The primary components of positive critical binomial ideals

Abstract

A natural candidate for a generating set of the (necessarily prime) defining ideal of an n-dimensional monomial curve, when the ideal is an almost complete intersection, is a full set of n critical binomials. In a somewhat modified and more tractable context, we prove that, when the exponents are all positive, critical binomial ideals in our sense are not even unmixed for n ≥ 4, whereas for n ≤ 3 they are unmixed. We further give a complete description of their isolated primary components as the defining ideals of monomial curves with coefficients. This answers an open question on the number of primary components of Herzog-Northcott ideals, which comprise the case n = 3. Moreover, we find an explicit, concrete description of the irredundant embedded component (for n ≥ 4) and characterize when the hull of the ideal, i.e., the intersection of its isolated primary components, is prime. Note that these last results are independent of the characteristic of the ground field. Our techniques involve the Eisenbud-Sturmfels theory of binomial ideals and Laurent polynomial rings, together with theory of Smith Normal Form and of Fitting ideals. This gives a more transparent and completely general approach, replacing the theory of multiplicities used previously to treat the particular case n = 3.Date: September 15, 2018. 2000 Mathematics Subject Classification. 13A05, 13A15 (primary), 13P05, 13P15 (secondary).

Degree and Algebraic Properties of Lattice and Matrix Ideals

“…. , s. The kernel of ϕ λ , denoted by a λ , is a prime ideal of S of height s − 1 [30]. This type of ideal was introduced in [30] to study the algebraic properties of PCB ideals.…”

“…Proof. Applying [30,Remark 3.4] to L, the PCB matrix associated to L, one gets the syzygy t b(1) f 1 + . .…”

“…By and large, we focus on the structure of the class of graded binomial ideals I that satisfy the vanishing condition V (I, t i ) = {0} for all i, where V (·) is the variety of (·). This class of ideals includes the graded lattice ideals of dimension 1 [20], the vanishing ideals over finite fields of algebraic toric sets [33], the toric ideals of monomial curves [16], the Herzog-Northcott ideals [29], the PCB ideals [30] and the Laplacian ideals of complete graphs [23]. We will also present some other interesting families of ideals that satisfy this hypothesis.…”

“…The matrix L is called: (a) a pure binomial matrix (PB matrix, for short) if a j,j > 0 for all j, and for each column of L at least one off-diagonal entry is non-zero; (b) a positive pure binomial matrix (PPB matrix) if all the entries of L are non-zero; (c) a critical binomial matrix (CB matrix) if L is a PB matrix and L1 ⊤ = 0; (d) a positive critical binomial matrix (PCB matrix [30]) if all the entries of L are non-zero and L1 ⊤ = 0; (e) a generalized critical binomial matrix (GCB matrix), if L is a PB matrix and there exists b ∈ N s + such that Lb ⊤ = 0; and (f) a generalized positive critical binomial matrix (GPCB matrix) if all the entries of L are non-zero and there exists b ∈ N s + such that Lb ⊤ = 0. If L is a PB matrix, we will call I(L) = (f 1 , .…”

“…The transpose of L is denoted by L ⊤ . If I(L ⊤ ) is graded, following [30], we study when I(L) is also graded.…”