Family Of Identities Research Articles

Local Okounkov bodies and limits in prime characteristic

This article is concerned with the asymptotic behavior of certain sequences of ideals in rings of prime characteristic. These sequences, which we call p-families of ideals, are ubiquitous in prime characteristic commutative algebra (e.g., they occur naturally in the theories of tight closure, Hilbert–Kunz multiplicity, and F-signature). We associate to each p-family of ideals an object in Euclidean space that is analogous to the Newton–Okounkov body of a graded family of ideals, which we call a p-body. Generalizing the methods used to establish volume formulas for the Hilbert–Kunz multiplicity and F-signature of semigroup rings, we relate the volume of a p-body to a certain asymptotic invariant determined by the corresponding p-family of ideals. We apply these methods to obtain new existence results for limits in positive characteristic, an analogue of the Brunn–Minkowski theorem for Hilbert–Kunz multiplicity, and a uniformity result concerning the positivity of a p-family.

T-clique ideal and t-independence ideal of a graph

ABSTRACTIn this paper, we introduce and study families of squarefree monomial ideals called clique ideals and independence ideals that can be associated to a finite graph. A family of clique ideals with linear resolutions has been characterized. Moreover, some families of graphs for which the quotient ring of their clique ideal is Cohen–Macaulay are introduced and some homological invariants of the clique ideal of a graph G, which is the complement of a path graph or a cycle graph, are obtained. Also some algebraic properties of the independence ideal of path graphs, cycle graphs and chordal graphs are studied.

Chordal Networks of Polynomial Ideals

We introduce a novel representation of structured polynomial ideals, which we refer to as chordal networks. The sparsity structure of a polynomial system is often described by a graph that captures the interactions among the variables. Chordal networks provide a computationally convenient decomposition into simpler (triangular) polynomial sets, while preserving the underlying graphical structure. We show that many interesting families of polynomial ideals admit compact chordal network representations (of size linear in the number of variables), even though the number of components is exponentially large. Chordal networks can be computed for arbitrary polynomial systems using a refinement of the chordal elimination algorithm from [Cifuentes-Parrilo-2016]. Furthermore, they can be effectively used to obtain several properties of the variety, such as its dimension, cardinality, and equidimensional components, as well as an efficient probabilistic test for radical ideal membership. We apply our methods to examples from algebraic statistics and vector addition systems; for these instances, algorithms based on chordal networks outperform existing techniques by orders of magnitude.

Ordinary and symbolic Rees algebras for ideals of Fermat point configurations

Fermat ideals define planar point configurations that are closely related to the intersection locus of the members of a specific pencil of curves. These ideals have gained recent popularity as counterexamples to some proposed containments between symbolic and ordinary powers [6]. We give a systematic treatment of the family of Fermat ideals, describing explicitly the minimal generators and the minimal free resolutions of all their ordinary powers as well as many symbolic powers. We use these to study the ordinary and the symbolic Rees algebra of Fermat ideals. Specifically, we show that the symbolic Rees algebras of Fermat ideals are Noetherian. Along the way, we give formulas for the Castelnuovo–Mumford regularity of powers of Fermat ideals and we determine their reduction ideals.

Minimal resolutions of dominant and semidominant ideals

We introduce new classes of monomial ideals: dominant, p-semidominant, and GNP ideals. The families of dominant and 1-semidominant ideals extend those of complete and almost complete intersections, while the family of GNP ideals extends that of generic ideals. We show that dominant ideals give a complete characterization of when the Taylor resolution is minimal, 1-semidominant ideals are Scarf, and the minimal resolutions of 2-semidominant ideals can be obtained from their Taylor resolutions by eliminating faces and facets of equal multidegree, in arbitrary order. We also generalize a theorem of Bayer, Peeva and Sturmfels by proving that GNP ideals are Scarf.

Realizations of pairs and Oka families in tensor triangulated categories

In this paper, we apply some methods from ring theory to the framework of prime ideals in tensor triangulated categories developed by Balmer. Given a thick tensor ideal $${\mathscr {A}}$$ and a multiplicatively closed family $${\mathscr {S}}$$ of objects in a tensor triangulated category $$({\mathscr {C}}, \otimes ,1)$$ , we say that a prime ideal $${\mathscr {P}}$$ realizes $$({\mathscr {A}},{\mathscr {S}})$$ if $${\mathscr {P}}\supseteq {\mathscr {A}}$$ and $${\mathscr {P}}\cap {\mathscr {S}}=\varnothing $$ . Analogously to the results of Bergman with ordinary rings, we show how to construct a realization of a family $$\{({\mathscr {A}}_i,{\mathscr {S}}_i)\}_{i\in I}$$ of such pairs indexed by a finite chain I, i.e., a collection $$\{{\mathscr {P}}_i\}_{i\in I}$$ of prime ideals such that each $${\mathscr {P}}_i$$ realizes $$({\mathscr {A}}_i,{\mathscr {S}}_i)$$ and for each $$i\leqslant j$$ in I. Thereafter we obtain conditions on a family $${\mathfrak {F}} $$ of thick tensor ideals of $$({\mathscr {C}},\otimes ,1)$$ so that any ideal that is maximal with respect to not being contained in $${\mathfrak {F}}$$ must be prime. This extends the Prime Ideal Principle of Lam and Reyes from commutative algebra. We also combine these methods to consider realizations of templates $$\{({\mathscr {A}}_i,{\mathfrak {F}}_i)\}_{i\in I}$$ , where each $${\mathscr {A}}_i$$ is a thick tensor ideal and each $${\mathfrak {F}}_i$$ is a family of thick tensor ideals that is also a monoidal semifilter.

Betti numbers of binomial ideals

Let us consider the family of binomial ideals B=I+J, where J is lattice ideal and I is a square-free quadratic monomial ideal. We give a formula for calculating the Betti numbers of B. Moreover we bound the Green–Lazarsfeld invariant of a family of quadratic binomial ideals B using this formula. This result extends a previous result of Eisenbud et al. for square-free quadratic monomial ideals and extends completely Fröberg's theorem. We describe also a subfamily where we can calculate the Green–Lazarsfeld invariant of any ideal B and we also compute its first non-linear Betti number.

The Waldschmidt constant for squarefree monomial ideals

Given a squarefree monomial ideal \(I \subseteq R =k[x_1,\ldots ,x_n]\), we show that \(\widehat{\alpha }(I)\), the Waldschmidt constant of I, can be expressed as the optimal solution to a linear program constructed from the primary decomposition of I. By applying results from fractional graph theory, we can then express \(\widehat{\alpha }(I)\) in terms of the fractional chromatic number of a hypergraph also constructed from the primary decomposition of I. Moreover, expressing \(\widehat{\alpha }(I)\) as the solution to a linear program enables us to prove a Chudnovsky-like lower bound on \(\widehat{\alpha }(I)\), thus verifying a conjecture of Cooper–Embree–Hà–Hoefel for monomial ideals in the squarefree case. As an application, we compute the Waldschmidt constant and the resurgence for some families of squarefree monomial ideals. For example, we determine both constants for unions of general linear subspaces of \(\mathbb P^n\) with few components compared to n, and we compute the Waldschmidt constant for the Stanley–Reisner ideal of a uniform matroid.

On intersections of complete intersection ideals

We prove that for certain families of toric complete intersection ideals, the arbitrary intersections of elements in the same family are again complete intersections.

Topology, intersections and flat modules

It is well known that, in general, multiplication by an ideal I I does not commute with the intersection of a family of ideals, but that this fact holds if I I is flat and the family is finite. We generalize this result by showing that finite families of ideals can be replaced by compact subspaces of a natural topological space, and that ideals can be replaced by submodules of an epimorphic extension of a base ring. As a particular case, we give a new proof of a conjecture by Glaz and Vasconcelos.

Connected Graphs Arising from Products of Veronese Varieties

Given a Segre squarefree Veronese configuration, following Bernd Sturmfels, we improve the study of the graphs associated to the configuration. We determine two special families of toric ideals and a finite set of moves for each of them, which still guarantee simultaneously the connection of all graphs arising from each family of moves.

On $C^*$-algebras associated to right LCM semigroups

We initiate the study of the internal structure of C*-algebras associated to a left cancellative semigroup in which any two principal right ideals are either disjoint or intersect in another principal right ideal; these are variously called right LCM semigroups or semigroups that satisfy Clifford's condition. Our main findings are results about uniqueness of the full semigroup C*-algebra. We build our analysis upon a rich interaction between the group of units of the semigroup and the family of constructible right ideals. As an application we identify algebraic conditions on S under which C*(S) is purely infinite and simple.

Growth of multiplicities of graded families of ideals

Let (R,m) be a Noetherian local ring of dimension d>0. Let I•={In}n∈N be a graded family of m-primary ideals in R. We examine how far off from a polynomial can the length function ℓR(R/In) be asymptotically. More specifically, we show that there exists a constant γ>0 such that for all n≥0,ℓR(R/In+1)−ℓR(R/In)<γnd−1.

When do L-fuzzy ideals of a ring generate a distributive lattice?

Abstract The notion of L-fuzzy extended ideals is introduced in a Boolean ring, and their essential properties are investigated. We also build the relation between an L-fuzzy ideal and the class of its L-fuzzy extended ideals. By defining an operator “⇝” between two arbitrary L-fuzzy ideals in terms of L-fuzzy extended ideals, the result that “the family of all L-fuzzy ideals in a Boolean ring is a complete Heyting algebra” is immediately obtained. Furthermore, the lattice structures of L-fuzzy extended ideals of an L-fuzzy ideal, L-fuzzy extended ideals relative to an L-fuzzy subset, L-fuzzy stable ideals relative to an L-fuzzy subset and their connections are studied in this paper.

Prime Ideals in Algebras Determined by Submonoids of Nilpotent Groups

The prime spectrum of the semigroup algebra K[S] of a submonoid S of a finitely generated nilpotent group is studied via the spectra of the monoid S and of the group algebra K[G] of the group G of fractions of S. It is shown that the classical Krull dimension of K[S] is equal to the Hirsch length of the group G provided that G is nilpotent of class two. This uses the fact that prime ideals of S are completely prime. An infinite family of prime ideals of a submonoid of a free nilpotent group of class three with two generators which are not completely prime is constructed. They lead to prime ideals of the corresponding algebra. Prime ideals of the monoid of all upper triangular n × n matrices with non-negative integer entries are described and it follows that they are completely prime and finite in number.

F-jumping and F-Jacobian ideals for hypersurfaces

We introduce two families of ideals, F-jumping ideals and F-Jacobian ideals, in order to study the singularities of hypersurfaces in positive characteristic. Both families are defined using the D-modules Mα that were introduced by Blickle, Mustaţă and Smith. Using strong connections between F-jumping ideals and generalized test ideals, we give a characterization of F-jumping numbers for hypersurfaces via D-modules and F-modules. In addition, we use F-Jacobian ideals to study intrinsic properties of the singularities of hypersurfaces. In particular, we give conditions for F-regularity. Moreover, we prove several properties of F-Jacobian ideals that resemble those of Jacobian ideals of polynomials.

Nonlinear absolutely summing operators revisited

In the last decades many authors have become interested in the study of multilinear and polynomial generalizations of families of operator ideals (such as, for instance, the ideal of absolutely summing operators). However, these generalizations must keep the essence of the given operator ideal and there seems not to be a universal method to achieve this. The main task of this paper is to discuss, study, and introduce multilinear and polynomial extensions of the aforementioned operator ideals taking into account the already existing methods of evaluating the adequacy of such generalizations. Besides this subject’s intrinsic mathematical interest, the main motivation is our belief (based on facts that shall be presented) that some of the already existing approaches are not adequate.

Asymptotic multiplicities

This paper is on graded families of ideals and filtrations and associated asymptotic multiplicities. We prove some new results, including a general Minkowski formula for limits and a proof that epsilon multiplicity exists as a limit for modules under rather general conditions. We also explore conditions under which various asymptotic limits exist.

Reduction of a family of ideals

In the paper we prove that there exists a simultaneous reduction of one-parameter family of $\mathfrak{m}_{n}$-primary ideals in the ring of germs of holomorphic functions. As a corollary we generalize the result of A. Ploski \cite{ploski} on the semicontinuity of the Łojasiewicz exponent in a multiplicity-constant deformation.

Mutually Exclusive Nuances of Truth in Moisil Logic

Moisil logic, having as algebraic counterpart Łukasiewicz-Moisil algebras, provide an alternative way to reason about vague information based on the following principle: a many-valued event is characterized by a family of Boolean events. However, using the original definition of Łukasiewicz-Moisil algebra, the principle does not apply for subalgebras. In this paper we identify an alternative and equivalent definition for the $n$-valued Ł ukasiewicz-Moisil algebras, in which the determination principle is also saved for arbitrary subalgebras, which are characterized by a Boolean algebra and a family of Boolean ideals. As a consequence, we prove a duality result for the $n$-valued Łukasiewicz-Moisil algebras, starting from the dual space of their Boolean center. This leads us to a duality for MV$_n$-algebras, since are equivalent to a subclass of $n$-valued Łukasiewicz-Moisil algebras.