On the Cohen-Macaulayness of fiber cones

Abstract

Abstract. Let A be a Noetherian local ring with the maximal ideal m and I an ideal of A. Denote by F A (I) = n≥0 (I n /mI n ) the fiber cone of I. This paper characterizes the multiplicity and the Cohen-Macaulayness of fiber cones in terms of minimal reductions of ideals.

Multiplicity and Cohen-Macaulayness of Fiber Cones of Good Filtrations

“…The theory of minimal reductions of good filtrations plays a crucial role in this paper (see Section 2). As consequences of the main result, we not only obtain an improvement on the results in [20] (Remark 4.3) but also show that: for any good filtration F = {I n } n≥0 with (F ) = 1, n≥0 (I Ln /mI Ln )t n is Cohen-Macaulay for all large L (Corollary 4.6). Moreover, we give an interesting result in the case that F has the reduction number r(F ) ≤ 1 (Corollary 4.4) which covers the results of Huneke and Sally [9] and Shah [17,18] to the case of good filtrations (Remark 4.5).…”

“…These problems have concerned many authors over the years. Using different approaches, the authors investigated the Cohen-Macaulayness and other properties of fiber cones F m (I ) and F J (F ) (see, for instance, [1][2][3][4][5][6][7]9,11,12,20]). Using weak-(FC)-sequences of ideals in local rings, the author of [20] characterized the multiplicity and the CohenMacaulayness of F m (I ) in terms of minimal reductions of ideals.…”

“…Using different approaches, the authors investigated the Cohen-Macaulayness and other properties of fiber cones F m (I ) and F J (F ) (see, for instance, [1][2][3][4][5][6][7]9,11,12,20]). Using weak-(FC)-sequences of ideals in local rings, the author of [20] characterized the multiplicity and the CohenMacaulayness of F m (I ) in terms of minimal reductions of ideals. The results of [20] recovered some earlier results of Huneke and Sally [9], Shah [17,18] and Cortadellas and Zarzuela [4].…”

“…Using weak-(FC)-sequences of ideals in local rings, the author of [20] characterized the multiplicity and the CohenMacaulayness of F m (I ) in terms of minimal reductions of ideals. The results of [20] recovered some earlier results of Huneke and Sally [9], Shah [17,18] and Cortadellas and Zarzuela [4].…”

On the multiplicity and the Cohen–Macaulayness of fiber cones of graded algebras

“…In recent years, the Cohen-Macaulayness and other properties of the special fiber cone F m (I) have attracted much attention (see for instance [1][2][3][4][5][6][7][8]). Using weak-(FC)-sequences of ideals in local rings, the author of [8] characterized the multiplicity and the Cohen-Macaulayness of F m (I) in terms of minimal reductions of ideals.…”