Degree and algebraic properties of lattice and matrix ideals.

*(English)*Zbl 1334.13017Let \(\mathcal{L}\) be a lattice in \(\mathbb{Z}^s\) of rank \(s-1\), generated by \(\alpha_1,\ldots,\alpha_m\) and let \(L\) be the \(s\times m\) matrix with column vectors \(\alpha_1,\ldots,\alpha_m\). Each \(\alpha_i\) can be written uniquely as \(\alpha_i=\alpha_i^+-\alpha_i^-\), where \(\alpha_i^+,\alpha_i^-\in \mathbb{N}^s\). The matrix ideal of \(L\), denoted by \(I(L)\), is generated by all the binomials \(t^{\alpha_i^+}-t^{\alpha_i^-}\), for all \(i=1,\ldots,m\).

Consider a polynomial ring \(\mathbb{K}[x_1]\) in one variable, where \(\mathbb{K}\) is a field containing the \(\gamma_{s-1}\)-th roots of unity with \(\text{char}(\mathbb{K})=0\) or \(\text{char}(\mathbb{K})=p\), where \(p\) is a prime number with \(p\nmid\gamma_{s-1}\). By \(\Lambda_i\) is denoted the set of \(\gamma_i\)-th roots of unity in \(\mathbb{K}\) and let \(\Lambda=\prod_{i=1}^{s-1}\Lambda_i\). For any \(\lambda=\left(\lambda_1,\ldots,\lambda_{s-1}\right)\in\Lambda\), there is an homomorphism of \(\mathbb{K}\)-algebras: \[ \phi_{\lambda}: S\longrightarrow\mathbb{K}[x_1],\;t_i\longmapsto \lambda_1^{p_{1,i}}\cdots\lambda_{s-1}^{p_{s-1,i}},\;i=1,\ldots,s. \] The kernel of \(\phi_{\lambda}\), denoted by \(\alpha_{\lambda}\), is a prime ideal of \(S\) and of height \(s-1\).

The authors in their first main result (Theorem 3.3) prove for a graded lattice ideal \(I(\mathcal{L})\) of dimension one, that \(I(\mathcal{L})=\bigcap_{\lambda\in\Lambda}\alpha_{\lambda}\) is the minimal primary decomposition of the ideal \(I(\mathcal{L})\) into exactly \(\gamma\) primary components. By this result, they are able to characterize when the graded binomial ideals, satisfying the vanishing condition \(V(I,t_i)=\{0\}\), for all \(i\), are lattice ideals (Proposition 5.3). Moreover, they apply this proposition to give a structure of graded matrix ideals \(I\) (Proposition 5.7), in terms of the notion of unmixed ideals; (ideals whose all associated primes have the same height) and the meaning of the intersection of the isolated primary components of \(I\), which is denoted by \(\text{Hull}(I)\).

In their next main result, the authors compute the degree of any lattice ideal (Theorem 4.6). More explicitly, they prove that if \(\mathcal{L}\subset\mathbb{Z}^s\) is a lattice of rank \(r\), then the following hold:

Consider a polynomial ring \(\mathbb{K}[x_1]\) in one variable, where \(\mathbb{K}\) is a field containing the \(\gamma_{s-1}\)-th roots of unity with \(\text{char}(\mathbb{K})=0\) or \(\text{char}(\mathbb{K})=p\), where \(p\) is a prime number with \(p\nmid\gamma_{s-1}\). By \(\Lambda_i\) is denoted the set of \(\gamma_i\)-th roots of unity in \(\mathbb{K}\) and let \(\Lambda=\prod_{i=1}^{s-1}\Lambda_i\). For any \(\lambda=\left(\lambda_1,\ldots,\lambda_{s-1}\right)\in\Lambda\), there is an homomorphism of \(\mathbb{K}\)-algebras: \[ \phi_{\lambda}: S\longrightarrow\mathbb{K}[x_1],\;t_i\longmapsto \lambda_1^{p_{1,i}}\cdots\lambda_{s-1}^{p_{s-1,i}},\;i=1,\ldots,s. \] The kernel of \(\phi_{\lambda}\), denoted by \(\alpha_{\lambda}\), is a prime ideal of \(S\) and of height \(s-1\).

The authors in their first main result (Theorem 3.3) prove for a graded lattice ideal \(I(\mathcal{L})\) of dimension one, that \(I(\mathcal{L})=\bigcap_{\lambda\in\Lambda}\alpha_{\lambda}\) is the minimal primary decomposition of the ideal \(I(\mathcal{L})\) into exactly \(\gamma\) primary components. By this result, they are able to characterize when the graded binomial ideals, satisfying the vanishing condition \(V(I,t_i)=\{0\}\), for all \(i\), are lattice ideals (Proposition 5.3). Moreover, they apply this proposition to give a structure of graded matrix ideals \(I\) (Proposition 5.7), in terms of the notion of unmixed ideals; (ideals whose all associated primes have the same height) and the meaning of the intersection of the isolated primary components of \(I\), which is denoted by \(\text{Hull}(I)\).

In their next main result, the authors compute the degree of any lattice ideal (Theorem 4.6). More explicitly, they prove that if \(\mathcal{L}\subset\mathbb{Z}^s\) is a lattice of rank \(r\), then the following hold:

- (a)
- if \(r=s\), then \(\deg\left(S/I\left(\mathcal{L}\right)\right)=\mid\mathbb{Z}^s/\mathcal{L}\mid\).
- (b)
- if \(r<s\), there is an integer matrix \(A\) of size \((s-r)\times s\) with \(\text{rank}(A)=s-r\) such that we have the containment of rank \(r\) lattices \(\mathcal{L}\subset\ker_{\mathbb{Z}(A)}\) with equality, if and only if \(\mathbb{Z}^s/\mathcal{L}\) is torsion free.
- (c)
- if \(r<s\) and \(v_1,\ldots,v_s\) are the columns of \(A\), then \[ \deg\left(S/I(\mathcal{L})\right)=\dfrac{\mid T(\mathbb{Z}^s/\mathcal{L})\mid(s-r)!\text{vol}(\text{conv}(0,v_1,\ldots,v_s))}{\mid T(\mathbb{Z}^{s-r}/ \langle v_1,\ldots,v_s\rangle) \mid}, \] where by \(\text{vol}(P)\) is denoted the relative volume of a lattice polytope \(P\) and by \(\text{conv}(A)\) the convex hull of a set \(A\). It is remarked here, that for the well known result (b), the authors give a different proof, from which there is a way to compute the degree of an arbitrary lattice ideal. Also, they present some applications of the above theorem, to compute the corresponding degrees of certain families of lattice ideals.

Reviewer: Christos Tatakis (Mitilini)

##### MSC:

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |

13A15 | Ideals and multiplicative ideal theory in commutative rings |

13H15 | Multiplicity theory and related topics |

13P05 | Polynomials, factorization in commutative rings |

05E40 | Combinatorial aspects of commutative algebra |