Mixed grading on polynomial rings (Q2785930)

The author considers weighted \({\mathbb{Z}}\)-gradings of a polynomial ring \(R\) in \(n\) variables over a field \(K\), with nonzero weights \(d_1,\ldots,d_n\). The grading is called \textit{mixed} if \(d_i<0\) and \(d_j>0\) for a couple of indices \(i,j\). NEWLINENEWLINENEWLINEA description is given of the maximal graded ideals of \(R\) that contain no variables, namely they are extended ideals of maximal ideals of \(R_0\) (the zero-th degree piece of \(R\)) that contain no monomials. Here an ideal is called maximal graded if it is maximal among the graded ideals (hence there will be many graded ideals which are graded maximal but not maximal!). The basic emphasis of the results is on the nature of the ring \(R_0\) which is a finitely generated \(K\)-algebra of dimension \(n-1\) (it is also Cohen--Macaulay by a result of Hochster--Huneke). It is shown that \(R_0\) is a toric ring and the ideal of relations is explicitly computed for \(n=3\). NEWLINENEWLINENEWLINEThe author also gives results on the number of generators of a maximal graded ideal and illustrates the theory with several examples.NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].