tt-Structures for Relative D\mathcal{D}-Modules and tt-Exactness of the de Rham Functor

This paper is a contribution to the study of relative holonomic $\mathcal{D}$-modules. Contrary to the absolute case, the standard $t$-structure on holonomic $\mathcal{D}$-modules is not preserved by duality and hence the solution functor is no longer $t$-exact with respect to the canonical, resp. middle-perverse, $t$-structures. We provide an explicit description of these dual $t$-structures. When the parameter space is 1-dimensional, we use this description to prove that the solution functor as well as the relative Riemann-Hilbert functor are $t$-exact with respect to the dual $t$-structure and to the middle-perverse one while the de Rham functor is $t$-exact for the canonical, resp. middle-perverse, $t$-structures and their duals.Comment: Final version to appear in Journal of Algebr

On tilted Giraud subcategories

Firstly we provide a technique to move torsion pairs in abelian categories via adjoint functors and in particular through Giraud subcategories. We apply this point in order to develop a correspondence between Giraud subcategories of an abelian category C and those of its tilt H(C) i.e., the heart of a t-structure on the derived category D(C)i nduced by a torsion pair

A classification theorem for t-structures

We give a classification theorem for a relevant class of t-structures in triangulated categories, which includes, in the case of the derived category of a Grothendieck category, a large class of t-structures whose hearts have at most nfixed consecutive non-zero cohomologies. Moreover, by this classification theorem, we deduce the construction of the t-tree, a new technique which generalizesthe filtration induced by a torsion pair. At last we apply our results in the tilting context generalizing the 1-tilting equivalence proved by Happel, Reiten and Smal\uf8. The last section provides applications to classical n-tilting objects, examples of t-trees for modules over a path algebra, and new developments on compatible t-structures

t-structures for relative D-modules and t-exactness of the de Rham functor

This paper is a contribution to the study of relative holonomic D-modules. Contrary to the absolute case, the standard t-structure on holonomic D-modules is not preserved by duality and hence the solution functor is no longer t-exact with respect to the canonical, resp. middle-perverse, t-structure. We provide an explicit description of these dual t-structures. We use this description to prove that the solution functor as well as the relative Riemann-Hilbert functor are t-exact with respect to the dual t-structure and to the middle-perverse one while the de Rham functor is t-exact for the canonical, resp. middle-perverse, t-structure and their duals

Derived equivalences induced by nonclassical tilting objects

Suppose that $\mathcal{A}$ is an abelian category whose derived category $\mathcal{D}(\mathcal{A})$ has $Hom$ sets and arbitrary (small) coproducts, let $T$ be a (not necessarily classical) ($n$-)tilting object of $\mathcal{A}$ and let $\mathcal{H}$ be the heart of the associated t-structure on $\mathcal{D}(\mathcal{A})$. We show that the inclusion functor $\mathcal{H}\hookrightarrow\mathcal{D}(\mathcal{A})$ extends to a triangulated equivalence of unbounded derived categories $\mathcal{D}(\mathcal{H})\stackrel{\cong}{\longrightarrow}\mathcal{D}(\mathcal{A})$. The result admits a straightforward dualization to cotilting objects in abelian categories whose derived category has $Hom$ sets and arbitrary products.Comment: The proof of Lemma 1.6 has been modified and the dual of Lemma 1.6 is now contained in the new Remark 1.8, which has been inserted at the end of Section 1. A clarification has been added at the end of the proof of Theorem 1.7. The present paper is going to appear in the Proceedings of the AMS. The authors thank the referee for her/his helpful comments and remark