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Graded analytic algebras over a field of positive characteristic. (English) Zbl 0790.13006
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 $$R$$.
Reviewer: A.Aramova (Sofia)

##### MSC:
 13J07 Analytical algebras and rings
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##### References:
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