Degree bounds in monomial subrings (Q1362376)

Let \(K\) be a field, let \(k\) and \(n\) be integers such that \(4\leq 2k\leq n\), let \({\mathcal A}\) denote the set of square-free monomials in the variables \(X_1, \dots, X_n\) of total degree \(k\), and let \(C\) denote the monoid of monomials generated by \({\mathcal A}\). Let \(R= \bigoplus^\infty_{i=0} R_i\) be the standard grading of \(K[X_1, \dots, X_n]\), and let \(R^{(k)} =\bigoplus^\infty_{i=1} R_{ki}\) be the \(k\)-th Veronese subring of \(R\) graded by \((R^{(k)})_i =R_{ki}\). Then \(K[C]\) is a graded subring of \(R^{(k)}\) with the normalized grading \((K[C])_i =K[C] \cap(R^{(k)})_i\). An explicit generating set is given for the canonical module \(\omega_{K[C]}\) of \(K[C]\), and the \(a\)-invariant is shown to be the negative of the least integer greater than or equal to \(n/k\). Results on generating sets of the normalization \(\overline A\) of certain graded rings generated by monomials are also given. For example let \(F\) be a finite set of monomials in \(K[X_1, \dots, X_n]\), and let \(F_0\) be the set of elements of \(F\) of lowest total degree. Let \(I\) and \(I_0\) denote the ideals of \(K[X_1, \dots, X_n]\) generated by \(F\) and \(F_0\) respectively. Let \({\mathcal R} =\bigoplus^\infty_{i=1} I^iT^i \subseteq R[T]\) and \({\mathcal R}_0= \bigoplus^\infty_{i=1} I^i_0T^i \subseteq R[T]\) denote the Rees algebras of \(I\) and \(I_0\). It is shown that the normalization \(\overline {\mathcal R}\) of \({\mathcal R}\) is generated as an \({\mathcal R}_0\)-module by monomials of degree \(\leq n\). If \(I_0\) is integrally closed, \(\overline {\mathcal R}\) is generated as an \({\mathcal R}_0\)-module by monomials of degree \(\leq n-1\). Similar results are obtained for subrings \(K[F]\) of \(K[X_1, \dots, X_n]\) generated by a set \(F\) of monomials of the same degree \(k\), and on \(a\)-invariants of \(\overline{K[F]}\) where \(F\) is a set of square-free monomials of degree \(k\).