Showing papers in "Journal of Commutative Algebra in 2022"
TL;DR: In this article , it was shown that if the small finitistic dimension of R is not zero, then it is not a DQ ring, and therefore not a Lucas module.
Abstract: Let R be a commutative ring with identity and let 𝒬 be the set of finitely generated semiregular ideals of R. A 𝒬-torsion-free R-module M is called a Lucas module if ExtR1(R∕J,M)=0 for any J∈𝒬. Moreover, R is called a DQ ring if every ideal of R is a Lucas module. We prove that if the small finitistic dimension of R is zero, then R is a DQ ring. In terms of a trivial extension, we construct a total ring of quotients of the type R=D∝H which is not a DQ ring. Thus in this case, the small finitistic dimension of R is not zero. This provides a negative answer to an open problem posed by Cahen et al.
8 citations
TL;DR: In this article , the set of possible Jordan types for a given Artinian algebra A or a finite A-module M is studied, where the Jordan type of a generic linear element ℓ in A1 is the partition giving the Jordan blocks of the multiplication map mℓ:M→M.
Abstract: There has been much work on strong and weak Lefschetz conditions for graded Artinian algebras A, especially those that are Artinian Gorenstein. A more general invariant of an Artinian algebra A or finite A-module M that we consider here is the set of Jordan types of elements of the maximal ideal 𝔪 of A, acting on M. Here, the Jordan type of ℓ∈𝔪A is the partition giving the Jordan blocks of the multiplication map mℓ:M→M. In particular, we consider the Jordan type of a generic linear element ℓ in A1, or in the case of a local ring 𝒜, that of a generic element ℓ∈𝔪𝒜, the maximum ideal. We often take M=A, the graded algebra, or M=𝒜 a local algebra. The strong Lefschetz property of an element, as well as the weak Lefschetz property can be expressed simply in terms of its Jordan type and the Hilbert function of M. However, there has not been until recently a systematic study of the set of possible Jordan types for a given Artinian algebra A or A-module M, except, importantly, in modular invariant theory, or in the study of commuting Jordan types. We first show some basic properties of the Jordan type. In a main result we show an inequality between the Jordan type of ℓ∈𝔪𝒜 and a certain local Hilbert function. In our last sections we give an overview of topics such as the Jordan types for Nagata idealizations, for modular tensor products, and for free extensions, including examples and some new results. We also propose open problems.
4 citations
TL;DR: In this article , the authors studied generalized power series that have a limited number of factorizations and gave several general results about (Laurent) power series rings, Laurent polynomial rings, and the (large) Halter-Koch rings.
Abstract: . Several past authors have studied questions related to unique factorization of generalized power series. Here we examine the broader topic of generalized power series that (in a sense we will make precise) have a limited number of factorizations. Special cases of our general results include new results about “limited factorization” in (Laurent) power series rings, (Laurent) polynomial rings, and the “large polynomial rings” of Halter-Koch. Along the way to our main results, we study Krull domains and Cohen-Kaplansky rings of generalized power series and give several slight extensions to the fundamental ring theory of generalized power series.
2 citations
TL;DR: A new class of abelian p-groups, the countably totally projective groups, was introduced in this paper , which contains the well-known class of totally-projective groups.
Abstract: A new class of abelian p-groups is introduced, the countably totally projective groups, that contains the well-known class of totally projective groups. A countably totally projective group is shown to have the property that every fully inert subgroup is commensurable with a fully invariant subgroup. This generalizes results of Goldsmith, Salce and Zanardo (2014), who proved that a direct sum of cyclic p-groups has this property. It also answers affirmatively two questions recently posed in the literature.
2 citations
TL;DR: In this article, it was shown that the conormal module of a power series ring over a hypersurface section is indecomposable in the Gorenstein case when the ideal is a hypersuran.
Abstract: For a licci ideal in a power series ring over a field, it is shown that its conormal module has a free summand precisely when the ideal is a hypersurface section. Using results of B. Ulrich, in the Gorenstein case one can show, up to deformation, that the conormal module is indecomposable.
1 citations
TL;DR: In this paper , the authors established a link between trace modules and rigidity in modules over Noetherian rings and identified classes of modules which must have self-extensions and used the theory of trace ideals to verify the Auslander-Reiten conjecture for syzygies of ideals over Artinian Gorenstein rings.
Abstract: We establish a link between trace modules and rigidity in modules over Noetherian rings. We identify classes of modules which must have self-extensions and use the theory of trace ideals to verify the Auslander-Reiten conjecture for syzygies of ideals over Artinian Gorenstein rings.
1 citations
TL;DR: In this article , the annihilators of positive Koszul homologies of a system of parameters of an almost complete intersection R were studied in terms of the acyclicity of certain (finite) residual approximation complexes whose zeroth homologies are the residue field of R.
Abstract: We propose a question on the annihilators of positive Koszul homologies of a system of parameters of an almost complete intersection R. The question can be stated in terms of the acyclicity of certain (finite) residual approximation complexes whose zeroth homologies are the residue field of R. We show that our question has an affirmative answer for the first Koszul homology of any almost complete intersection, as well as for all positive Koszul homologies of certain system of parameters which exist in some almost complete intersection rings with small multiplicities. The statement about the first Koszul homology is shown to be equivalent to the monomial conjecture and thus follows from its validity.
1 citations
TL;DR: In this article , Dutta et al. employed similar ideas as in [Alexandru, Popescu and Zaharescu 1990] to provide an estimate of IC(K(X)∣K,v) when (K,X) ∣K and v) is a valuation algebraic extension.
Abstract: This article is a natural continuation of our previous works [Dutta 2021] and [Dutta 2022]. In this article, we employ similar ideas as in [Alexandru, Popescu and Zaharescu 1990] to provide an estimate of IC(K(X)∣K,v) when (K(X)∣K,v) is a valuation algebraic extension. Our central result is an analogue of [Dutta 2022, Theorem 1.3]. We further provide a natural construction of a complete sequence of key polynomials for v over K in the setting of valuation algebraic extensions.
1 citations
TL;DR: In this article , the authors give a connection between complete sets of ABKPs for a valued field of arbitrary rank and an extension of the field to arbitrary rank, for a given field k, v, and an ABKP for k(k, v) = k(X) = v(X).
Abstract: In this paper, for a valued field $(K, v)$ of arbitrary rank and an extension $w$ of $v$ to $K(X),$ we give a connection between complete sets of ABKPs for $w$ and MacLane-Vaqui\'e chains of $w.$
1 citations
TL;DR: In this paper , the splitting of a graph is studied and compared with algebraic properties of the edge ideals of graphs and those of their splitting graphs, and the splitting graphs are compared.
Abstract: We introduce and study the concept which we call the splitting of a graph and compare algebraic properties of the edge ideals of graphs and those of their splitting graphs.
1 citations
TL;DR: In this paper , the authors give a resolution N of P¯∕𝔠P¯ by free P¯-modules, which is built from a differential graded algebra resolution of P∕(𝔦:f) by free p-modules.
Abstract: Let P be a commutative Noetherian ring, 𝔠 be an ideal of P which is generated by a regular sequence of length four, f be a regular element of P, and P¯ be the hypersurface ring P∕(f). Assume that 𝔠:f is a grade four Gorenstein ideal of P. We give a resolution N of P¯∕𝔠P¯ by free P¯-modules. The resolution N is built from a differential graded algebra resolution of P∕(𝔠:f) by free P-modules, together with one homotopy map. In particular, we give an explicit form for the matrix factorization which is the infinite tail of the resolution N.
TL;DR: In this paper , Keshari and Lokhande this paper showed that projective modules over a commutative Noetherian ring of dimension d and a projective cancellative torsion-free seminormal monoid are free.
Abstract: Let R be a commutative Noetherian ring of dimension d and M a commutative cancellative torsion-free seminormal monoid. Then: (1) Let A be a ring of type R[d,m,n] and P be a projective A[M]-module of rank r≥max {2,d+1}. Then the action of E(A[M]⊕P) on (A[M]⊕P) is transitive and (2) Assume (R,m,K) is a regular local ring containing a field k such that either char k=0 or char k=p and tr-deg K∕𝔽p≥1. Let A be a ring of type R[d,m,n]∗ and f∈R be a regular parameter. Then all finitely generated projective modules over A[M], A[M]f and A[M]⊗RR(T) are free. When M is free both results are due to Keshari and Lokhande (2014).
TL;DR: In this paper , a Noetherian ring of prime characteristic p, whose Frobenius morphism is locally finite but not finite, was constructed using an old example of Nagata.
Abstract: Using an old example of Nagata, we construct a Noetherian ring of prime characteristic p, whose Frobenius morphism is locally finite, but not finite.
TL;DR: In this article , it was shown that the divisor class group of any open Richardson variety in the Grassmannian is trivial, using Nagata's criterion, localizing the coordinate ring at a suitable set of Plücker coordinates.
Abstract: We prove that the divisor class group of any open Richardson variety in the Grassmannian is trivial. Our proof uses Nagata’s criterion, localizing the coordinate ring at a suitable set of Plücker coordinates. We prove that these Plücker coordinates are prime elements by showing that the subscheme they define is an open subscheme of a positroid variety. Our results hold over any field and over the integers.
TL;DR: In this paper , it was shown that the Castelnuovo-Mumford regularity of the intersection of t linear ideals is at most t. The same authors also showed that analogous results hold when we work over the exterior algebra ∧ (W ∗) over a field of characteristic 0.
Abstract: Given a collection of t subspaces in an ndimensional vector space W we can associate to them t linear ideals in the symmetric algebra S(W ∗). Conca and Herzog showed that the Castelnuovo-Mumford regularity of the product of t linear ideals is equal to t. Derksen and Sidman showed that the Castelnuovo-Mumford regularity of the intersection of t linear ideals is at most t. In this paper we show that analogous results hold when we work over the exterior algebra ∧ (W ∗) (over a field of characteristic 0). To prove these results we rely on the functoriality of equivariant free resolutions and construct a functor Ω from the category of polynomial functors to itself. The functor Ω transforms resolutions of polynomial functors associated to subspace arrangements over the symmetric algebra to resolutions over the exterior algebra.
TL;DR: In this article , an interpolation result for homogeneous polynomials over the integers was proved for PIDs with finite residue fields, and an independent proof using elementary techniques.
Abstract: We prove an interpolation result for homogeneous polynomials over the integers, or more generally for PIDs with finite residue fields. Previous proofs of this result use the well-known but nontrivial fact that class groups of rings of integers are torsion. We provide an independent proof using elementary techniques.
TL;DR: In this article , direct sum decompositions of pure projective torsion free modules over one-dimensional commutative noetherian domains are studied, and a satisfactory criterion is given for analytically unramified reduced local rings and for Bass domains.
Abstract: We study direct sum decompositions of pure projective torsion free modules over one-dimensional commutative noetherian domains. Having an inspiration in the representation theory of orders in separable algebras we study when every pure projective torsion free module is a direct sum of finitely generated modules. A satisfactory criterion is given for analytically unramified reduced local rings and for Bass domains.
TL;DR: The Waring rank of the generic d×d determinant is bounded by d⋅d in this paper , which is the best known upper bound for the determinant.
Abstract: The Waring rank of the generic d×d determinant is bounded above by d⋅d!. This improves previous upper bounds, which were of the form an exponential times the factorial. Our upper bound comes from an explicit power sum decomposition. We describe some of the symmetries of the decomposition and set-theoretic defining equations for the terms of the decomposition.
TL;DR: In this paper , it was shown that the atomic density of the polynomial ring is zero for any finite field ǫ ≥ 0 for any numerical semigroup S. The atomic density is a measure of the proportion of irreducible elements in a given ring or semigroup, for semigroup algebras.
Abstract: A numerical semigroup S is a cofinite, additively closed subset of the nonnegative integers that contains 0. We initiate the study of atomic density, an asymptotic measure of the proportion of irreducible elements in a given ring or semigroup, for semigroup algebras. It is known that the atomic density of the polynomial ring 𝔽q[x] is zero for any finite field 𝔽q; we prove that the numerical semigroup algebra 𝔽q[S] also has atomic density zero for any numerical semigroup S. We also examine the particular algebra 𝔽2[x2,x3] in more detail, providing a bound on the rate of convergence of the atomic density as well as a counting formula for irreducible polynomials using Möbius inversion, comparable to the formula for irreducible polynomials over a finite field 𝔽q.
TL;DR: In this paper , a criterion of Cohen-Macaulayness of the form module GM(𝔮) in terms of (non)vanishing of a variation of local cohomology introduced by P. Schenzel and the author (J. Commut.
Abstract: Let 𝔮 be an ideal of a Noetherian local ring (A,𝔪) and M a nonzero finitely generated A-module. We present a criterion of Cohen–Macaulayness of the form module GM(𝔮) in terms of (non)vanishing of a variation of local cohomology introduced by P. Schenzel and the author (J. Commut. Algebra 12:3 (2020), 353-370).
TL;DR: The theory of mixed multiplicities of (not necessarily Noetherian) filtrations of mR-primary ideals in a noetherian local ring R has been developed by Cutkosky, Sarkar and Srinivasan as discussed by the authors .
Abstract: The theory of mixed multiplicities of (not necessarily Noetherian) filtrations of mR-primary ideals in a Noetherian local ring R, has been developed by Cutkosky, Sarkar and Srinivasan. The objective of this article is to generalise a Minkowski-type inequality given in their paper. We also recover a result of Cutkosky, Srinivasan and Verma as a simple consequence of our inequality.
TL;DR: In this paper , a 0-dimensional affine K-algebra R = K[x1,…,xn]∕I is considered, where I is an ideal in a polynomial ring K[1,..,x n] over a field K, and R is a complete intersection at a maximal ideal.
Abstract: Given a 0-dimensional affine K-algebra R=K[x1,…,xn]∕I, where I is an ideal in a polynomial ring K[x1,…,xn] over a field K, or, equivalently, given a 0-dimensional affine scheme, we construct effective algorithms for checking whether R is a complete intersection at a maximal ideal, whether R is locally a complete intersection, and whether R is a strict complete intersection. These algorithms are based on Wiebe’s characterization of 0-dimensional local complete intersections via the 0-th Fitting ideal of the maximal ideal. They allow us to detect which generators of I form a regular sequence resp. a strict regular sequence, and they work over an arbitrary base field K. Using degree filtered border bases, we can detect strict complete intersections in certain families of 0-dimensional ideals.
TL;DR: In this paper , the authors provide a direct computation of the $F$-pure threshold of degree four homogeneous polynomials in two variables and, more generally, of certain homogenous polynomorphisms with four distinct roots.
Abstract: We provide a direct computation of the $F$-pure threshold of degree four homogeneous polynomial in two variables and, more generally, of certain homogeneous polynomials with four distinct roots. The computation depends on whether the cross ratio of the roots satisfies a specific M\"{o}bius transformation of a Legendre polynomial. We then make a connection between a long lasting open question, involving the relationship between the $F$-pure and the log canonical threshold, and roots of Legendre polynomials over $\mathbb{F}_p$.
TL;DR: In this paper , the authors provided a formula to compute the big Cohen-Macaulay test ideal for triples ((R, Δ),𝔞t) where R is a mixed characteristic toric ring and ǫ is a monomial ideal.
Abstract: We provide a formula to compute the big Cohen–Macaulay test ideal for triples ((R,Δ),𝔞t) where R is a mixed characteristic toric ring and 𝔞 is a monomial ideal. Of particular interest is that this result is consistent with the formulas for test ideals in positive characteristic and multiplier ideals in characteristic zero.
TL;DR: In this paper , the authors studied grade 3 homogeneous ideals I⊆R defining compressed rings with socle k(−s)⊕k(−2s+1), where s ≥ 3 is some integer.
Abstract: Let R=k[x,y,z] be a standard graded 3-variable polynomial ring, where k denotes any field. We study grade 3 homogeneous ideals I⊆R defining compressed rings with socle k(−s)⊕k(−2s+1), where s≥3 is some integer. We prove that all such ideals are obtained by a trimming process introduced by Christensen, Veliche, and Weyman (J. Commut. Algebra 11:3 (2019), 325–339). We also construct a general resolution for all such ideals which is minimal in sufficiently generic cases. Using this resolution, we give bounds on the minimal number of generators μ(I) of I depending only on s; moreover, we show these bounds are sharp by constructing ideals attaining the upper and lower bounds for all s≥3. Finally, we study the Tor-algebra structure of R∕I. It is shown that these rings have Tor algebra class G(r) for s≤r≤2s−1. Furthermore, we produce ideals I for all s≥3 and all r with s≤r≤2s−1 such that Soc(R∕I)=k(−s)⊕k(−2s+1) and R∕I has Tor-algebra class G(r), partially answering a question of realizability posed by Avramov (J. Pure Appl. Algebra 216:11 (2012), 2489–2506).
TL;DR: In this article , the authors use the theory of D-modules to deduce information about the defining equations of the Rees algebra of I. They prove that the whole bigraded structure of ǫ is characterized by the integral roots of certain b-functions, and that certain de Rham cohomology groups can give partial information about 𝒦.
Abstract: Let I⊂R=𝔽[x1,x2] be a height two ideal minimally generated by three homogeneous polynomials of the same degree d, where 𝔽 is a field of characteristic zero. We use the theory of D-modules to deduce information about the defining equations of the Rees algebra of I. Let 𝒦 be the kernel of the canonical map α: Sym(I)→ Rees(I) from the symmetric algebra of I onto the Rees algebra of I. We prove that 𝒦 can be described as the solution set of a system of differential equations, that the whole bigraded structure of 𝒦 is characterized by the integral roots of certain b-functions, and that certain de Rham cohomology groups can give partial information about 𝒦.
TL;DR: In this article , it was shown that positive integer sequences of positive integers can be realized as divisor sequences of irreducible elements in Krull monoids, which gives a means for studying nonunique direct-sum decompositions of modules over local Noetherian rings for which the Krull-Remak-Schmidt property fails.
Abstract: The divisor sequence of an irreducible element (atom) a of a reduced monoid H is the sequence (sn)n∈ℕ where, for each positive integer n, sn denotes the number of distinct irreducible divisors of an. We investigate which sequences of positive integers can be realized as divisor sequences of irreducible elements in Krull monoids. In particular, this gives a means for studying nonunique direct-sum decompositions of modules over local Noetherian rings for which the Krull–Remak–Schmidt property fails.
TL;DR: In this paper , the authors consider homogeneously licci ideals in a polynomial ring and focus on the degrees of the forms generating the regular sequences, using a sequentially bounded condition on these degrees, E. Chong discovered a large class of IL ideals satisfying the Eisenbud-Green-Harris conjecture.
Abstract: We consider homogeneously licci ideals in a polynomial ring and focus on the degrees of the forms generating the regular sequences. Using a sequentially bounded condition on these degrees, E. Chong discovered a large class of licci ideals satisfying the Eisenbud–Green–Harris conjecture (among them, grade 3 Gorenstein ideals). He raised the question of whether these sequentially bounded links were possible for all homogeneously licci ideals. We answer his question in the negative, and in doing so answer a question of C. Huneke and B. Ulrich about strongly licci ideals. The structure of certain Betti tables plays a central role in our proof.
TL;DR: In this paper , the local Bertini theorem for weak normality and seminormality in mixed characteristic Noetherian rings has been established under a certain condition, where the normality is defined in terms of the number of semesters in the ring.
Abstract: We study certain properties of Noetherian rings with weak normality and seminormality in mixed characteristic. It is known that the two concepts can differ in the equal prime characteristic case, while they coincide by definition in the equal characteristic zero case. We exhibit some examples in the mixed characteristic case. We also establish the local Bertini theorem for weak normality in mixed characteristic under a certain condition.
TL;DR: In this paper , the authors investigated the Jordan-Hölder property for ring extensions of unital rings and showed that many types of ring extensions are catenarian, and they reduced the characterization of such ring extensions to the case of field extensions.
Abstract: If R⊆S is a ring extension of commutative unital rings, the poset [R,S] of R-subalgebras of S is called catenarian if it verifies the Jordan–Hölder property. This property has already been studied by Dobbs and Shapiro for finite extensions of fields. We investigate this property for arbitrary ring extensions, showing that many types of extensions are catenarian. We reduce the characterization of catenarian extensions to the case of field extensions, an unsolved question at this time.