Homogenising differential operators

We consider the Weyl algebra A (=A_n(k)) and its Rees algebra B with respect to the Bernstein filtration. The homogenisation of a differential operator in A is an element in B. In this paper we establish the validity of the division theorem for homogenized differential operators and Buchbergers algorithm for computing Groebner (or standard) bases in B. As an application we describe an algorithm for computing delta-standard bases in the Weyl algebra A.

Algebraic computation of some intersection D-modules

Let $X$ be a complex analytic manifold, $Dsubset X$ 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.

On the logarithmic comparison theorem for integrable logarithmic connections

Let $X$ be a complex analytic manifold, $Dsubset X$ a free divisor with jacobian ideal of linear type (e.g. a locally quasi-homogeneous free divisor), $j: U=X-D to X$ the corresponding open inclusion, $E$ an integrable logarithmic connection with respect to $D$ and $L$ the local system of the horizontal sections of $E$ on $U$. In this paper we prove that the canonical morphisms between the logarithmic de Rham complex of $E(kD)$ and $R j_* L$ (resp. the logarithmic de Rham complex of $E(-kD)$ and $j_!L$) are isomorphisms in the derived category of sheaves of complex vector spaces for $kgg 0$ (locally on $X$)

Coefficient fields and scalar extension in positive characteristic

Let k be a perfect field of positive characteristic, k(t)_{per} the perfect closure of k(t) and A=k[[X_1,...,X_n]]. We show that for any maximal ideal N of A=k(t)_{per}otimes_k A, the elements in hat{A_N} which are annihilated by the Taylor Hasse-Schmidt derivations with respect to the X_i form a coefficient field of hat{A_N}.

Dualite et comparaison sur les complexes de de Rham logarithmiques par rapport aux diviseurs libres

Let X be a complex analytic manifold and D subset X a free divisor. Integrable logarithmic connections along D can be seen as locally free {cal O}_X-modules endowed with a (left) module structure over the ring of logarithmic differential operators {cal D}_X(log D). In this paper we study two related results: the relationship between the duals of any integrable logarithmic connection over the base rings {cal D}_X and {cal D}_X(log D), and a differential criterion for the logarithmic comparison theorem. We also generalize a formula of Esnault-Viehweg in the normal crossing case for the Verdier dual of a logarithmic de Rham complex.

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 two-strata case, but our results extend to the general case.

The module $D f^s$ for locally quasi-homogeneous free divisors

We find explicit free resolutions for the $scr D$-modules ${scr D} f^s$ and ${scr D}[s] f^s/{scr D}[s] f^{s+1}$, where $f$ is a reduced equation of a locally quasi-homogeneous free divisor. These results are based on the fact that every locally quasi-homogeneous free divisor is Koszul free, which is also proved in this paper

Hasse-Schmidt Derivations and Coefficient Fields in Positive Characteristics

We show how to express any Hasse-Schmidt derivation of an algebra in terms of a finite number of them under natural hypothesis. As an application, we obtain coefficient fields of the completion of a regular local ring of positive characteristic in terms of Hasse-Schmidt derivations

Conservation of the noetherianity by perfect transcendental field extensions

Let $k$ be a perfect field of characteristic $p>0$, $k(t)_{per}$ the perfect closure of $k(t)$ and $A$ a $k$-algebra. We characterize whether the ring $Aotimes_k k(t)_{per}$ is noetherian or not. As a consequence, we prove that the ring $Aotimes_k k(t)_{per}$ is noetherian when $A$ is the ring of formal power series in $n$ indeterminates over $k$.