Using a result of J.-M. Bony, we prove the weak involutivity of truncated microsupports. More precisely, given a sheaf $F$ on a real manifold and $k\in \mathbb{Z}$, if two functions vanish on ${SS}_k\left(F\right)$, then so does their Poisson bracket.

This paper lays the foundations for the global theory of irreducible components of rigid analytic spaces over a complete field $k$. We prove the excellence of the local rings on rigid spaces over $k$. This is used to prove the standard existence theorems and to show compatibility with the notion of irreducible components for schemes and formal schemes. Behavior with respect to extension of the base field is also studied. It is often necessary to augment scheme-theoretic techniques with other algebraic...

In $𝒟$-modules theory, Gauss-Manin systems are defined by the direct image of the structure sheaf $𝒪$ by a morphism. A major theorem says that these systems have only regular singularities. This paper examines the irregularity of an analogue of the Gauss-Manin systems. It consists in the direct image complex $f_{+}\left(𝒪{\mathrm{e}}^g\right)$ of a $𝒟$-module twisted by the exponential of a polynomial $g$ by another polynomial $f$, where $f$ and $g$ are two polynomials in two variables. The analogue of the Gauss-Manin systems can have irregular...