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preprint2014arXiv

On the vanishing of Hochster's theta invariant

Hochster's theta invariant is defined for a pair of finitely generated modules on a hypersurface ring having only an isolated singularity. Up to a sign, it agrees with the Euler invariant of a pair of matrix factorizations. Working over the complex numbers, Buchweitz and van Straten have established an interesting connection between Hochster's theta invariant and the classical linking form on the link of the singularity. In particular, they establish the vanishing of the theta invariant if the hypersurface is even-dimensional by exploiting the fact that the (reduced) cohomology of the Milnor fiber is concentrated in odd degrees in this situation. In this paper, we give purely algebraic versions of some of these results. In particular, we establish the vanishing of the theta invariant for isolated hypersurface singularities of odd dimension in characteristic p > 0, under some mild extra assumptions. This confirms, in a large number of cases, a conjecture of Hailong Dao.

preprint2017arXiv

Irreducibility of the Cayley-Menger determinant, and of a class of related polynomials

If $S$ is a given regular $n$-simplex, $n \ge 2$, of edge length $a$, then the distances $a_1$, $\cdots$, $a_{n+1}$ of an arbitrary point in its affine hull to its vertices are related by the fairly known elegant relation $ϕ_{n+1} (a,a_1,\cdots,a_{n+1})=0$, where $$ϕ= ϕ_t (x, x_1,\cdots,x_{n+1}) = \left( x^2+x_1^2+\cdots+x_{n+1}^2\right)^2 - t\left( x^4+x_1^4+\cdots+x_{n+1}^4\right).$$ The natural question whether this is essentially the only relation is answered positively by M. Hajja, M. Hayajneh, B. Nguyen, and Sh. Shaqaqha in a recently submitted paper entitled "Distances from the vertices of a regular simplex." In that paper, the authors made use of the irreducibility of the polynomial $ϕ$ in the case when $n \ge 2$, $t=n+1$, $x= a \ne 0$, and $k = \mathbb{R}$, but supplied no proof, promising to do so in another paper that is turning out to be this one. It is thus the main aim of this paper to establish that irreducibility. In fact, we treat the irreducibility of $ϕ$ without restrictions on $t$, $x$, $a$, and $k$. As a by-product, we obtain new proofs of results pertaining to the irreducibility of the general Cayley-Menger determinant that are more general than those

preprint2013arXiv

s-Lecture Hall Partitions, Self-Reciprocal Polynomials, and Gorenstein Cones

In 1997, Bousquet-Melou and Eriksson initiated the study of lecture hall partitions, a fascinating family of partitions that yield a finite version of Euler's celebrated odd/distinct partition theorem. In subsequent work on s-lecture hall partitions, they considered the self-reciprocal property for various associated generating functions, with the goal of characterizing those sequences s that give rise to generating functions of the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$. We continue this line of investigation, connecting their work to the more general context of Gorenstein cones. We focus on the Gorenstein condition for s-lecture hall cones when s is a positive integer sequence generated by a second-order homogeneous linear recurrence with initial values 0 and 1. Among such sequences s, we prove that the n-dimensional s-lecture hall cone is Gorenstein for all n greater than or equal to 1 if and only if s is an l-sequence. One consequence is that among such sequences s, unless s is an l-sequence, the generating function for the s-lecture hall partitions can have the form $((1-q^{e_1})(1-q^{e_2})...(1-q^{e_n}))^{-1}$ for at most finitely many n. We also apply the res

preprint2013arXiv

Brauer-Thrall for totally reflexive modules over local rings of higher dimension

Let $R$ be a commutative Noetherian local ring. Assume that $R$ has a pair $\{x,y\}$ of exact zerodivisors such that $\dim R/(x,y)\ge2$ and all totally reflexive $R/(x)$-modules are free. We show that the first and second Brauer--Thrall type theorems hold for the category of totally reflexive $R$-modules. More precisely, we prove that, for infinitely many integers $n$, there exists an indecomposable totally reflexive $R$-module of multiplicity $n$. Moreover, if the residue field of $R$ is infinite, we prove that there exist infinitely many isomorphism classes of indecomposable totally reflexive $R$-modules of multiplicity $n$.

preprint2016arXiv

Exploiting chordal structure in polynomial ideals: a Gröbner bases approach

Chordal structure and bounded treewidth allow for efficient computation in numerical linear algebra, graphical models, constraint satisfaction and many other areas. In this paper, we begin the study of how to exploit chordal structure in computational algebraic geometry, and in particular, for solving polynomial systems. The structure of a system of polynomial equations can be described in terms of a graph. By carefully exploiting the properties of this graph (in particular, its chordal completions), more efficient algorithms can be developed. To this end, we develop a new technique, which we refer to as chordal elimination, that relies on elimination theory and Gröbner bases. By maintaining graph structure throughout the process, chordal elimination can outperform standard Gröbner basis algorithms in many cases. The reason is that all computations are done on "smaller" rings, of size equal to the treewidth of the graph. In particular, for a restricted class of ideals, the computational complexity is linear in the number of variables. Chordal structure arises in many relevant applications. We demonstrate the suitability of our methods in examples from graph colorings, crypt

preprint2000arXiv

Growth of sumsets in abelian semigroups

Let S be an abelian semigroup, written additively. Let A be a finite subset of S. We denote the cardinality of A by |A|. For any positive integer h, the sumset hA is the set of all sums of h not necessarily distinct elements of A. We define 0A = {0}. If A_1,...,A_r, and B are finite sumsets of A and h_1,...,h_r are nonnegative integers, the sumset h_1A + ... + h_rA_r + B is the set of all elements of S that can be represented in the form u_1 + ... + u_r + b, where u_i \in h_iA_i and b \in B. The growth function of this sumset is γ(h_1,...,h_r) = |h_1A + ... + h_rA_r + B|. Applying the Hilbert function for graded modules over graded algebras, where the grading is over the semigroup of r-tuples of nonnegative integers, we prove that there is a polynomial p(t_1,...,t_r) such that γ(h_1,...,h_r) = p(t_1,...,t_r) if min(h_1,...,h_r) is sufficienlty large.

preprint2016arXiv

Invertibility of ideals in Prufer extensions

Using the general approach to invertibility for ideals in ring extensions given by Knebush-Zhang, we investigate about connections between faithfully flatness and invertibility for ideals in rings with zero divisors.

preprint2016arXiv

Valuations and Frobenius

The behavior of the Frobenius map is investigated for valuation rings of prime characteristic. We show that valuation rings are always F-pure. We introduce a generalization of the notion of strong F-regularity, which we call F-pure regularity, and show that a valuation ring is F-pure regular if and only if it is Noetherian. For valuations on function fields, we show that the Frobenius map is finite if and only if the valuation is divisorial; in this case the valuation ring is Frobenius split. For Noetherian valuation rings in function fields, we show that the valuation ring is Frobenius split if and only if Frobenius is finite, or equivalently, if and only if the valuation ring is excellent.

preprint2016arXiv

Finite Commutative Rings with a MacWilliams Type Relation for the m-Spotty Hamming Weight Enumerators

Let $R$ be a finite commutative ring. We prove that a MacWilliams type relation between the m-spotty weight enumerators of a linear code over $R$ and its dual hold, if and only if, $R$ is a Frobenius (equivalently, Quasi-Frobenius) ring, if and only if, the number of maximal ideals and minimal ideals of $R$ are the same, if and only if, for every linear code $C$ over $R$, the dual of the dual $C$ is $C$ itself. Also as an intermediate step, we present a new and simpler proof for the commutative case of Wood's theorem which states that $R$ has a generating character if and only if $R$ is a Frobenius ring.

preprint2016arXiv

Spaces of Sums of Powers and Real Rank Boundaries

We investigate properties of Waring decompositions of real homogeneous forms. We study the moduli of real decompositions, so-called Space of Sums of Powers, naturally included in the Variety of Sums of Powers. Explicit results are obtained for quaternary quadrics, relating the algebraic boundary of ${\rm SSP}$ to various loci in the Hilbert scheme of four points in $\mathbb{P}^3$. Further, we study the locus of general real forms whose real rank coincides with the complex rank. In case of quaternary quadrics the boundary of this locus is a degree forty hypersurface $J(σ_3(v_3(\mathbb{P}^3)),τ(v_3(\mathbb{P}^3)))$.

preprint2013arXiv

Flat local morphisms of rings with prescribed depth and dimension

For pairs of integers (n,m) and (d,e) satisfying some nedesary conditions, we construct a local flat ring morphism of noetherian local rings u:A -->B such that dim(A)=n, depth(A)=d, dim(B)=m, depth(B)=e.

preprint2016arXiv

Depth and Stanley depth of the path ideal associated to an $n$-cyclic graph

We compute the depth and Stanley depth for the quotient ring of the path ideal of length $3$ associated to a $n$-cyclic graph, given some precise formulas for depth when $n\not\equiv 1\,(\mbox{mod}\ 4)$, tight bounds when $n\equiv 1\,(\mbox{mod}\ 4)$ and for Stanley depth when $n\equiv 0,3\,(\mbox{mod}\ 4)$, tight bounds when $n\equiv 1,2\,(\mbox{mod}\ 4)$. Also, we give some formulas for depth and Stanley depth of a quotient of the path ideals of length $n-1$ and $n$.

preprint2012arXiv

Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

We give an expression for the Łojasiewicz exponent of a wide class of n-tuples of ideals $(I_1,..., I_n)$ in $Ø_n$ using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation of Łojasiewicz exponents in terms of Rees mixed multiplicities. As a consequence, we obtain a wide class of semi-weighted homogeneous functions $(\mathbb{C}^n,0)\to (\mathbb{C},0)$ for which the Łojasiewicz of its gradient map $\nabla f$ attains the maximum possible value.

preprint2016arXiv

Polynomial invariants and moduli of generic two-dimensional commutative algebras

Let $V$ be a two-dimensional vector space over a field $\mathbb F$ of characteristic not $2$ or $3$. We show there is a canonical surjection $ν$ from the set of suitably generic commutative algebra structures on $V$ modulo the action of $GL(V)$ onto the plane $\mathbb F^2$. In these coordinates, which are quotients of invariant quartic polynomials, properties such as associativity and the existence of zero divisors are described by simple algebraic conditions. The map $ν$ is a bijection over the complement of a degenerate elliptic curve $Γ$ and over $Γ$ we give an explicit parametrisation of the fibre in terms of Galois extensions of $\mathbb F$. Algebras in $ν^{-1}(Γ)$ are exactly those which admit non-trivial automorphisms. We show how $ν$ can be lifted to a map from the $SL(V)-$moduli space to an algebraic hypersurface $Γ'$ in a four-dimensional vector space whose equation is essentially the classical Eisenstein equation for the covariants of a binary cubic. This map is the restriction of a surjective map from the set of stable commutative algebras on $V$ modulo the action of $SL(V)$ onto $Γ'$.

preprint2016arXiv

Matlis' semi-regularity in trivial ring extensions issued from integral domains

This paper contributes to the study of homological aspects of trivial ring extensions (also called Nagata idealizations). Namely, we investigate the transfer of the notion of (Matlis') semi-regular ring (also known as IF-ring) along with related concepts, such as coherence, in trivial ring extensions issued from integral domains. All along the paper, we put the new results in use to enrich the literature with new families of examples subject to semi-regularity.

preprint2016arXiv

A Homological Approach to Factorization

Mott noted a one-to-one correspondence between saturated multiplicatively closed subsets of a domain D and directed convex subgroups of the group of divisibility D. With this, we construct a functor between inclusions into saturated localizations of D and projections onto partially ordered quotient groups of G(D). We use this functor to construct many cochain complexes of o-homomorphisms of po-groups. These complexes naturally lead to some fundamental structure theorems and some natural homology theory that provide insight into the factorization behavior of D.

preprint2016arXiv

Spontaneous atomicity for polynomial rings with zero-divisors

In this paper, we show that it is possible for a commutative ring with identity to be non-atomic (that is, there exist non-zero nonunits that cannot be factored into irreducibles) and yet have a strongly atomic polynomial extension. In particular, we produce a commutative ring with identity, R, that is antimatter (that is, R has no irreducibles whatsoever) such that R[t] is strongly atomic. What is more, given any nonzero nonunit f(t) in R[t] then there is a factorization of f(t) into irreducibles of length no more than deg(f(t)) + 2.

preprint2010arXiv

Asymptotic Behavior of Ext functors for modules of finite complete intersection dimension

Let $R$ be a local ring, and let $M$ and $N$ be finitely generated $R$-modules such that $M$ has finite complete intersection dimension. In this paper we define and study, under certain conditions, a pairing using the modules $\Ext_R^i(M,N)$ which generalizes Buchweitz's notion of the Herbrand diference. We exploit this pairing to examine the number of consecutive vanishing of $\Ext_R^i(M,N)$ needed to ensure that $\Ext_R^i(M,N)=0$ for all $i\gg 0$. Our results recover and improve on most of the known bounds in the literature, especially when $R$ has dimension at most two.

preprint2016arXiv

Higher Hochschild homology is not a stable invariant

Higher Hochschild homology is the analog of the homology of spaces, where the context for the coefficients -- which usually is that of abelian groups -- is that of commutative algebras. Two spaces that are equivalent after a suspension have the same homology. We show that this is not the case for higher Hochschild homology, providing a counterexample to a behavior so far observed in stable homotopy theory.

preprint2016arXiv

The Sally Modules of Ideals: A Survey

The Sally module of a Rees algebra $\BB$ relative to one of its Rees subalgebras $Å$ is a construct that can be used as a mediator for the trade-off of cohomological (e.g. depth) information between $\BB$ and the corresponding associated graded ring for several types of filtrations. While originally devised to deal with filtrations of finite colength, here we treat aspects of these developments for filtrations in higher dimensions as well.

preprint2010arXiv

Hochster's Theta Pairing and Algebraic Equivalence

We define a variant of Hochster's theta pairing and prove that it is constant in flat families of modules over hypersurfaces with isolated singularities. As a consequence, we show that the theta pairing factors through the Grothendieck group modulo algebraic equivalence. Moreover, our result allows us, in certain situations, to translate the properties of the theta pairing in characteristic zero to the characteristic p setting. We also give an application of our result to the rigidity of Tor over hypersurfaces.

preprint2011arXiv

Discrepancies of non-$\Q$-Gorenstein varieties

We give an example of a non $\Q$-Gorenstein variety which is canonical but not klt, and whose canonical divisor has an irrational valuation. We also give an example of an irrational jumping number and we prove that there are no accumulation points for the jumping numbers of normal non-$\Q$-Gorenstein varieties with isolated singularities.

preprint2016arXiv

Cohen-Macaulay properties under the amalgamated construction

Let $A$ and $B$ be commutative rings with unity, $f:A\to B$ a ring homomorphism and $J$ an ideal of $B$. Then the subring $A\bowtie^fJ:=\{(a,f(a)+j)|a\in A$ and $j\in J\}$ of $A\times B$ is called the amalgamation of $A$ with $B$ along $J$ with respect to $f$. In this paper, we study the property of Cohen-Macaulay in the sense of ideals which was introduced by Asgharzadeh and Tousi, a general notion of the usual Cohen-Macaulay property (in the Noetherian case), on the ring $A\bowtie^fJ$. Among other things, we obtain a generalization of the well-known result that when the Nagata's idealization is Cohen-Macaulay.

preprint2016arXiv

Fitness, Apprenticeship, and Polynomials

This article discusses the design of the Apprenticeship Program at the Fields Institute, held 21 August - 3 September 2016. Six themes from combinatorial algebraic geometry were selected for the two weeks: curves, surfaces, Grassmannians, convexity, abelian combinatorics, parameters and moduli. The activities were structured into fitness, research and scholarship. Combinatorics and concrete computations with polynomials (and theta functions) empowers young scholars in algebraic geometry, and it helps them to connect with the historic roots of their field. We illustrate our perspective for the threefold obtained by blowing up six points in $\mathbb{P}^3$.

301 works