to a direct summand of arbitrary direct sums of copies of \(T\).NEWLINENEWLINEIf \(T\) is finitely presented, then we say that \(T\) is a classical tilting module. If the modules \(T_0\) and \(T_1\) in \((T3)\) are isomorphic to direct summands of finite direct sums of copies of \(T\), then we say that \(T\) is a good tilting module. Actually, each classical tilting module is good.NEWLINENEWLINEThe theory of finitely generated tilting modules has been successfully applied, in the representation theory of algebras and groups, to understanding different aspects of algebraic structure and homological features of (algebraic) groups, algebras and modules. Recently, infinitely generated tilting modules over arbitrary associated rings have become of interest in and attracted increasingly attentions toward to understanding derived categories and equivalences of general rings. In this general situation, many classical results in tilting theory appear in a very different new fashion. This more general context of tilting theory not only renews our view on features of finitely generated tilting modules, but also provides us with completely different information about the whole tilting theory.NEWLINENEWLINEAs in the theory of classical tilting modules, a natural context for studying infinitely generated tilting modules is the relationship of derived categories and equivalences induced by infinitely generated tilting modules.NEWLINENEWLINELet \(T\) 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, the authors proved that if \(T\) is good then there exists a ring \(C\), a homological ring epimorphism \(B\rightarrow C\) and a recollement among the (unbounded) derived module categories \(\mathcal{D}(C)\) of \(C\), \(\mathcal{D}(B)\) of \(B\), and \(\mathcal{D}(A)\) 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 \(\mathcal{D}(C)\). Conversely, if the functor \(T\otimes_B^{\mathbb L}-\) admits a fully faithful left adjoint functor, then \(T\) is a good tilting module. Moreover, if \(T\) arises from an injective ring epimorphisms, 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 products of two commutative rings. Consequently, the authors produced a large variety of 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.