Structure theorems for rings under certain coactions of a Hopf algebra. (Q2862218)

Let \(k\) be a field of characteristic \(p\) for a prime integer \(p\) and \(A\) a \(k\)-algebra with a system of derivations \(\{D_1,\ldots,D_n\}\) for some integer \(n\). Let \(H=k[X_1,\ldots,X_n]/(X_1^{p^{s_1}},\ldots,X_n^{p^{s_n}})\) such that \(s_1\geq s_2\geq\cdots\geq s_n\geq 1\). Then \(H\) is the Hopf algebra of the additive group with the comultiplication \(\triangle(x_i)=x_i\otimes 1+1\otimes x_i\) for each \(x_i=X_i+(X_1^{p^{s_1}},\ldots,X_n^{p^{s_n}})\) and the Hopf algebra of the multiplicative group with \(\triangle(x_i)=x_i\otimes 1+1\otimes x_i+x_i\otimes x_i\). In the Lie algebra case of the additive group, \(H=k[X_1,\ldots,X_n]/(X_1^p,\ldots,X_n^p)\), \(H=H_0\) and \(H=H_1\) of the multiplicative group.NEWLINENEWLINE Let \(H_c\), \(c=0,1\), be the Hopf algebras in the Lie algebra cases and \(A\) a right \(H_c\)-comodule algebra with a structure map \(\delta\colon A\to A\otimes H_c\). The authors obtain elements \(y_1,\ldots,y_n\in A\) such that \(D_i(y_j)=\delta_{ij}(1+cy_i)\) and show a structure theorem for a commutative \(k\)-algebra \(A\) in the additive case.NEWLINENEWLINE Theorem. Let \(A\) be a commutative \(k\)-algebra over a field of nonzero characteristic \(p\) and \(\{D_1,\ldots,D_n\}\subset\text{Der}_k(A)\) such that \(D_iD_j=D_jD_i\), \(D_i^p=0\) for all \(i,j=1,\ldots,n\). Suppose that (1) there exist \(z_1,\ldots,z_{n-1}\in A\) such that \(D_i(z_j)=\delta_{i,j}\), \(i,j=1,\ldots,n-1\), and (2) there exists \(y\in A\) such that \(D_n(y)=1\). Then there exists \(t\in A\) such that \(D_n(t)=1\) and \(D_i(t)=0\) for all \(i=1,\ldots,n-1\).