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The aim of this note is threefold. The first is to obtain a simple characterization of relative constructible sheaves when the parameter space is projective. The second is to study the relative Fourier-Mukai for relative constructible sheaves and for relative regular holonomic $\mathcal D$-modules and prove they induce relative equivalences of categories. The third is to introduce and study the notions of relative constructible functions and relative Euler-Poincar\'e index. We prove that the relative Euler-Poincar\'e index provides an isomorphism between the Grothendieck group of the derived category of complexes with bounded relative $\mathbb R$-constructible cohomology and the ring of relative constructible functions., Comment: 25 pages. This version corrects an error found in the proof of old Lemma 1.4 which became Lemma 1.11 in this new version

We study, through new recurrence relations for certain binomial coefficients modulo a power of a prime, the evolution of the primitives of a modular periodic sequence. We prove that we can reduce to study primitives of constant sequences and that the latter are controlled by modular binomial coefficients. Finally we apply our results to describe the dynamics of the primitives of the sequence considered by the Romanian composer Vieru in his "Book of Modes".

We develop the theory of relative regular holonomic D-modules with a smooth complex manifold S of arbitrary dimension as parameter space, together with their main functorial properties. In particular, we establish in this general setting the relative Riemann-Hilbert correspondence proved in a previous work in the one-dimensional case., Comment: V2: 58 pages, revised version following referee's comments; the arXiv version contains a last section with details. V3: Statement of Propositions 5.4(1) and A.3 corrected. V4: 61 pages Statement and proof of Proposition 5.4 corrected after a referee's remark; details of proofs added in the appendix. V5: 54 pages Final published version in the Compositio style. V6: = V5 plus details of proofs of V4

This is a systematic study of the behaviour of finite coverings of (affine) schemes with regard to two Grothendieck topologies: the canonical topology and the fpqc topology. The history of the problem takes roots in the foundations of Grothendieck topologies, passes through main strides in Commutative Algebra and leads to new Mathematics up to perfectoids and prisms. We first review the canonical topology of affine schemes and show, keeping with Olivier's lost work, that it coincides with the effective descent topology; covering maps are given by universally injective ring maps, which we discuss in detail. We then give a "catalogue raisonn\'e" of examples of finite coverings which separate the canonical, fpqc and fppf topologies. The key result is that finite coverings of regular schemes are coverings for the canonical topology, and even for the fpqc topology (but not necessarily for the fppf topology). We discuss a "weakly functorial" aspect of this result. "Splinters" are those affine Noetherian schemes for which every finite covering is a covering for the canonical topology. We also investigate their mysterious fpqc analogs, and prove that in prime characteristic, they are all regular. This leads us to the problem of descent of regularity by (non-necessarily flat) morphisms $f$ which are coverings for the fpqc topology, which is settled thanks to a recent theorem of Bhatt-Iyengar-Ma., Comment: Final version, accepted in Ann. Sc. Norm. Super. Pisa Cl. Sci

On the product of a complex manifold $X$ by a complex curve $S$ considered as a parameter space, we show a Riemann-Hilbert correspondence between regular holonomic relative $\mathcal D$-modules (resp. complexes) on the one hand and relative perverse complexes (resp. $S$-$\mathbb{C}$-constructible complexes) on the other hand., Comment: 22 pages. This article improves and supersedes arXiv:1811.07151. V2: 28 pages, various corrections, new Th.2, improvement of the presentation. V3: final version to appear in the Proceedings of the London Mathematical Society. V4 Version post-publication, corrects an error in the proof of Prop.3.3 of the published version, Sections 3.2-3.4 modified

We introduce the notion of strong regular holonomic ${\mathcal{D}}_{{X\times S}/S}$-module and we prove that the functor ${\mathrm{RH}}^S$ introduced by T. Monteiro Fernandes and C. Sabbah in [14] takes image in ${\mathsf{D}}^{\mathrm{b}}_{\mathrm{srhol}}({\mathcal{D}}_{{X\times S}/S})$ (complexes of ${\mathcal{D}}_{{X\times S}/S}$-module whose cohomologies are strongly regular). We prove that for $\dim X=\dim S=1$ the functor solution functor ${}^\mathrm{p}{\mathrm{Sol}}$ restricted to ${\mathsf{D}}^{\mathrm{b}}_{\mathrm{srhol}}({\mathcal{D}}_{{X\times S}/S})$ is an equivalence of categories with quasi-inverse ${\mathrm{RH}}^S$., Comment: 13 pages

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 Algebra

It is a well established fact that the notions of quasi-abelian categories and tilting torsion pairs are equivalent. This equivalence fits in a wider picture including tilting pairs of $t$-structures. Firstly, we extend this picture into a hierarchy of $n$-quasi-abelian categories and $n$-tilting torsion classes. We prove that any $n$-quasi-abelian category admits a derived category endowed with a $n$-tilting pair of $t$-structures such that the respective hearts are derived equivalent. Secondly, we describe the hearts of these $t$-structures as quotient categories of coherent functors, generalizing Auslander's Formula. Thirdly, we apply our results to Bridgeland's theory of perverse coherent sheaves for flop contractions. In Bridgeland's work, the relative dimension $1$ assumption guaranteed that $f_*$-acyclic coherent sheaves form a $1$-tilting torsion class, whose associated heart is derived equivalent to $D(Y)$. We generalize this theorem to relative dimension $2$., Comment: Completely revised version taking into account some new developments in the field

This paper is devoted to the comparison of the notions of regularity for algebraic connections and (holonomic) regularity for algebraic $\mathcal D$-modules.

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 remarks

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, the $t$-structures whose hearts have at most $n$ fixed consecutive non-zero cohomologies. Moreover, by this classification theorem, we deduce the construction of the $t$-tree, a new technique which generalises the 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{\o} [HRS96]. 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 [KeV88b], [Ke07]., Comment: 38 pages

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 $D(C)$ induced by a torsion pair., Comment: Updated: dedication to Alberto Facchini on the occasion of his 60th Birthday added in front matter; 15 pages

In this paper we introduce the category of stratified Pro-modules and the notion of induced object in this category. We propose a translation of a Morihiko Saito equivalence result using the dual language of Pro-objects. So we prove an equivalence between the derived category of stratified Pro-modules and the category of Pro-differential complexes. We also supply a comparison with the notion of Crystal in Pro-module (introduced by P. Deligne in 1960)., Comment: 16 pages

In the dictionary between the language of (algebraic integrable) connections and that of (algebraic) $\cD$-modules, to compare the definitions of inverse images for connections and $\cD$-modules is easy. But the comparison between direct images for connections (the classical construction of the Gauss-Manin connection for smooth morphisms) and for $\cD$-modules, although known to specialists, has been explicitly proved only recently in a paper of Dimca, Maaref, Sabbah and Saito in 2000, where the authors' main technical tool was M. Saito's equivalence between the derived category of $\cD$-modules and a localized category of differential complexes. The aim of this short paper is to give a simplified summary of the [DMSS] argument, and to propose an alternative proof of this comparison which is simpler, in the sense that it does not use Saito equivalence. Moreover, our alternative strategy of comparison works in a context which is a precursor to the Gauss-Manin connection (at the level of $f^{-1}\cD_Y$-modules, for a morphism $f:X\to Y$), and may be of some intrinsic interest., Comment: 11 pages

This paper is devoted to the comparison of different localized categories of differential complexes. The first result is that the canonical functor from the category of complexes of differential operators of order one (defined by Herrera and Lieberman) to the category of differential complexes (of any order, defined by M. Saito), both localized with respect to a suitable notion of quasi-isomorphism, is an equivalence of categories. Then we prove a similar result for a filtered version of the previous categories (defined respectively by Du Bois and M.Saito), localized with respect to graded-quasi-isomorphisms, thus answering a question posed by M. Saito., Comment: 13 pages