Gelfand-Kirillov dimension of Noetherian semigroup algebras
Let \(K\) be a field, let \(S\) be a semigroup, and let \(K[S]\) be a semigroup algebra of \(S\) over \(K\). The author proves some theorems concerning the Gelfand-Kirillov dimension of \(K[S]\). His approach is based on his own results about the same problem for \(S\) cancellative, and on Grigorchuk's theorem describing semigroups with polynomial growth. The main tool used by the author is the connection between \(K[S]\) and \(K[\overline{S}]\), where \(\overline{S}\) is the image of \(S\) under the natural homomorphism \(K[S] \to K[S] / B(K[S])\) (\(B(R)\) is the prime radical of a ring \(R\)). The following proposition (Theorem 16) shows the character of the author's results. Denote by \(\text{GK}(S)\) the Gelfand-Kirillov dimension of \(K[S]\), and assume that \(K[S]\) is right Noetherian. Then the following conditions (i)-(iii) are equivalent: (i) \(\text{GK}(S) < \infty\), (ii) \(\text{GK}(T) < \infty\) for every cancellative subsemigroup \(T\) of \(S\), (iii) every cancellative subsemigroup of \(S\) has a finitely generated nilpotent-by-finite group of fractions. Moreover, in this case \(\text{GK}(S) \leq rq\), where \(r\) denotes the nilpotency index of \(B(K[S])\) and \(q\) the maximum of GK-dimensions of the cancellative subsemigroups of \(S\).
scientific article; zbMATH DE number 554224