On actions of infinitesimal group schemes
Let \(H\) be a finite-dimensional commutative Hopf algebra over a perfect field \(k\) with characteristic \(p>0\). Such a Hopf algebra is necessarily a truncated polynomial algebra -- we will call the variables \(x_1,x_2,\dots,x_n\). Given a right \(H\)-comodule algebra \(A\) via \(\delta\colon A\to A\otimes H\) there exist derivations \(D_i\colon A\to A\) for \(1\leq i\leq n\) such that \(\delta(a)=a\otimes 1+1\otimes a+\sum D_i(a)\otimes x_i+\) terms of higher total degree. Let \(R=A^{\text{co }H}\) be the subalgebra of coinvariants under \(\delta \). The authors show that \(A/R\) is an \(H\)-Galois extension if and only if there are elements \(y_1,y_2,\dots,y_n\in A\) such that the matrix \((D_i(y_j))\) is invertible. Such \(y\)'s are found by considering elements that map to \(1\otimes x_i\) for all \(i\) under the canonical map \(A\otimes_RA\to A\otimes H\) (which is surjective when \(A/R\) is \(H\)-Galois). The final theorem gives results in special cases. Suppose that \(A\) is a regular local ring of dimension \(d\) and once again that \(A/R\) is \(H\)-Galois with \(y_i\)'s as above. Then \(R\) is regular local. If, for all \(1\leq i\leq n\), we have that \(y^{p^{s_i}}\in R\) then there are parameters \(z_{n+1},z_{n+2},\dots,z_d\) such that \(\{y_1,\dots,y_n,z_{n+1},\dots,z_d\}\) is a regular system of parameters for \(A\), \(\{y_1^{p^{s_1}},\dots,y_n^{p^{s_n}},z_{n+1},\dots,z_d\}\) is a regular system of parameters for \(R\). If \(A\) is also complete then there is a field \(F\) with \(k\subset F\subset R\) which is isomorphic to the residue fields of both \(A\) and \(R\); and \(A\cong F[[y_1,\dots,y_n,z_{n+1},\dots,z_d]]\) and \(R\cong F[[y_1^{p^{s_1}},\dots,y_n^{p^{s_n}},z_{n+1},\dots,z_d]]\).
scientific article; zbMATH DE number 1903594