A subsheaf of the sheaf ${ℰ}_{\Omega }$ of germs $C^{\infty }$ functions over an open subset $\Omega$ of ${\mathbf{R}}^n$ is called a sheaf of sub $C^{\infty }$ function. Comparing with the investigations of sheaves of ideals of ${ℰ}_{\Omega }$, we study the finite presentability of certain sheaves of sub $C^{\infty }$-rings. Especially we treat the sheaf defined by the distribution of Mather’s $𝒞$-classes of a $C^{\infty }$ mapping.

Frobenius modules are difference modules with respect to a Frobenius operator. Here we show that over non-archimedean complete differential fields Frobenius modules define differential modules with the same Picard-Vessiot ring and the same Galois group schemes up to extension by constants. Moreover, these Frobenius modules are classified by unramified Galois representations over the base field. This leads among others to the solution of the inverse differential Galois problem for $p$-adic differential...