Showing papers in "Journal of Commutative Algebra in 2017"

Powers of edge ideals of regularity three bipartite graphs

Abstract: In this paper, we prove that, if $I(G)$ is the edge ideal of a connected bipartite graph with regularity 3, then, for all $s\geq 2$, the regularity of $I(G)^s$ is exactly $2s+1$.

The realization problem for delta sets of numerical semigroups

Abstract: The delta set of a numerical semigroup $S$, denoted $\Delta (S)$, is a factorization invariant that measures the complexity of the sets of lengths of elements in~$S$. We study the following problem: Which finite sets occur as the delta set of a numerical semigroup $S$? It is known that $\min \Delta (S) = \gcd \Delta (S)$ is a necessary condition. For any two-element set $\{d,td\}$ we produce a semigroup~$S$ with this delta set. We then show that, for $t\ge 2$, the set $\{d,td\}$ occurs as the delta set of some numerical semigroup of embedding dimension~3 if and only if $t=2$.

Simple polyominoes are prime

Abstract: IntroductionPolyominoes are two dimensional objects which are originally rooted in recre-ational mathematics and combinatorics. They have been widely discussed in con-nection with tiling problems of the plane. Typically, a polyomino is plane figureobtained by joining squares of equal sizes, which are known as cells. In connectionwith commutative algebra, polyominoes were first discussed in [8] by assigning eachpolyomino the ideal of its inner 2-minors or the polyominoideal. The study of idealof t-minors of an m × n matrix is a classical subject in commutative algebra. Theclass of polyomino ideal widely generalizes the class of ideals of 2-minors of m × nmatrix as well as the ideal of inner 2-minors attached to a two-sided ladder.Let P be a polyomino and K be a field. We denote by I

Bi-amalgamated algebras along ideals

Abstract: Let $f: A\rightarrow B$ and $g: A\rightarrow C$ be two commutative ring homomorphisms, and let $J$ and $J'$ be two ideals of $B$ and $C$, respectively, such that $f^{-1}(J)=g^{-1}(J')$. The \textit {bi-amalgamation} of $A$ with $(B, C)$ along $(J, J')$ with respect to $(f,g)$ is the subring of $B\times C$ given by \[ A\bowtie ^{f,g}(J,J'):=\big \{(f(a)+j,g(a)+j') \mid a\in A, (j,j')\in J\times J'\big \}. \] In this paper, we investigate ring-theoretic properties of \textit {bi-amalgamations} and capitalize on previous work carried out on various settings of pullbacks and amalgamations. In the second and third sections, we provide examples of bi-amalgamations and show how these constructions arise as pullbacks. The fourth section investigates the transfer of some basic ring theoretic properties to bi-amalgamations, and the fifth section is devoted to the prime ideal structure of these constructions. All new results agree with recent studies in the literature on D'Anna, Finocchiaro and Fontana's amalgamations and duplications.

Some remarks on biequidimensionality of topological spaces and Noetherian schemes

Abstract: The space of Cohen–Macaulay curves is a compactification of the space of curves that are embedded in a given projective space Pn. The idea is similar to that of the Hilbert scheme but instead of adding degenerated curves, one considers only curves without embedded or isolated points. However, the curves need not be embedded into the projective space. Instead, they come with a finite morphism to Pn that is generically a closed immersion. More precisely, the space CM of Cohen–Macaulay curves parameterizes flat families of pairs where is a curve without embedded or isolated points and is a finite morphism that is an isomorphism onto its image away from finitely many closed points and such that has Hilbert polynomial p(t) with respect to the map .In Paper A we show that the moduli functor CM is represented by a proper algebraic space. This is done by constructing a smooth, surjective cover and by verifying the valuative criterion for properness.Paper B studies the moduli space CM in the particular case n = 3 and p(t) = 3t + 1, that is, the Cohen–Macaulay space of twisted cubics. We de- scribe the points of CM and show that they are in bijection with the points on the 12-dimensional component H0 of the Hilbert scheme of twisted cu- bics. Knowing the points of CM, we can then show that the moduli space is irreducible, smooth and has dimension 12.Paper C concerns the notion of biequidimensionality of topological spaces and Noetherian schemes. In EGA it is claimed that a topological space X is equidimensional, equicodimensional and catenary if and only if all maximal chains of irreducible closed subsets in X have the same length. We construct examples of topological spaces and Noetherian schemes showing that the sec- ond property is strictly stronger. This gives rise to two different notions of biequidimensionality, and we show how they relate to the dimension formula and the existence of a codimension function.

Arithmetical rank of strings and cycles

Abstract: Let $R$ be a polynomial ring over a field $K$. To a given squarefree monomial ideal $I \subset R$, one can associate a hypergraph $H(I)$. In this article, we prove that the arithmetical rank of $I$ is equal to the projective dimension of $R/I$ when $H(I)$ is a string or a cycle hypergraph.

Frobenius Betti numbers and modules of finite projective dimension

Abstract: Let $(R,\mathfrak{m} ,K)$ be a local ring, and let $M$ be an $R$-module of finite length. We study asymptotic invariants, $\beta ^F_i(M,R)$, defined by twisting with Frobenius the free resolution of $M$. This family of invariants includes the Hilbert-Kunz multiplicity ($e_{HK}(\mathfrak{m} ,R)=\beta ^F_0(K,R)$). We discuss several properties of these numbers that resemble the behavior of the Hilbert-Kunz multiplicity. Furthermore, we study when the vanishing of $\beta ^F_i(M,R)$ implies that $M$ has finite projective dimension. In particular, we give a complete characterization of the vanishing of $\beta ^F_i(M,R)$ for one-dimensional rings. As a consequence of our methods we give conditions for the non-existence of syzygies of finite length.

Star-reductions of ideals and Prufer v-multiplication domains

Abstract: Let $R$ be a commutative ring and $I$ an ideal of $R$. An ideal $J\subseteq I$ is a reduction of $I$ if $JI^{n}=I^{n+1}$ for some positive integer~$n$. The ring~$R$ has the (finite) basic ideal property if (finitely generated) ideals of $R$ do not have proper reductions. Hays characterized (one-dimensional) Pr\"ufer domains as domains with the finite basic ideal property (basic ideal property). We extend Hays's results to Pr\"ufer $v$-multiplication domains by replacing ``basic'' with ``$w$-basic,'' where $w$ is a particular star operation. We also investigate relations among $\star $-basic properties for certain star operations $\star $.

Lattice-ordered abelian groups finitely generated as semirings

Abstract: A lattice-ordered group (an $\ell $-group) $G(\oplus , \vee , \wedge )$ can naturally be viewed as a semiring $G(\vee ,\oplus )$. We give a full classification of (abelian) $\ell $-groups which are finitely generated as semirings by first showing that each such $\ell $-group has an order-unit so that we can use the results of Busaniche, Cabrer and Mundici~\cite {BCM}. Then, we carefully analyze their construction in our setting to obtain the classification in terms of certain $\ell $-groups associated to rooted trees (Theorem \ref {classify}). This classification result has a number of interesting applications; for example, it implies a classification of finitely generated ideal-simple (commutative) semirings $S(+, \cdot )$ with idempotent addition and provides important information concerning the structure of general finitely generated ideal-simple (commutative) semirings, useful in obtaining further progress towards Conjecture~\ref {main-conj} discussed in \cite {BHJK, JKK}.

On canonical modules of idealizations

Abstract: Let $(R,\mathfrak {m})$ be a Noetherian local ring which is a quotient of a Gorenstein local ring. Let $M$ be a finitely generated $R$-module. In this paper, we study the structure of the canonical module $K(R\ \mathbb {n}\ M)$ of the idealization $R\ \mathbb {n}\ M$ via the polynomial type introduced by Cuong~\cite {C}. In particular, we give a characterization for $K(R\ \mathbb {n}\ M)$ being Cohen-Macaulay and generalized Cohen-Macaulay.

Factoring ideals and stability in integral domains

Abstract: In an integral domain $R$, a nonzero ideal is called a \textit {weakly $ES$-stable ideal} if it can be factored into a product of an invertible ideal and an idempotent ideal of $R$; and $R$ is called a \textit {weakly $ES$-stable domain} if every nonzero ideal is a weakly $ES$-stable ideal This paper studies the notion of weakly $ES$-stability in various contexts of integral domains such as Noetherian and Mori domains, valuation and Pr\"ufer domains, pullbacks and more In particular, we establish strong connections between this notion and well-known stability conditions, namely, Lipman, Sally-Vasconcelos and Eakin-Sathaye stabilities

Graded version of local cohomology with respect to a pair of ideals

Abstract: In this paper, we prove some well-known results on local cohomology with respect to a pair of ideals in graded version, such as the Independence theorem, Lichtenbaum-Harshorne vanishing theorem, Basic finiteness and vanishing theorem, among others. In addition, we present a generalized version of the Melkersson theorem regarding the Artinianness of modules, and a result concerning Artinianness of local cohomology modules.

Direct summands of infinite-dimensional polynomial rings

Abstract: Let k be a field and R a pure subring of the infinite-dimensional polynomial ring k[X1;...]. If R is generated by monomials, then we show that the equality of height and grade holds for all ideals of R. Also, we show R satisfies the weak Bourbaki unmixed property. As an application, we give the Cohen-Macaulay property of the invariant ring of the action of a linearly reductive group acting by k-automorphism on k[X1;...]. This provides several examples of non-Noetherian Cohen-Macaulay rings (e.g. Veronese, determinantal and Grassmanian rings).

Boij-Söderberg and Veronese decompositions

Abstract: Boij-S\"oderberg theory has had a dramatic impact on commutative algebra. We determine explicit Boij-S\"oderberg coefficients for ideals with linear resolutions and illustrate how these arise from the usual Eliahou-Kervaire computations for Borel ideals. In addition, we explore a new numerical decomposition for resolutions based on a row-by-row approach; here, the coefficients of the Betti diagrams are not necessarily positive. Finally, we demonstrate how the Boij-S\"oderberg decomposition of an arbitrary homogeneous ideal with a pure resolution changes when multiplying the ideal by a homogeneous polynomial.

On the Loewy length of modules of finite projective dimension

Abstract: Let (A,m) be a local Gorenstein local ring and let M be an A module of finite length and finite projective dimension. We prove that the Lowey length of M is greater than or equal to order of A. This generalizes a result of Avramov, Buchweitz, Iyengar and Miller (2, 1.1).

Serre dimension and Euler class groups of overrings of polynomial rings

Abstract: Let $R$ be a commutative Noetherian ring of dimension~$d$ and \[ B=R[X_1,\ldots ,X_m,Y_1^{\pm 1},\ldots ,Y_n^{\pm 1}] \] a Laurent polynomial ring over $R$. If $A=B[Y,f^{-1}]$ for some $f\in R[Y]$, then we prove the following results: (i) if $f$ is a monic polynomial, then the Serre dimension of $A$ is $\leq d$. The case $n=0$ is due to Bhatwadekar, without the condition that $f$ is a monic polynomial. (ii) The $p$th Euler class group $E^p(A)$ of $A$, defined by Bhatwadekar and Sridharan, is trivial for $p\geq \max \{d+1,\dim A -p+3\}$. The case $m=n=0$ is due to Mandal and Parker.

Ideal class groups of monoid algebras

Abstract: Let $A\subset B$ be an extension of commutative reduced rings and $M\subset N$ an extension of positive commutative cancellative torsion-free monoids. We prove that $A$ is subintegrally closed in $B$ and $M$ is subintegrally closed in $N$ if and only if the group of invertible $A$-submodules of $B$ is isomorphic to the group of invertible $A[M]$-submodules of $B[N]$. In case $M=N$, we prove the same without the assumption that the ring extension is reduced.

Postulation and reduction vectors of multigraded filtrations of ideals

Abstract: We study the relationship between postulation and reduction vectors of admissible multigraded filtrations $\mathcal{F}= \{\mathcal{F} (\underline{n})\}_{\underline{n} \in \mathbb{Z} ^s}$ of ideals in Cohen-Macaulay local rings of dimension at most two. This is enabled by a suitable generalization of the Kirby-Mehran complex. An analysis of its homology leads to an analogue of Huneke's fundamental lemma which plays a crucial role in our investigations. We also clarify the relationship between the Cohen-Macaulay property of the multigraded Rees algebra of $\mathcal{F} $ and reduction vectors with respect to complete reductions of $\mathcal{F} $.

A structure theorem for most unions of complete intersections

Abstract: Using the connections among almost complete intersection schemes, arithmetically Gorenstein schemes and schemes that are a union of complete intersections, we give a structure theorem for the arithmetically Cohen-Macaulay union of two complete intersections of codimension~2, of type $(d_1,e_1)$ and $(d_2,e_2)$ such that $\min \{d_1,e_1\} e \min \{d_2,e_2\}$. We apply the results for computing Hilbert functions and graded Betti numbers for such schemes.

Quasi-gorensteinness of extended rees algebras

Abstract: Let $R$ be a Noetherian local ring and $I$ an $R$-ideal. It is well known that, if the associated graded ring ${gr} _I(R)$ is Cohen-Macaulay (Gorenstein), then so is $R$, but in general, the converse is not true. In this paper, we investigate the Cohen-Macaulayness and Gorensteinness of the associated graded ring ${gr} _I(R)$ under the hypothesis that the extended Rees algebra $R[It,t^{-1}]$ is quasi-Gorenstein or the associated graded ring ${gr} _I(R)$ is a domain.