On the Cohen-Macaulay property of diagonal subalgebras of the Rees algebra
In chronological order, \textit{A. Simis, Ngo Viet Trung} and \textit{G. Valla} [J. Pure Appl. Algebra 125, No. 1, 305-328 (1998; see the preceding review)], \textit{A. Conca, J. Herzog, Ngo Viet Trung} and \textit{G. Valla} [``Diagonal subalgebras of bigraded algebras and embeddings of blow-ups of projective spaces'', Am. J. Math. 199, No. 4, 859-901 (1997)], and \textit{S. D. Cutkosky} and \textit{J. Herzog} [``Cohen-Macaulay coordinate rings of blowup schemes'', Comment. Math. Helv. 72, No. 4, 605-617 (1997)] all studied the Cohen-Macaulayness of rings of the form \(k[(I^e)_c]\), where \(k\) is a field, \(I\) a homogeneous ideal in the polynomial ring \(R = k[x_1, \ldots, x_n]\), and \((I^e)_c\) the ideal in \(R\) generated by all elements of \(I^e\) of degree at least \(c\). These are homogeneous coordinate rings of projective embeddings of a blowing up of \(\mathbb{P}^{n-1}\) along a closed subscheme defined by \(I\). These authors proved that in many cases, for somewhat explicit \(c\) and \(e\), \(k[(I^e)_c]\) is Cohen-Macaulay. Conca, Herzog, Trung and Valla conjectured in their paper (loc. cit.) that the Cohen-Macaulayness of \(R[It]\) implies that for all large \(e\) and \(c\), \(k[(I^e)_c]\) is Cohen-Macaulay. In the paper under review this conjecture is proved and the author even gives an explicit description of \(c\) in terms of \(e\), degrees and the number of generator of \(I\) and \(n\). The main tools are the multigraded \(a\)-invariants and ``good'' resolutions, all developed in the paper.
scientific article; zbMATH DE number 1135224
scientific article