Algebraic connections vs. Algebraic $\mathcal D$-modules: regularity conditions
This paper is devoted to the comparison of the notions of regularity for algebraic connections and (holonomic) regularity for algebraic $\mathcal D$-modules.
math.AG↗
arXiv subjects
Maurizio Cailotto
Publications and source records attributed to Maurizio Cailotto.
Classification of compact real surfaces: a quick proof
The aim of this note is to give a quick algebraic proof of (the combinatorial part of) the classification theorem for compact real surfaces, whose classical proofs (as in the Massey book and in the Conway ZIP proof) are based on surgery (and pictures) and look more intricate.
math.AT↗
Algebraic Connections vs. Algebraic {$\cD$}-modules: inverse and direct images
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.
math.AG↗
$p$-adic formulas and unit root $F$-subcrystals of the hypergeometric system
We define the notion of {\it Dwork family of logarithmic $F$-crystals}, a typical example of which is the family of Gauss hypergeometricdifferential systems, viewed as parametrized by their exponents of algebraic monodromy. The $p$-adic analytic dependence of the Frobenius operation upon those exponents, is Dwork's "Boyarsky Principle". We discuss, in favorable cases, the $p$-adic analytic continuation of the unit root $F$-subcrystal in the open tube of a singularity, uniformly w.r.t. the exponents. We obtain a conceptual proof of the Koblitz-Diamond formula $p$-adically analog to Gauss' evaluation of $F(a,b,c;1)$.
math.NT↗
Deligne Localized Functors
In this paper we present the notion of ``Deligne localized functors'', an avatar of the derived functors, whose definition is inspired by Deligne in [SGA 4,XVII]. Their definition involves the notions of Ind and Pro categories, they always exist and are characterized in terms of universal properties. The classical localized functor, in the sense of Grothendieck and Verdier, exists if suitable conditions are verified for the Deligne localized functors. We apply these notions to triangulated and derived categories.
math.CT↗