Showing papers in "Journal of Commutative Algebra in 2020"
TL;DR: In this paper, a minimal free resolution of the ideal generated by all square-free monomials of a given degree is given, which relies upon and exhibits the natural action of the symmetric group on the syzygy modules.
Abstract: An explicit construction is given of a minimal free resolution of the ideal generated by all squarefree monomials of a given degree. The construction relies upon and exhibits the natural action of the symmetric group on the syzygy modules. The resolution is obtained over an arbitrary coefficient ring; in particular, it is characteristic free. Two applications are given: an equivariant resolution of De Concini–Procesi rings indexed by hook partitions, and a resolution of FI-modules.
27 citations
TL;DR: In this paper, a finite unordered sequence S =g1 ∙⋯ ∙gl of terms from G, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals 1G, the identity element of the group.
Abstract: Let G be a finite group. A finite unordered sequence S=g1 ∙⋯ ∙gl of terms from G, where repetition is allowed, is a product-one sequence if its terms can be ordered such that their product equals 1G, the identity element of the group. As usual, we consider sequences as elements of the free abelian monoid ℱ(G) with basis G, and we study the submonoid ℬ(G)⊂ℱ(G) of all product-one sequences. This is a finitely generated C-monoid, which is a Krull monoid if and only if G is abelian. In case of abelian groups, ℬ(G) is a well-studied object. In the present paper we focus on nonabelian groups, and we study the class semigroup and the arithmetic of ℬ(G).
19 citations
TL;DR: In this article, the elasticity of Puiseux monoids is studied in terms of the number of atoms and the ratio of their largest factorization length to their shortest factorization lengths.
Abstract: Let M be an atomic monoid and let x be a non-unit element of M. The elasticity of x, denoted by ρ(x), is the ratio of its largest factorization length to its shortest factorization length, and it measures how far x is from having all its factorizations of the same length. The elasticity ρ(M) of M is the supremum of the elasticities of all non-unit elements of M. In this paper, we study the elasticity of Puiseux monoids (i.e., additive submonoids of ℚ≥0). We characterize, in terms of the atoms, which Puiseux monoids M have finite elasticity, giving a formula for ρ(M) in this case. We also classify when ρ(M) is achieved by an element of M. When M is a primary Puiseux monoid (that is, a Puiseux monoid whose atoms have prime denominator), we describe the topology of the set of elasticities of M, including a characterization of when M is a bounded factorization monoid. Lastly, we give an example of a Puiseux monoid that is bifurcus (that is, every nonzero element has a factorization of length at most 2).
19 citations
TL;DR: In this paper, it was shown that in a commutative ring R with unity, for any n > 0, if I is an n -absorbing ideal of R, then ( I ) n ⊆ I.
Abstract: In this note, we show that in a commutative ring R with unity, for any n > 0 , if I is an n -absorbing ideal of R , then ( I ) n ⊆ I .
11 citations
TL;DR: In this paper, the following holds for all sufficiently large primes p: if ϕ and ψ are any splittings on 𝒳× Spec𝔽p, such that the restriction of α is compatible with ϕ, ψ, and V is any compatibly split subvariety of (𝒴×Spec&# 120125,p,ϕ), then the reduction α−1(V)red is a compatibly-split subvariate of &# 119988,× Spec
Abstract: Let 𝒳 and 𝒴 be schemes of finite type over Specℤ and let α:𝒴→𝒳 be a finite map. We show the following holds for all sufficiently large primes p: If ϕ and ψ are any splittings on 𝒳× Spec𝔽p and 𝒴× Spec𝔽p, such that the restriction of α is compatible with ϕ and ψ, and V is any compatibly split subvariety of (𝒳× Spec𝔽p,ϕ), then the reduction α−1(V)red is a compatibly split subvariety of (𝒴× Spec𝔽p,ψ). This is meant as a tool to aid in listing the compatibly split subvarieties of various classically split varieties.
7 citations
TL;DR: In this paper, the authors studied the sets of semistar and star operations on a semilocal Prufer domain, with an emphasis on which properties of the domain are enough to determine them.
Abstract: We study the sets of semistar and star operations on a semilocal Prufer domain, with an emphasis on which properties of the domain are enough to determine them. In particular, we show that these sets depend chiefly on the properties of the spectrum and of some localizations of the domain; we also show that, if the domain is h-local, the number of semistar operations grows as a polynomial in the number of semistar operations of its localizations.
6 citations
TL;DR: The existence of a general strongly stable submodule of a finitely generated graded free S -module, which preserves values and positions of the extremal Betti numbers of M, is proved in this article.
Abstract: Let S be a polynomial ring in n variables over a field K of any characteristic. Let M be a strongly stable submodule of a finitely generated graded free S -module F , with all basis elements of F of the same degree. The existence of a general strongly stable submodule M ˜ of a finitely generated graded free S -module F ˜ , rank F ˜ ≥ rank F , which preserves values and positions of the extremal Betti numbers of M , is proved.
6 citations
TL;DR: In this paper, the authors define an extension of the no broken circuit complex of a matroid and use it to give a direct Grobner basis argument that the polynomials extracted from the Hilbert series in these two ways agree.
Abstract: Given the complement of a hyperplane arrangement, let Γ be the closure of the graph of the map inverting each of its defining linear forms. The characteristic polynomial manifests itself in the Hilbert series of Γ in two different-seeming ways, one due to Terao and the other to Huh and Katz. We define an extension of the no broken circuit complex of a matroid and use it to give a direct Grobner basis argument that the polynomials extracted from the Hilbert series in these two ways agree.
5 citations
TL;DR: In this paper, the authors consider an integral domain, where R is a prime domain, and A, A(1), A(2),..., A s be overrings of R, where A s is of the form S-1 R, and S = R \ p1 boolean OR... boolean OR p(n) for for some prime ideals p(i), and whe...
Abstract: Let R be an integral domain, and let A, A(1), A(2), ..., A s be overrings of R, where A is of the form S-1 R, where S = R \ p1 boolean OR ... boolean OR p(n) for for some prime ideals p(i), and whe ...
4 citations
TL;DR: In this article, the authors examined the case of nonhomologically isolated singularities, providing many examples in which the Stanley-Reisner ring modulo two generic linear forms is a quasi-Buchsbaum but not Buchsbaum ring.
Abstract: We study properties of the Stanley–Reisner rings of simplicial complexes with isolated singularities modulo two generic linear forms. Miller, Novik, and Swartz proved that if a complex has homologically isolated singularities, then its Stanley–Reisner ring modulo one generic linear form is Buchsbaum. Here we examine the case of nonhomologically isolated singularities, providing many examples in which the Stanley–Reisner ring modulo two generic linear forms is a quasi-Buchsbaum but not Buchsbaum ring.
4 citations
TL;DR: Azarpanah, Namdari and Olfati as discussed by the authors showed that the sum of z-ideals in a subalgebra of C(X) is indeed a zA-ideal in A(X).
Abstract: Let X be a Tychonoff space and A(X) be an intermediate subalgebra of C(X), i.e., C∗(X)⊆A(X)⊆C(X). We show that such subrings are precisely absolutely convex subalgebras of C(X). An ideal I in A(X) is said to be a zA-ideal if Z(f)⊆Z(g), f∈I and g∈A(X) imply that g∈I. We observe that the coincidence of zA-ideals and z-ideals of A(X) is equivalent to the equality A(X)=C(X). This shows that every z-ideal in A(X) need not be a zA-ideal and this is a point which is not considered by D. Rudd in Theorem 4.1 of Michigan Math. J. 17 (1970), 139–141, or by G. Mason in Theorem 3.3 and Proposition 3.5 of Canad. Math. Bull. 23:4 (1980), 437-443. We rectify the induced misconceptions by showing that the sum of z-ideals in A(X) is indeed a z-ideal in A(X). Next, by studying the sum of z-ideals in subrings of the form I+ℝ of C(X), where I is an ideal in C(X), we investigate a wide class of examples of subrings of C(X) in which the sum of z-ideals need not be a z-ideal. It is observed that, for every ideal I in C(X), the sum of any two z-ideals in I+ℝ is a z-ideal in I+ℝ or all of I+ℝ if and only if X is an F-space. This result answers a question raised by Azarpanah, Namdari and Olfati in J. Commut. Algebra 11:4 (2019), 479–509.
TL;DR: In this paper, the Castelnuovo-Mumford regularity of the quotient S∕I(X) of a bipartite graph was shown to be invariant to the combinatorial structure of the graph, where S is the polynomial ring on the edges of G and I is the vanishing ideal of the projective toric subset parameterized by G.
Abstract: Let G be a graph obtained by taking r≥2 paths and identifying all first vertices and identifying all last vertices. We compute the Castelnuovo–Mumford regularity of the quotient S∕I(X), where S is the polynomial ring on the edges of G and I(X) is the vanishing ideal of the projective toric subset parameterized by G. This invariant is known for several special families of graphs such as trees, cycles, complete graphs and complete bipartite graphs. For bipartite graphs, it is also known that the computation of the regularity can be reduced to the 2-connected case. Thus, we focused on the first case of a bipartite graph where the regularity was unknown. We also prove new inequalities relating the Castelnuovo–Mumford regularity of S∕I(X) with the combinatorial structure of G, for a general graph.
TL;DR: In this article, the authors introduce a class of graded R -modules called Grobner-coherent modules, which are coherent as ungraded modules because they admit an adequate theory of Grobners bases.
Abstract: Let R be a graded ring. We introduce a class of graded R -modules called Grobner-coherent modules. Roughly, these are graded R -modules that are coherent as ungraded modules because they admit an adequate theory of Grobner bases. The class of Grobner-coherent modules is formally similar to the class of coherent modules: for instance, it is an abelian category closed under extension. However, Grobner-coherent modules come with tools for effective computation that are not present for coherent modules.
TL;DR: In this paper, a construction of a complex of finitely generated free R-modules having the same homology is given, where the original complex is an R-free resolution of a finitely-generated R-module over a commutative local Noetherian ring.
Abstract: Let (Q,𝔫,k) be a commutative local Noetherian ring, f1,…,fc a Q-regular sequence in 𝔫, and R=Q∕(f1,…,fc). Given a complex of finitely generated free R-modules, we give a construction of a complex of finitely generated free Q-modules having the same homology. A key application is when the original complex is an R-free resolution of a finitely generated R-module. In this case our construction is a sort of converse to a construction of Eisenbud and Shamash which yields a free resolution of an R-module M over R given one over Q.
TL;DR: In this paper, the authors use the concept of multiple convex body semigroup to characterize Buchsbaum circle and convex polygonal semigroups and describe algorithms to check such characterizations.
Abstract: In this work, using the concept of multiple convex body semigroup, we present new families of Buchsbaum semigroups. We characterize Buchsbaum circle and convex polygonal semigroups and we describe algorithmic methods to check such characterizations.
TL;DR: Support and rank varieties of modules over a group algebra of an elementary abelian p -group have been studied in this article, and it has been shown that the rank and support varieties are equivalent.
Abstract: Support and rank varieties of modules over a group algebra of an elementary abelian p -group have been well studied. In particular, Avrunin and Scott showed that in this setting, the rank and support varieties are equivalent. Avramov and Buchweitz proved an analogous result for pairs of modules over arbitrary commutative local complete intersection rings. In this paper we study support and rank varieties in the triangulated category of totally acyclic chain complexes over a complete intersection ring and show that these varieties are also equivalent.
TL;DR: In this article, it was shown that a module M over a Noetherian ring R of positive characteristic p has finite flat dimension if there exists an integer t ≥ 0 such that Tor i R (M, f e R ) = 0 for t ≤ i ≤ t + dim R and infinitely many e.
Abstract: It is proved that a module M over a Noetherian ring R of positive characteristic p has finite flat dimension if there exists an integer t ≥ 0 such that Tor i R ( M , f e R ) = 0 for t ≤ i ≤ t + dim R and infinitely many e . This extends results of Herzog, who proved it when M is finitely generated. It is also proved that when R is a Cohen–Macaulay local ring, it suffices that the Tor vanishing holds for one e ≥ log p e ( R ) , where e ( R ) is the multiplicity of R .
TL;DR: In this article, a subcomplex and a factor complex of the Cech complex are investigated for a finitely generated A-module M. The authors prove the Artinianness of these cohomology modules and characterize the last nonvanishing among them.
Abstract: Let 𝔮 denote an ideal of a local ring (A,𝔪). For a system of elements a¯=a1,…,at such that ai∈𝔮ci,i=1,…,t, and n∈ℤ we investigate a subcomplex and a factor complex of the Cech complex Ca¯⊗AM for a finitely generated A-module M. We start with the inspection of these cohomology modules that approximate in a certain sense the local cohomology modules Ha¯i(M) for all i∈ℕ. In the case of an 𝔪-primary ideal a¯A we prove the Artinianness of these cohomology modules and characterize the last nonvanishing among them.
TL;DR: In this paper, the authors investigated the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field and showed that the Betti number can be encoded by finitely many polynomials.
Abstract: Using the concept of vector partition functions, we investigate the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. Our main results state that if the polynomial ring is equipped with a positive ℤ d -grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials. Specially, in the case of ℤ -grading, for each homological degree i we can split ℤ 2 = { ( μ , t ) ∣ t , μ ∈ ℤ } in a finite number of regions such that for each region there is a polynomial in μ and t that computes dim k ( Tor i S ( I t , k ) μ ) . This refines, in a graded situation, the result of Kodiyalam on Betti numbers of powers of ideals. Our main statement treats the case of a power products of homogeneous ideals in a ℤ d -graded algebra, for a positive grading.
TL;DR: In this article, it was shown that any quadratic binomial complete intersection can have the set of square-free monomials as a vector space basis if the generators are put in a normal form.
Abstract: We compute the resultants for quadratic binomial complete intersections. As an application we show that any quadratic binomial complete intersection can have the set of square-free monomials as a vector space basis if the generators are put in a normal form.
TL;DR: In this article, it was shown that F can be given the structure of a differential graded algebra with divided powers, and furthermore, multiplication on F exhibits Poincare duality.
Abstract: Let P be a commutative Noetherian ring and F be a self-dual acyclic complex of finitely generated free P-modules. Assume that F has length four and F0 has rank one. We prove that F can be given the structure of a differential graded algebra with divided powers; furthermore, the multiplication on F exhibits Poincare duality. This result is already known if P is a local Gorenstein ring and F is a minimal resolution. The purpose of the present paper is to remove the unnecessary hypotheses that P is local, P is Gorenstein, and F is minimal.
TL;DR: In this article, the Hilbert functions of bigraded algebras in k[1,x2,y1,y2] were classified by introducing a numerical function called a Ferrers function.
Abstract: We classify the Hilbert functions of bigraded algebras in k[x1,x2,y1,y2] by introducing a numerical function called a Ferrers function.
TL;DR: In this paper, the authors examined under what conditions I is a surjective image of a projective R[T]-module P of rank n with determinant L, and they showed that I∕I2 is a SURJECTIVE image of L⊕R[T]n−1.
Abstract: Let R be an affine domain of dimension n≥3 over a field of characteristic 0. Let L be a projective R[T]-module of rank 1 and I⊂R[T] a local complete intersection ideal of height n. Assume that I∕I2 is a surjective image of L⊕R[T]n−1. This paper examines under what conditions I is a surjective image of a projective R[T]-module P of rank n with determinant L.
TL;DR: The p-radical closure R prc as discussed by the authors is the largest purely inseparable R -subalgebra inside the formal completion of a local noetherian ring whose formal completion is integral.
Abstract: Given a local noetherian ring R whose formal completion is integral, we introduce and study the p -radical closure R prc . Roughly speaking, this is the largest purely inseparable R -subalgebra inside the formal completion R . It turns out that the finitely generated intermediate rings R ⊂ A ⊂ R prc have rather peculiar properties. They can be used in a systematic way to provide examples of integral local rings whose normalization is nonfinite, that do not admit a resolution of singularities, and whose formal completion is nonreduced.
TL;DR: In this article, it was shown that the secant ideal in I 2, n { r } is always prime, thus giving a new class of prime initial ideals of the Pfaffian ideals.
Abstract: We explore the relationship between secant ideals and initial ideals of I 2 , n , the ideal of the Grassmannian, Gr ( 2 , ℂ n ) . The ( r − 1 ) -secant of I 2 , n is the ideal generated by the 2 r × 2 r subpfaffians of a generic n × n skew-symmetric matrix. It has been conjectured that for a weight vector ω in the tropical Grassmannian, the secant of the initial ideal of I 2 , n with respect to ω is equal to the initial ideal of the secant. We show that this conjecture is not true in general. Using the correspondence between weight vectors in the tropical Grassmannian and binary leaf-labeled trees, we also give necessary and sufficient conditions for the conjecture to hold in terms of the topology of the tree associated to ω . In the course of proving this result, we show that the ideal in ω ( I 2 , n { r } ) is always prime, thus giving a new class of prime initial ideals of the Pfaffian ideals.