The total symbol theorem of a \(p\)-adic differential echelon operator \(h\geq 0\)
Let \(V\) be a complete discrete valuation ring of unequal characteristic \((0,p)\), let \(A^{\dagger}\) be a weakly complete (the author writes: \(\dagger\)-adic) \(V\)-algebra with coordinate functions \(x_1,\ldots,x_n\) such that their differentials form a basis of the \(A^{\dagger}\)-module of separated differential forms. The subject matter of the present paper is an in depth study of the structure of the ring \(D^{\dagger, h}_{A^{\dagger}/V}\) of \(p\)-adic differential operators of degree \(h\), for any integer \(h\geq0\). The definition of \(D^{\dagger, h}_{A^{\dagger}/V}\) reflects, in the present mixed characteristic setting, the \(p\)-filtration of the ring of differential operators of an affine algebraic scheme over a field of characteristic \(p\). The main result, which in part builds on previous joint work of the author with L. Narváez-Macarro, can be stated as follow: there is a natural isomorphism of left \(A^{\dagger}\)-modules -- the assignment of the \textit{symbol} to an element of \(D^{\dagger, h}_{A^{\dagger}/V}\) -- between \(D^{\dagger, h}_{A^{\dagger}/V}\) and a certain commutative algebra \((A^{\dagger}[\xi_1^0,\ldots,\xi^h_n])^{\dagger}\), \[ D^{\dagger, h}_{A^{\dagger}/V}\cong(A^{\dagger}[\xi_1^0,\ldots,\xi^h_n])^{\dagger}. \] The latter is the weak completion of \(A^{\dagger}[\xi_1^0,\ldots,\xi^h_n]\), the quotient of the polynomial ring \(A^{\dagger}[\Xi_1^0,\ldots,\Xi_n^h]\) by the ideal generated by the expressions \((\Xi_i^j)^p-u_j\Xi_i^{j+1}\) for \(0\leq j\leq h-1\) and \(1\leq i\leq n\), where \(u_j=\frac{p^{j+1}!}{(p^j!)^p}\). If one assumes in addition that \(A^{\dagger}\) arises from a smooth \(V\)-algebra of finite type admitting global coordinates, then one can deduce that \(D^{\dagger, h}_{A^{\dagger}/V}\) is noetherian.
scientific article; zbMATH DE number 5883101
scientific article