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Register a new userOn the generalized Scarf complex of lattice ideals
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Let $k$ be a field, $ mathcal{L}subset mathbb{Z}^n$ be a lattice such that $Lcap mathbb{N}^n={{bf 0}}$, and $I_Lsubset Bbbk[x_1,..., x_n]$ the corresponding lattice ideal. We present the generalized Scarf complex of $I_L$ and show that it is indispensable in the sense that it is contained in every minimal free resolution of $R/I_L$.
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In this paper we completely characterize lattice ideals that are complete intersections or equivalently complete intersections finitely generated semigroups of $bz^noplus T$ with no invertible elements, where $T$ is a finite abelian group. We also characterize the lattice ideals that are set-theoretic complete intersections on binomials.
In this article we associate to every lattice ideal $I_{L,rho}subset K[x_1,..., x_m]$ a cone $sigma $ and a graph $G_{sigma}$ with vertices the minimal generators of the Stanley-Reisner ideal of $sigma $. To every polynomial $F$ we assign a subgraph $G_{sigma}(F)$ of the graph $G_{sigma}$. Every expression of the radical of $I_{L,rho}$, as a radical of an ideal generated by some polynomials $F_1,..., F_s$ gives a spanning subgraph of $G_{sigma}$, the $cup_{i=1}^s G_{sigma}(F_i)$. This result provides a lower bound for the minimal number of generators of $I_{L,rho}$ and therefore improves the generalized Krulls principal ideal theorem for lattice ideals. But mainly it provides lower bounds for the binomial arithmetical rank and the $A$-homogeneous arithmetical rank of a lattice ideal. Finally we show, by a family of examples, that the bounds given are sharp.
We characterize the graphs $G$ for which their toric ideals $I_G$ are complete intersections. In particular we prove that for a connected graph $G$ such that $I_G$ is complete intersection all of its blocks are bipartite except of at most two. We prove that toric ideals of graphs which are complete intersections are circuit ideals. The generators of the toric ideal correspond to even cycles of $G$ except of at most one generator, which corresponds to two edge disjoint odd cycles joint at a vertex or with a path. We prove that the blocks of the graph satisfy the odd cycle condition. Finally we characterize all complete intersection toric ideals of graphs which are normal.
The Qth-power algorithm for computing structured global presentations of integral closures of affine domains over finite fields is modified to compute structured presentations of integral closures of ideals in affine domains over finite fields relative to a local monomial ordering. A non-homogeneous version of the standard (homogeneous) Rees algebra is introduced as well.
Let $R=mathbf{C}[xi_1,xi_2,ldots]$ be the infinite variable polynomial ring, equipped with the natural action of the infinite symmetric group $mathfrak{S}$. We classify the $mathfrak{S}$-primes of $R$, determine the containments among these ideals, and describe the equivariant spectrum of $R$. We emphasize that $mathfrak{S}$-prime ideals need not be radical, which is a primary source of difficulty. Our results yield a classification of $mathfrak{S}$-ideals of $R$ up to copotency. Our work is motivated by the interest and applications of $mathfrak{S}$-ideals seen in recent years.
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