Mixed Multiplicities of Modules over Noetherian Local Rings

On some multiplicity and mixed multiplicity formulas

“…, x p is a weak-(FC)-sequence of N with respect to (J, I). Then by [21,22](see [11,Proposition 3.3(iii) Since p is maximal, e(J [q−i] , I [i] ; N) = 0 if and only if 0 i p by [21]. Consequently by [21](see [11, by Remark 2.7.…”

“…In past years, using different sequences, one expressed mixed multiplicities into Hilbert-Samuel multiplicity, for instance: Risler-Teissier in 1973 [17] by superficial sequences and Rees in 1984 [13] by joint reductions; Viet in 2000 [21] by (FC)-sequences (see e.g. [3,11,25]). Definition 4.4 [21].…”

A note on joint reductions and mixed multiplicities

“…In past years, the positivity and the relationship between mixed multiplicities and Hilbert-Samuel multiplicity of ideals have attracted much attention (see e.g. [2,3,4,7,8,9,13,14,17,19,20,21,22,23]). …”

“…And he proved that mixed multiplicities of m-primary ideals are multiplicities of ideals generated by joint reductions. This concept was extended to the set of arbitrary ideals by [9,11,21].…”

The Euler-Poincaré Characteristic and Mixed Multiplicities

“…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]). …”