Hilbert coefficients and depth of fiber cones

Abstract

Criteria are given in terms of certain Hilbert coefficients for the fiber cone F (I ) of an m-primary ideal I in a Cohen-Macaulay local ring (R, m) so that it is Cohen-Macaulay or has depth at least dim(R) − 1. A version of Huneke's fundamental lemma is proved for fiber cones. Goto's results concerning Cohen-Macaulay fiber cones of ideals with minimal multiplicity are obtained as consequences.

Fiber Cones of Ideals with Almost Minimal Multiplicity

“…Therefore,ȳ o is regular in F (Ī). Hence Sally machine for fiber cone, [20,Lemma 2.7], yields that x o ∈ F (I) is regular and hence depth F (I) ≥ 1. For j ≥ k, consider the following exact sequence:…”

Multiplicity and Cohen-Macaulayness of Fiber Cones of Good Filtrations

“…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.…”

About K-Positivity Properties of Time-Invariant Linear Systems Subject to Point Delays

“…Positive systems have an important relevance since the input, state, and output signals in many physical or biological systems are necessarily positive [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Therefore, important attention has been paid to such systems in the last decades.…”

“…Nonnegativity/positivity properties apply for both continuoustime and discrete-time systems and are commonly formulated on the first orthant which is an important case in applications [7-15, 18, 20]. However, there are also studies of characterizations of the nonnegativity/positivity properties in more abstract spaces in terms of the solutions belonging to appropriate K-cones [3][4][5][6]. On the other hand, positive solutions of singular problems including nonlinearities have been studied in [1,14].…”