Positivity of mixed multiplicities

Abstract

IntroductionLet R = ⊕ (u,v)∈N 2 R (u,v) be a standard bigraded algebra over an artinian local ring K = R (0,0) . Standard means R is generated over K by a finite number of elements of degree (1, 0) and (0, 1). Since the length ℓ(R (u,v) ) of R (u,v) is finite, we can consider ℓ(R (u,v) ) as a function in two variables u and v. This function was first studied by van der Waerden [W] and Bhattacharya [B] who proved that there is a polynomial P R (u, v) of degree ≤ dim R − 2 such that ℓ(R (u,v) ) = P R (u, v) for u and v large enough. Katz, Mandal and Verma [KMV] found out that the degree of P R (u, v) is equal to rdim R − 2, where rdim R is the relevant dimension of R defined as follows. Let R ++ denote the ideal generated by the homogeneous elements of degree (u, v) with u ≥ 1, v ≥ 1. Let Proj R be the set of all homogeneous prime ideals ℘ ⊇ R ++ of R. Then rdim R := max{dim R/℘| ℘ ∈ Proj R} if Proj R = ∅ and rdim R can be any negative integer if Proj R = ∅. If we writethen a ij are non-negative integers for i + j = rdim R − 2. Let us denote a ij by e ij (R) for all i, j ≥ 0 with e ij (R) = 0 for i + j > rdim R − 2.We call P R (u, v) the Hilbert polynomial and the numbers e ij (R) with i + j = rdim R − 2 the mixed multiplicities of R. These notions seem to be of fundamental importance. But we would be surprised to learn how little is known on mixed multiplicities of an arbitrary standard bigraded algebra, especially on their positivity [KMV]The main motivation for the study of mixed multiplicities comes from the following situation. Let (A, m) be a local ring. Given an m-primary ideal I and an ideal J of A, we can consider the function ℓ(I v J u /I v+1 J u ) of the standard bigraded algebraover the artinian local ring A/I. Let r = deg P R(I|J) (u, v). Then we putand call it the ith mixed multiplicity of I and J, i = 0, . . . , r. To extend Teissier's result to hypersurfaces with non-isolated singularities we need to consider the case J is not a m-primary ideal. But this case has remained mysterious. A characterization of e i (I|J) as the multiplicity of A with respect to sufficiently general elements or to joint reductions of ideals as in the m-primary case has not been known. This paper will solve this problem for both mixed multiplicities of bigraded algebras and of ideals. We shall see that a mixed multiplicity is positive if and only if a certain ring has maximal dimension and that the positive mixed multiplicities can be expressed as Samuel's multiplicities. Our main tool is the notion of filter-regular sequences in a standard bigraded algebra R (see Section 1). This notion originated from the theory of Buchsbaum rings which have their roots in intersection theory [SV2]. Now we are going to present the main results of this paper. For this we shall need the following notation. For any pair of ideals a, b of a commutative ring S let a : b ∞ := {x ∈ S| there is a positive integer n such that xb n ⊆ a}.¿From the formula deg P R (u, v) = rdim R − 2 we can easily deduce that deg P R (u, v) = dim R/0 : R ∞ +...

Arithmetic degree and associated graded modules

“…We need to extend this idea to modules with a new definition, which is still close to the graded one. Another property of a relevant module is the following generalization of a result of N.V. Trung [17]. Proposition 2.6.…”

“…[17]) in two variables. As in the graded case some important informations are encoded in this polynomial, e.g.…”

On some multiplicity and mixed multiplicity formulas

“…The theory of filter-regular sequences became an important tool to study some classes of singular rings and has been continually developed (see e.g. [2,6,18,19,24]). …”

“…[7,8,9,11,14,16,19,21,22,23,25]). 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.…”

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]). …”