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Linear gradings of polynomial algebras. (English) Zbl 1137.13001

Let \(K\) be a field. A grading of the polynomial algebra \(K[X]=K[X_1,\ldots,X_m]\) is linear if it is in some natural sense compatible with the grading defined by the degree of a polynomial. Let \(G\) be a finite group. The author describes the linear \(G\)-grading of the polynomial algebra \(K[X]\) such that the unit coponent is a polynomial \(K\)-algebra.
More precisely, we have the following theorem: Let \(G\) be a finite commutative group. Consider a linear grading \(\bigoplus_{c\in G}K[X]^c\) of the polynomial algebra \(K[X]\) over a field \(K\). Let \(y_1,\ldots,y_m\) be a \(K\)-linear basis of \(K[X]_1\), consisting of \(G\)-homogeneous elements. Let \(y_j\in K[X]^{c_j}\) for some \(c_j\in G\), let \(r_j=| <c_j>| \), where \(j=1,\ldots,m\). Then the following conditions are equivalent:
(i) \(K[X]^0\) is a polynomial \(K\)-algebra.
(ii) \(K[X]^0=K[y_1^{r_1},\ldots,y_m^{r_m}]\).
(iii) the sum of subgroups \(<c_1>+\ldots+<c_m>\) is direct.
(iv) \(G_0=\{0\}\).

MSC:

13A02 Graded rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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References:

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