Hasse--Schmidt derivations versus classical derivations

In this paper we survey the notion and basic results on multivariate Hasse--Schmidt derivations over arbitrary commutative algebras and we associate to such an object a family of classical derivations. We study the behavior of these derivations under the action of substitution maps and we prove that, in characteristic $0$, the original multivariate Hasse--Schmidt derivation can be recovered from the associated family of classical derivations. Our constructions generalize a previous one by M. Mirzavaziri in the case of a base field of characteristic $0$.Comment: Dedicated to L\^e D\~ung Tr\'ang; final version; 2 references added; minor corrections. arXiv admin note: text overlap with arXiv:1807.10193, arXiv:1903.0898

Hasse--Schmidt modules versus integrable connections

We prove that, in characteristic 0, any Hasse-Schmidt module structure can be recovered from its underlying integrable connection, and consequently Hasse--Schmidt modules and modules endowed with an integrable connection coincide.Comment: 20 pages; comments are welcome. arXiv admin note: text overlap with arXiv:1810.08075, arXiv:1807.1019

Algebraic computation of some intersection D-modules

Let $X$ be a complex analytic manifold, $D\subset X$ a locally quasi-homogeneous free divisor, $E$ an integrable logarithmic connection with respect to $D$ and $L$ the local system of the horizontal sections of $E$ on $X-D$. In this paper we give an algebraic description in terms of $E$ of the regular holonomic D-module whose de Rham complex is the intersection complex associated with $L$. As an application, we perform some effective computations in the case of quasi-homogeneous plane curves.Comment: 18 page

Explicit models for perverse sheaves

We consider categories of generalized perverse sheaves, with relaxed constructibility conditions, by means of the process of gluing t-structures and we exhibit explicit abelian categories defined in terms of standard sheaves categories which are equivalent to the former ones. In particular, we are able to realize perverse sheaves categories as non full abelian subcategories of the usual bounded complexes of sheaves categories. Our methods use induction on perversities. In this paper, we restrict ourselves to the twostrata case, but our results extend to the general case