On some multiplicity and mixed multiplicity formulas

Abstract

This paper gives the additivity and reduction formulas for mixed multiplicities of multi-graded modules M and mixed multiplicities of arbitrary ideals, and establishes the recursion formulas for the sum of all the mixed multiplicities of M. As an application of these formulas we get the recursion formulas for the multiplicity of multi-graded Rees modules.

The Euler-Poincaré Characteristic and Mixed Multiplicities

“…The notion of filterregular sequences was introduced by Stuckrad and Vogel in [17] (see [2]). The theory of filter-regular sequences became an important tool for studying some classes of singular rings and has been continually developed (see, for example, [2,8,20,29,31] (ii) Let a ∈ S i . Since the following exact sequence…”

“…Using different sequences, one can express mixed multiplicities of ideals in terms of the Hilbert-Samuel multiplicity. For instance, in the case of n-primary ideals, Risler-Teissier [19] in 1973 showed that each mixed multiplicity is the multiplicity of an ideal generated by a superficial sequence and Rees [14] in 1984 proved that mixed multiplicities are multiplicities of ideals generated by joint reductions; for the case of arbitrary ideals, Viet [23] Note that [23] defines weak-(FC)-sequences in the condition I √ Ann R N (see, for example, [5,6,13,24,25,26,28,30,31]). In Definition 4.12, we omitted this condition.…”

“…Moreover, by using filterregular sequences, Manh and Viet [29] in 2013 characterized mixed multiplicities of multigraded modules in terms of the length of modules. In past years, the theory of mixed multiplicities has attracted much attention and has been continually developed (see, for example, [4,5,[11][12][13][14][15][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]). …”

A note on joint reductions and mixed multiplicities

“…It should be noted that this theorem does not hold in general if one omits the assumption k 1 + · · · + k s < h (see Remark 3.5). Our approach, which is based on the results in [18] and [24], links the multiplicity of (FC)-sequences in [18] and the multiplicity of joint reductions via our studies on these sequences (see Proposition 2.3 and Lemma 2.5). As an application of the main theorem, we interpret mixed multiplicities as Hilbert-Samuel multiplicities of Rees's superficial sequences (see Remark 3.4) and recover Rees's theorem in [12,Theorem 2.4] (see Corollary 3.6).…”

A note on formulas transmuting mixed multiplicities

“…In a recent paper [20], by a new approach, the authors gave the additivity and reduction formulas for mixed multiplicities of multi-graded modules and mixed multiplicities of arbitrary ideals; and they also showed that mixed multiplicities of arbitrary ideals are additive on exact sequences.…”