Mathematics - Commutative Algebra Publications (50)


Mathematics - Commutative Algebra Publications

In this work we define a primary spectrum of a commutative ring R with its Zariski topology $\mathfrak{T}$. We introduce several properties and examine some topological features of this concept. We also investigate differences between the prime spectrum and our primary spectrum. Read More

In this paper, we state criteria of a Hibi ring to be level, non-Gorenstein and almost Gorenstein and to be non-level and almost Gorenstein in terms of the structure of the partially ordered set defining the Hibi ring. We also state a criterion of a ladder determinantal ring defined by 2-minors to be non-Gorenstein and almost Gorenstein in terms of the shape of the ladder. Read More

This chapter of the forthcoming Handbook of Graphical Models contains an overview of basic theorems and techniques from algebraic geometry and how they can be applied to the study of conditional independence and graphical models. It also introduces binomial ideals and some ideas from real algebraic geometry. When random variables are discrete or Gaussian, tools from computational algebraic geometry can be used to understand implications between conditional independence statements. Read More

The structure of the defining ideal of the semigroup ring $k[H]$ of a numerical semigroup $H$ over a field $k$ is described, when the pseudo-Frobenius numbers of $H$ are multiples of a fixed integer. Read More

Given a finitely generated and projective Lie-Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of this convolution algebra is here clarified, by providing an explicit description of its topological antipode as well as of its other structure maps. Conditions under which that homomorphism becomes an homeomorphism are also discussed. Read More

We study Cohen-Macaulay non-Gorenstein local rings $(R,m,k)$ admitting certain totally reflexive modules. We describe the Bass series of $R$, which especially shows that it is rational. We also give a description of the Poincare series of $k$ by using the Poincare series of the totally reflexive module. Read More

Let $\H$ be a unimodular hypergraph over the vertex set $[n]$ and let $J(\H)$ be the cover ideal of $\H$ in the polynomial ring $R=K[x_1,\ldots,x_n]$. We show that $\reg J(\H)^s$ is a linear function in $s$ for all $s\geqslant r\left\lceil \frac{n}{2}\right\rceil+1$ where $r$ is the rank of $\H$. Moreover for every $i$, $a_i(R/J(\H)^s)$ is also a linear function in $s$ for $s \geqslant n^2$. Read More

Let $\text{Red}(M)$ be the sum of all reduced submodules of a module $M$. For modules over commutative rings, $\text{Soc}(M)\subseteq \text{Red}(M)$. By drawing motivation from how $\text{Soc}$-injective modules were defined by Amin et. Read More

Let $X_{P}$ be the projective toric surface associated to a lattice polytope $P$. If the number of lattice points lying on the boundary of $P$ is at least $4$, it is known that $X_{P}$ is embeddable into a suitable projective space as zero set of finitely many quadrics. In this case, the determination of a minimal generating system of the toric ideal defining $X_{P}$ is reduced to a simple Gaussian elimination. Read More

This article consists of two parts. In Part 1, we present a formulation of two-dimensional topological quantum field theories in terms of a functor from a category of Ribbon graphs to the endofuntor category of a monoidal category. The key point is that the category of ribbon graphs produces all Frobenius objects. Read More

We extend to one dimensional quotients the result of A. Conca and S. Murai on the convexity of the regularity of Koszul cycles. Read More

In this work, we define the face ring for a matroid over Z. Its Hilbert series is, indeed, the expected specialization of the Grothendieck-Tutte polynomial defined by Fink and Moci. Read More

Let $S$ be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if $R$ is a non-Gorenstein quotient of $S$ of small colength, then every totally reflexive $R$-module is free. Indeed, the second syzygy of the canonical module of $R$ has a direct summand $T$ which is a test module for freeness over $R$ in the sense that if $\mathrm{Tor}_+^R(T,N)=0$, for some finitely generated $R$-module $N$, then $N$ is free. Read More

Macaulay's Inverse System gives an effective method to construct Artinian Gorenstein k-algebras. To date a general structure for Gorenstein k-algebras of any dimension (and codimension) is not understood. In this paper we extend Macaulay's correspondence characterizing the submodules of the divided power ring in one-to-one correspondence with Gorenstein d-dimensional k-algebras. Read More

In this work, the set of quasi-primary ideals of a commutative ring with identity is equipped with a topology and is called quasi-primary spectrum. Some topological properties of this space are examined. Further, a sheaf of rings on the quasi-primary spectrum is constructed and it is shown that this sheaf is the direct image sheaf with respect to the inclusion map from the prime spectrum of a ring to the quasi-primary spectrum of the same ring. Read More

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. While generalizing the classical theory of Gr{\"o}bner bases, it is not clear how modern algorithms for computing Gr{\"o}bner bases can be adapted to the tropical case. Read More

Let R be a Cohen-Macaulay local ring with a canonical module. We consider Auslander's (higher) delta invariants of powers of certain ideals of R. Firstly, we shall provide some conditions for an ideal to be a parameter ideal in terms of delta invarints. Read More

Let $R$ be a polynomial ring over a field $k$ with irrelevant ideal $\frak m$ and dimension $d$. Let $I$ be a homogeneous ideal in $R$. We study the asymptotic behavior of the length of the modules $H^{i}_{\frak m}(R/I^n)$ for $n\gg 0$. Read More

In this short note we compute length, support and dimension of syzygy modules of certain modules. Read More

By virtue of Balmer's celebrated theorem, the classification of thick tensor ideals of a tensor triangulated category $\T$ is equivalent to the topological structure of its Balmer spectrum $\spc \T$. Motivated by this theorem, we discuss connectedness and noetherianity of the Balmer spectrum of a right bounded derived category of finitely generated modules over a commutative ring. Read More

The catenary degree is an invariant that measures the distance between factorizations of elements within an atomic monoid. In this paper, we classify which finite subsets of $\mathbb Z_{\ge 0}$ occur as the set of catenary degrees of a numerical monoid (i.e. Read More

Let $\mathfrak{q}$ denote an ideal in a Noetherian local ring $(A,\mathfrak{m})$. Let $\underline{a}=a_1,\ldots,a_d \subset \mathfrak{q}$ denote a system of parameters in a finitely generated $A$-module $M$. This note investigate an improvement of the inequality $c_1\cdot \ldots \cdot c_d \cdot e_0(\mathfrak{q};M) \leq \ell_A(M/\underline{a}\,M)$, where $c_i$ denote the initial degrees of $a_i$ in the form ring $G_A(\mathfrak{q})$. Read More

In this paper, we introduce the notion of Auslander and Z-Auslander modules, inspired from Auslander's Zero-Divisor Conjecture and give some interesting results for these modules. We also introduce and investigate the concept of modules having property (Z), a concept that is based on a property of flat modules. Read More

We prove that the Newton-Okounkov body of the flag $E_{\bullet}:= \left\{ X=X_r \supset E_r \supset \{q\} \right\}$, defined by the surface $X$ and the exceptional divisor $E_r$ given by any divisorial valuation of the complex projective plane $\mathbb{P}^2$, with respect to the pull-back of the line-bundle $\mathcal{O}_{\mathbb{P}^2} (1)$ is either a triangle or a quadrilateral, characterizing when it is a triangle or a quadrilateral. We also describe the vertices of that figure. Finally, we introduce a large family of flags for which we determine explicitly their Newton-Okounkov bodies which turn out to be triangular. Read More

We call shifted power a polynomial of the form $(x-a)^e$. The main goal of this paper is to obtain broadly applicable criteria ensuring that the elements of a finite family $F$ of shifted powers are linearly independent or, failing that, to give a lower bound on the dimension of the space of polynomials spanned by $F$. In particular, we give simple criteria ensuring that the dimension of the span of $F$ is at least $c. Read More

We prove an if-and-only-if criterion for direct sum decomposability of a smooth homogeneous form in terms of the factorization properties of the Macaulay inverse system of its Milnor algebra. This criterion leads to an algorithm for computing direct sum decompositions over any field either of characteristic zero, or of sufficiently large positive characteristic, for which polynomial factorization algorithms exist. Read More

In this paper, we describe improved algorithms to compute Janet and Pommaret bases. To this end, based on the method proposed by Moller et al., we present a more efficient variant of Gerdt's algorithm (than the algorithm presented by Gerdt-Hashemi-M. Read More

The space of linearly recursive sequences of complex numbers admits two distinguished topologies. Namely, the adic topology induced by the ideal of those sequences whose first term is $0$ and the topology induced from the Krull topology on the space of complex power series via a suitable embedding. We show that these topologies are not equivalent. Read More

Let $(R, \mathfrak m)$ be a commutative noetherian local ring and $I$ an ideal of $R$. Let $\mathcal{P}$ be the class of all $I$-generated $R$-modules $M$ (i.e. Read More

We begin the study of the notion of diameter of an ideal I of a polynomial ring S over a field, an invariant measuring the distance between the minimal primes of I. We provide large classes of Hirsch ideals, i.e. Read More

We consider several notions of genericity appearing in algebraic geometry and commutative algebra. Special emphasis is put on various stability notions which are defined in a combinatorial manner and for which a number of equivalent algebraic characterisations are provided. It is shown that in characteristic zero the corresponding generic positions can be obtained with a simple deterministic algorithm. Read More

We improve certain degree bounds for Grobner bases of polynomial ideals in generic position. We work exclusively in deterministically verifiable and achievable generic positions of a combinatorial nature, namely either strongly stable position or quasi stable position. Furthermore, we exhibit new dimension- (and depth-)dependent upper bounds for the Castelnuovo-Mumford regularity and the degrees of the elements of the reduced Grobner basis (w. Read More

Let $m$, $n$ and $p$ be integers with $3\leq m\leq n$ and $(m-1)(n-1)+1\leq p\leq (m-1)m$. We showed in previous papers that if $p\geq (m-1)(n-1)+2$, then typical ranks of $p\times n\times m$-tensors over the real number field are $p$ and $p+1$ if and only if there exists a nonsingular bilinear map $\mathbb{R}^m\times \mathbb{R}^n\to\mathbb{R}^{mn-p}$. We also showed that the "if" part also valid in the case where $p=(m-1)(n-1)+1$. Read More

This work concerns the moment map $\mu$ associated with the standard representation of a classical Lie algebra. For applications to deformation quantization it is desirable that $S/(\mu)$, the coordinate algebra of the zero fibre of $\mu$, be Koszul. The main result is that this algebra is not Koszul for the standard representation of $\mathfrak{sl}_{n}$, and of $\mathfrak{sp}_{n}$. Read More

We show that existence of nonzero nilpotent elements in the $\Z$-module $\Z/(p_1^{k_1}\times \cdots \times p_n^{k_n})\Z$ inhibits the module from possessing good structural properties. In particular, it stops it from being semisimple and from admitting certain good homological properties. Read More

We prove that, on a large cone containing the constant multiplicities, the only free multiplicities on the braid arrangement are those identified in work of Abe, Nuida, and Numata (2009). We also give a conjecture on the structure of all free multiplicities on braid arrangements. Read More

Using perfectoid algebras, we introduce a mixed characteristic analog of the multiplier ideal, respectively test ideal, from characteristic zero, respectively $p > 0$, in the case of a regular ambient ring. We prove several properties about this ideal such as subadditivity. We then use these techniques to derive a uniform bound on the growth of symbolic powers of radical ideals in all excellent regular rings. Read More

We give a sufficient criterion for a lower bound of the cactus rank of a tensor. Then we refine that criterion in order to be able to give an explicit sufficient condition for a non-redundant decomposition of a tensor to be minimal and unique. Read More

Let $k$ be an algebraically closed field of any characteristic. We apply the Hamburger-Noether process of successive quadratic transformations to show the equivalence of two definitions of the {\L}ojasiewicz exponent $\mathfrak{L}(\mathfrak{a})$ of an ideal $\mathfrak{a}\subset k[[x,y]]$. Read More

In this article, we discuss the rationality of the Betti Series of the universal module of nth order derivations of R_{m} where m is a maximal ideal of R. We proved that if R is a coordinate ring of an affine irreducible curve and if it has at most one singularity point, then the Betti series is rational. Read More

The present paper deals with permutations induced by tame automorphisms over finite fields. The first main result is a formula for determining the sign of the permutation induced by a given elementary automorphism over a finite field. The second main result is a formula for determining the sign of the permutation induced by a given affine automorphism over a finite field. Read More

Let $R$ be the homogeneous coordinate ring of the Grassmannian $\mathbb{G}=\operatorname{Gr}(2,n)$ over an algebraically closed field of characteristic $p>0$. In this paper we give a completely characteristic free description of the decomposition of $R$, considered as a graded $R^p$-module, into indecomposables ("Frobenius summands"). As a corollary we obtain a similar decomposition for the Frobenius pushforward of the structure sheaf of $\mathbb{G}$ and we obtain in particular that this pushforward is almost never a tilting bundle. Read More

We build a new bridge between geometric group theory and commutative algebra by showing that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley--Reisner ring of its nerve. Using this connection and techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo--Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily many steps. Read More

Let R denote a commutative Noetherian ring and let I be an ideal of R such that H_i^I(R) = 0, for all integers i greater than or equal to 2. In this paper we shall prove some results concerning the homological properties of I. Read More

We prove that any finite-degree polynomial functor is topologically Noetherian. This theorem is motivated by the recent resolution of Stillman's conjecture and a recent Noetherianity proof for the space of cubics. Via forthcoming work by others, our theorem and Stillman's conjecture together imply boundedness of a wide class of invariants of ideals in polynomial rings with a fixed number of generators of prescribed degrees. Read More

In 2012, Ananthnarayan, Avramov and Moore gave a new construction of Gorenstein rings from two Gorenstein local rings, called their connected sum. In this article, we investigate conditions on the associated graded ring of a Gorenstein Artin local ring Q, which force it to be a connected sum over its residue field. In particular, we recover some results regarding short, and stretched, Gorenstein Artin rings. Read More

Let $V$ be a minimal valuation overring of an integral domain $D$ and let $\mathrm{Zar}(D)$ be the Zariski space of the valuation overrings of $D$. Starting from a result in the theory of semistar operations, we prove a criterion under which the set $\mathrm{Zar}(D)\setminus\{V\}$ is not compact. We then use it to prove that, in many cases, $\mathrm{Zar}(D)$ is not a Noetherian space, and apply it to the study of the spaces of Kronecker function rings and of Noetherian overrings. Read More

In this article, we present the basic definitions of modules and Lie semialgebras over semirings with a negation map. Our main example of a semiring with a negation map is ELT algebras, and some of the results in this article are formulated and proved only in the ELT theory. When dealing with modules, we focus on linearly independent sets and spanning sets. Read More

Working over an infinite field of positive characteristic, an upper bound is given for the nilpotency index of a finitely generated nil algebra of bounded nil index $n$ in terms of the maximal degree in a minimal homogenous generating system of the ring of simultaneous conjugation invariants of tuples of $n$ by $n$ matrices. This is deduced from a result of Zubkov. As a consequence, a recent degree bound due to Derksen and Makam for the generators of the ring of matrix invariants yields an upper bound for the nilpotency index of a finitely generated nil algebra that is polynomial in the number of generators and the nil index. Read More

We study the effects on $D$ of assuming that the power series ring $D[[X]]$ is a $v$-domain or a PVMD. We show that a PVMD $D$ is completely integrally closed if and only if $\bigcap_{n=1}^{\infty }(I^{n})_{v}=(0)$ for every proper $t$-invertible $t$-ideal $I$ of $D$. Using this, we show that if $D$ is an AGCD domain, then $D[[X]]$ is integrally closed if and only if $D$ is a completely integrally closed PVMD with torsion $t$-class group. Read More