If $X$ is a smooth scheme over a perfect field of characteristic $p$, and if ${𝒟}_X^{\left(\infty \right)}$ is the sheaf of differential operators on $X$ [7], it is well known that giving an action of ${𝒟}_X^{\left(\infty \right)}$ on an ${𝒪}_X$-module $ℰ$ is equivalent to giving an infinite sequence of ${𝒪}_X$-modules descending $ℰ$ via the iterates of the Frobenius endomorphism of $X$ [5]. We show that this result can be generalized to any infinitesimal deformation $f:X\to S$ of a smooth morphism in characteristic $p$, endowed with Frobenius liftings. We also show that it extends to adic...

Let $k\subseteq k^{\text{'}}$ be a field extension. We give relations between the kernels of higher derivations on $k\left[X\right]$ and $k^{\text{'}}\left[X\right]$, where $k\left[X\right]:=k[x_1,\cdots ,x_n]$ denotes the polynomial ring in $n$ variables over the field $k$. More precisely, let $D={\left\{D_n\right\}}_{n=0}^{\infty }$ a higher $k$-derivation on $k\left[X\right]$ and $D^{\text{'}}={\left\{D_n^{\text{'}}\right\}}_{n=0}^{\infty }$ a higher $k^{\text{'}}$-derivation on $k^{\text{'}}\left[X\right]$ such that $D_m^{\text{'}}\left(x_i\right)=D_m\left(x_i\right)$ for all $m\ge 0$ and $i=1,2,\cdots ,n$. Then (1) $k{\left[X\right]}^D=k$ if and only if $k^{\text{'}}{\left[X\right]}^{D^{\text{'}}}=k^{\text{'}}$; (2) $k{\left[X\right]}^D$ is a finitely generated $k$-algebra if and only if $k^{\text{'}}{\left[X\right]}^{D^{\text{'}}}$ is a finitely generated $k^{\text{'}}$-algebra. Furthermore, we also show that the kernel $k{\left[X\right]}^D$ of a higher derivation $D$ of $k\left[X\right]$ can be generated by a set...

A. Crachiola and L. Makar-Limanov [J. Algebra 284 (2005)] showed the following: if X is an affine curve which is not isomorphic to the affine line ${¹}_k$, then ML(X×Y) = k[X]⊗ ML(Y) for every affine variety Y, where k is an algebraically closed field. In this note we give a simple geometric proof of a more general fact that this property holds for every affine variety X whose set of regular points is not k-uniruled.