arXiv subjects
Publications and source records attributed to Luis Narváez-Macarro.
In this paper we study the comparison between the logarithmic and the meromorphic de Rham complexes along a divisor in a complex manifold. We focus on the case of free divisors, starting with the case of locally quasihomogeneous divisors, and we explain how D-module theory can be used for this comparison.
We prove that any multi-variate Hasse-Schmidt derivation can be decomposed in terms of substitution maps and uni-variate Hasse-Schmidt derivations. As a consequence we prove that the bracket of two $m$-integrable derivations is also $m$-integrable, for $m$ a positive integer or infinity.
We study the action of substitution maps between power series rings as an additional algebraic structure on the groups of Hasse--Schmidt derivations. This structure appears as a counterpart of the module structure on classical derivations.
Let $k$ be a commutative ring and $A$ a commutative $k$-algebra. Given a positive integer $m$, or $m=\infty$, we say that a $k$-linear derivation $δ$ of $A$ is $m$-integrable if it extends up to a Hasse--Schmidt derivation $D=(\Id,D_1=δ,D_2,...,D_m)$ of $A$ over $k$ of length $m$. This condition is automatically satisfied for any $m$ under one of the following orthogonal hypotheses: (1) $k$ contains the rational numbers and $A$ is arbitrary, since we can take $D_i=\frac{δ^i}{i!}$; (2) $k$ is arbitrary and $A$ is a smooth $k$-algebra. The set of $m$-integrable derivations of $A$ over $k$ is an $A$-module which will be denoted by $\Ider_k(A;m)$. In this paper we prove that, if $A$ is a finitely presented $k$-algebra and $m$ is a positive integer, then a $k$-linear derivation $δ$ of $A$ is $m$-integrable if and only if the induced derivation $δ_{\mathfrak{p}}:A_{\mathfrak{p}} \to A_{\mathfrak{p}} $ is $m$-integrable for each prime ideal $\mathfrak{p}\subset A$. In particular, for any locally finitely presented morphism of schemes $f:X \to S$ and any positive integer $m$, the $S$-derivations of $X$ which are locally $m$-integrable form a quasi-coherent submodule $\fIder_S(\OO_X;m)\subset \fDer_S(\OO_X)$ such that, for any affine open sets $U=\Spec A \subset X$ and $V=\Spec k \subset S$, with $f(U)\subset V$, we have $Γ(U,\fIder_S(\OO_X;m))=\Ider_k(A;m)$ and $\fIder_S(\OO_X;m)_p = \Ider_{\OO_{S,f(p)}}(\OO_{X,p};m)$ for each $p\in X$. We also give, for each positive integer $m$, an algorithm to decide whether all derivations are $m$-integrable or not.