Graded analytic algebras over a field of positive characteristic
Let \(A\) be a formal power series ring over an algebraically closed field \(k\) of characteristic \(p>0\). We call \(f \in A\) homogeneous of degree \(\alpha\in k\), if there exist coordinates \(X_ 1,\dots,X_ n\) and elements \(\alpha_ 1,\dots,\alpha_ n \in k\) not all zero such that for all monomials \(X_ 1^{i_ 1} \dots X_ n^{i_ n}\) occurring in \(f\), one has \(i_ 1 \alpha_ 1 +\cdots+i_ n\alpha_ n=\alpha\). It is shown: \(f \in A\) is homogeneous of nonzero degree if and only if \(f \in {\mathfrak m}_ Aj(f)\), where \(j(f)\) is the ideal generated by the partial derivatives of \(f\) and \({\mathfrak m}_ A\) is the maximal ideal of \(A\). For an analytic \(k\)-algebra \(R=A/{\mathfrak a}\), it is proved \(R\) is graded, i.e. \(R=\oplus_{\lambda \in G} R_ \lambda\) with \(G\) a subgroup of \(k\) and \(R_ \lambda R_ \mu \subset R_{\lambda+\mu}\), if and only if there exists a derivation \(d\) of \(R\), which induces a non-nilpotent map \(\overline d: {\mathfrak m}/{\mathfrak m}^ 2 \to {\mathfrak m}/{\mathfrak m}^ 2\), \({\mathfrak m}\) being the maximal ideal of \(R\).
scientific article; zbMATH DE number 567466