Abstract: Publication date: Available online 3 April 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Dirceu Bagio, Antonio Paques, Héctor Pinedo

Abstract: Publication date: Available online 1 April 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Pau Enrique Moliner, Chris Heunen, Sean Tull

Abstract: Publication date: Available online 31 March 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Eduardo Martinez-Pedroza, Luis Jorge Sánchez Saldaña

Abstract: Publication date: Available online 31 March 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Marino Gran, Diana Rodelo, Idriss Tchoffo Nguefeu

Abstract: Publication date: Available online 12 March 2020Source: Journal of Pure and Applied AlgebraAuthor(s): S.N. Hosseini, A.R. Shir Ali Nasab, W. Tholen

Abstract: Publication date: Available online 2 March 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Momonari Kudo, Shushi Harashita, Hayato Senda

Abstract: Publication date: Available online 24 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Huanchen Bao, Weiqiang Wang, Hideya Watanabe

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Abstract: Publication date: Available online 17 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Maria Virginia Catalisano, Elena Guardo, Yong-Su Shin We find the Waldschmidt constant of special configurations, such as star configurations in Pn and standard k-configurations in P2 of type (1,2,…,s) with 3≤s≤6. We also find the resurgence for those configurations as an application.

Abstract: Publication date: Available online 17 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Manfred Dugas, Daniel Herden, Jack Rebrovich R. D. Sorkin showed in [8] how to recover a finite poset (P,≤) by algebraic means from its incidence algebra I(P). We generalize this result to finitary incidence algebras FI(P) and arbitrary posets (P,≤).

Abstract: Publication date: Available online 13 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Takao Satoh We consider a certain subgroup An+ of the automorphism group of a free group of rank n. It can be regarded as a free group analogue of the group Λn of integral lower-triangular matrices. We call An+ the lower-triangular automorphism group of a free group. The first aim of the paper is to give a finite presentation for An+.The abelianization of the free group induces the surjective homomorphism ρ+ from An+ to Λn. In our previous paper [19], we introduced the lower-triangular IA-automorphism group IAn+. Here we show that IAn+ coincides with the kernel of ρ+. The second aim of the paper is to give an infinite presentation for IAn+.Finally, we study a relation of the second homology groups between An+ and Λn. In particular, we compute the second homology group H2(Λn,L) by using Magnus's presentation where L is a principal ideal domain in which 2 is invertible. For example, L=Q,Z/pZ for prime p≥3. This gives a lower bound on the integral second homology group of An+.

Abstract: Publication date: Available online 13 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): ZhiHao Bian, Mingqiang Liu The classification of little q-Schur algebras of finite representation type at the odd roots of unity case was given in [15]. In this paper, we give the complete classification of representation type of little q-Schur algebras in the any roots of unity case.

Abstract: Publication date: Available online 11 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Eugenio Giannelli, Noelia Rizo, A.A. Schaeffer Fry We prove a variation of Thompson's Theorem. Namely, if the first column of the character table of a finite group G contains only two distinct values not divisible by a given prime number p>3, then Opp′pp′(G)=1. This is done by using the classification of finite simple groups.

Abstract: Publication date: Available online 11 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Ivan Kaygorodov, Mykola Khrypchenko, Samuel A. Lopes We give algebraic and geometric classifications of 6-dimensional complex nilpotent anticommutative algebras. Specifically, we find that, up to isomorphism, there are 14 one-parameter families of 6-dimensional nilpotent anticommutative algebras, complemented by 130 additional isomorphism classes. The corresponding geometric variety is irreducible and determined by the Zariski closure of a one-parameter family of algebras. In particular, there are no rigid 6-dimensional complex nilpotent anticommutative algebras.

Abstract: Publication date: Available online 11 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Hiroyuki Minamoto, Kota Yamaura This paper is the first one of a series of papers in which we study a finitely graded Iwanaga-Gorenstein (IG) algebras A=⨁i=0ℓAi and their representation theory. Every finitely graded algebra is graded Morita equivalent to a trivial extension algebra A=Λ⊕C with the grading degΛ=0,degC=1. Hence, in principle, representation theoretic study of finitely graded algebras is reduced to that of trivial extension algebras. The main result of this paper is a necessary and sufficient condition that a trivial extension algebra A=Λ⊕C to be an IG-algebra. We also give a description of the kernel Ker ϖ of a canonical functor ϖ:Db(modΛ)→SingZA. These results play important roles in the subsequent works.To this end, we study complexes of graded projective (injective) A-modules and establish formulas for the projective (injective) dimension of a graded A-modules in terms of Λ and C. As a byproducts, we obtain a modern expression of the classical formula for the global dimension of a trivial extension algebra by Palmer-Roos and Löfwall that is written in complicated classical derived functors.

Abstract: Publication date: Available online 11 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Johannes Cuno, Gerald Williams We consider groups defined by non-empty balanced presentations with the property that each relator is of the form R(x,y), where x and y are distinct generators and is determined by some fixed cyclically reduced word R(a,b) that involves both a and b. To every such presentation we associate a directed graph whose vertices correspond to the generators and whose arcs correspond to the relators. Under the hypothesis that the girth of the underlying undirected graph is at least 4, we show that the resulting groups are non-trivial and cannot be finite of rank 3 or higher. Without the hypothesis on the girth it is well known that both the trivial group and finite groups of rank 3 can arise.

Abstract: Publication date: Available online 11 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Jordan McMahon Over a finite-dimensonal algbera A, simple A-modules that have projective dimension one have special properties. For example, Geigle-Lenzing studied them in connection to homological epimorphisms of rings and they have also appeared in work concerning the finitistic dimension conjecture. If we however work in a d-cluster-tilting subcategory, then not all simples are contained in the subcategory. In this context, a replacement might be to work with idempotents instead, and utilise the theory of Auslander-Platzeck-Todorov. We introduce the notion of a fabric idempotent as an analogue of the localisable objects studied by Chen-Krause, and show that they provide rich combinatorial properties to illustrate the theory. In particular, we use fabric idempotents to extend the classification of singularity categories of Nakayama algebras by Chen-Ye to higher Nakayama algebras.

Abstract: Publication date: Available online 11 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Luca Scala Let X be a smooth quasi-projective algebraic surface and let Δn the big diagonal in the product variety Xn. We study cohomological properties of the ideal sheaves IΔnk and their invariants (IΔnk)Sn by the symmetric group, seen as ideal sheaves over the symmetric variety SnX. In particular we obtain resolutions of the sheaves of invariants (IΔn)Sn for n=3,4 in terms of invariants of sheaves over Xn whose cohomology is easy to calculate. Moreover, we relate, via the Bridgeland-King-Reid equivalence, powers of determinant line bundles over the Hilbert scheme to powers of ideals of the big diagonal Δn. We deduce applications to the cohomology of double powers of determinant line bundles over the Hilbert scheme with 3 and 4 points and we give universal formulas for their Euler-Poincaré characteristic. Finally, we obtain upper bounds for the regularity of the sheaves IΔnk over Xn with respect to very ample line bundles of the form L⊠⋯⊠L and of their sheaves of invariants (IΔnk)Sn on the symmetric variety SnX with respect to very ample line bundles of the form DL.

Abstract: Publication date: Available online 11 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Daniel Irving Bernstein, Cameron Farnsworth, Jose Israel Rodriguez A finite unit norm tight frame is a collection of r vectors in Rn that generalizes the notion of orthonormal bases. The affine finite unit norm tight frame variety is the Zariski closure of the set of finite unit norm tight frames. Determining the fiber of a projection of this variety onto a set of coordinates is called the algebraic finite unit norm tight frame completion problem. Our techniques involve the algebraic matroid of an algebraic variety, which encodes the dimensions of fibers of coordinate projections. This work characterizes the bases of the algebraic matroid underlying the variety of finite unit norm tight frames in R3. Partial results towards similar characterizations for finite unit norm tight frames in Rn with n≥4 are also given. We provide a method to bound the degree of the projections based off of combinatorial data.

Abstract: Publication date: Available online 10 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Peter F. Faul, Graham R. Manuell Artin glueings provide a way to reconstruct a frame from a closed sublocale and its open complement. We show that Artin glueings can be described as the weakly Schreier split extensions in the category of frames with finite-meet preserving maps. These extensions correspond to meet-semilattice homomorphisms between frames, yielding an extension bifunctor. Finally, we discuss Baer sums, the induced order structure on extensions and the failure of the split short five lemma.

Abstract: Publication date: Available online 10 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Yeongrak Kim, Frank-Olaf Schreyer In this paper, we compute an explicit matrix factorization of a rank 9 Ulrich sheaf on a general cubic hypersurface of dimension at most 7, whose existence was proved by Manivel. Instead of using invariant theory, we use Shamash's construction with a cone over the spinor variety. We also describe an algebro-geometric interpretation of our matrix factorization which connects the spinor tenfold and the Cartan cubic.

Abstract: Publication date: Available online 10 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): María-Cruz Fernández-Fernández We prove that the space of Gevrey solutions of an A–hypergeometric system along a coordinate subspace is contained in a space of formal Nilsson solutions. Moreover, under some additional condition, both spaces are equal. In the process we prove some other results about formal Nilsson solutions.

Abstract: Publication date: Available online 10 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Łucja Farnik, Krishna Hanumanthu, Jack Huizenga, David Schmitz, Tomasz Szemberg Let X be a projective surface and let L be an ample line bundle on X. The global Seshadri constant ε(L) of L is defined as the infimum of Seshadri constants ε(L,x) as x∈X varies. It is an interesting question to ask if ε(L) is a rational number for any pair (X,L). We study this question when X is a blow up of P2 at r≥0 very general points and L is an ample line bundle on X. For each r we define a submaximality threshold which governs the rationality or irrationality of ε(L). We state a conjecture which strengthens the SHGH Conjecture and assuming that this conjecture is true we determine the submaximality threshold.

Abstract: Publication date: Available online 10 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Thomas Hudson, Dennis Peters In this paper we express the class of the structure sheaves of the closures of Deligne–Lusztig varieties as explicit double Grothendieck polynomials in the first Chern classes of appropriate line bundles on the ambient flag variety. This is achieved by viewing such closures as degeneracy loci of morphisms of vector bundles.

Abstract: Publication date: Available online 7 February 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Michele Bolognesi, Claudio Pedrini In this note we introduce the transcendental part t(X) of the motive of a cubic fourfold X and prove that it is isomorphic to the (twisted) transcendental part h2tr(F(X)) in a suitable Chow-Künneth decomposition for the motive of the Fano variety of lines F(X). Then we prove that t(X) is isomorphic to the Prym motive associated to the surface Sl⊂F(X) of lines meeting a general line l. If X is very general (non special), then we prove that t(X) can not be isomorphic to t2(S) for any smooth projective surface, where t2(S) is the transcendental motive. We relate the existence of an isomorphism t(X)≃t2(S)(1), with S a K3 surface, to conjectures by Hassett and Kuznetsov on the rationality of a special cubic fourfold. Finally we consider the case of cubic fourfolds X admitting a certain projective involoution, and prove for this specific class the isomorphism t2(S)(1)≃t(X), with S a K3 surface.

Abstract: Publication date: Available online 21 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): H. Meng, A. Ballester-Bolinches Assume that A and G are finite groups of coprime orders such that A acts on G via automorphisms. Let p be a prime. The following coprime action version of a well-known theorem of Itô about the structure of a minimal non-p-nilpotent groups is proved: if every maximal A-invariant subgroup of G is p-nilpotent, then G is p-soluble. If, moreover, G is not p-nilpotent, then G must be soluble. Some earlier results about coprime action are consequences of this theorem.

Abstract: Publication date: Available online 11 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Wolfgang Rump The structure group GX of a non-degenerate involutive set-theoretic solution X to the Yang-Baxter equation was introduced by Etingof et al. (Duke Math. J., 1999). We prove that GX coincides with the structure group of an L-algebra, a quantum structure that arises, e. g., in the context of Artin groups and operator algebras. The connection unifies the concept of structure group and exhibits new operations associated to set-theoretic solutions X.

Abstract: Publication date: Available online 10 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): A.V. Jayanthan, Rajiv Kumar In this article, we prove that for several classes of graphs, the Castelnuovo-Mumford regularity of symbolic powers of their edge ideals coincide with that of their ordinary powers.

Abstract: Publication date: Available online 9 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): F. Ávila, G. Bezhanishvili, P.J. Morandi, A. Zaldívar For a frame L, let XL be the Esakia space of L. We identify a special subset YL of XL consisting of nuclear points of XL, and prove the following results:•L is spatial iff YL is dense in XL.•If L is spatial, then N(L) is spatial iff YL is weakly scattered.•If L is spatial, then N(L) is boolean iff YL is scattered. As a consequence, we derive the well-known results of Beazer and Macnab [1], Simmons [22], Niefield and Rosenthal [13], and Isbell [10].

Abstract: Publication date: Available online 8 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): John Guaschi, Carolina de Miranda e Pereiro For an arbitrary semi-direct product, we give a general description of its lower central series and an estimation of its derived series. In the second part of the paper, we study these series for the full braid group Bn(M) and pure braid group Pn(M) of a compact surface M, orientable or non-orientable, the aim being to determine the values of n for which Bn(M) and Pn(M) are residually nilpotent or residually soluble. We first solve this problem in the case where M is the 2-torus. We then use the results of the first part of the paper to calculate explicitly the lower central series of Pn(K), where K is the Klein bottle. Finally, if M is a non-orientable, compact surface without boundary, we determine the values of n for which Bn(M) is residually nilpotent or residually soluble in the cases that were not already known in the literature.

Abstract: Publication date: Available online 7 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Toshinori Kobayashi, Justin Lyle, Ryo Takahashi We say that a Cohen–Macaulay local ring has finite CM+-representation type if there exist only finitely many isomorphism classes of indecomposable maximal Cohen–Macaulay modules that are not locally free on the punctured spectrum. In this paper, we consider finite CM+-representation type from various points of view, relating it with several conjectures on finite/countable Cohen–Macaulay representation type. We prove in dimension one that the Gorenstein local rings of finite CM+-representation type are exactly the local hypersurfaces of countable CM-representation type, that is, the hypersurfaces of type (A∞) and (D∞). We also discuss the closedness and dimension of the singular locus of a Cohen–Macaulay local ring of finite CM+-representation type.

Abstract: Publication date: Available online 7 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Patrick Wegener We call an element of a Coxeter group a parabolic quasi-Coxeter element if it has a reduced decomposition into a product of reflections that generate a parabolic subgroup. We show that for a parabolic quasi-Coxeter element in an affine Coxeter group the Hurwitz action on its set of reduced decompositions into a product of reflections is transitive.

Abstract: Publication date: Available online 7 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Cristóbal Gil Canto, Daniel Gonçalves We study relative Cohn path algebras, also known as Leavitt-Cohn path algebras, and we realize them as partial skew group rings. To do this we prove uniqueness theorems for relative Cohn path algebras. Furthermore, given any graph E we define E-relative branching systems and prove how they induce representations of the associated relative Cohn path algebra. We give necessary and sufficient conditions for faithfulness of the representations associated to E-relative branching systems. This improves previous results known to Leavitt path algebras of row-finite graphs with no sinks. To prove this last result we show first a version, for relative Cohn-path algebras, of the reduction theorem for Leavitt path algebras.

Abstract: Publication date: Available online 7 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Luigi Ferraro, Ellen Kirkman, W. Frank Moore, Robert Won Let H be a semisimple Hopf algebra acting on an Artin-Schelter regular algebra A, homogeneously, inner-faithfully, preserving the grading on A, and so that A is an H-module algebra. When the fixed subring AH is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that H is a reflection Hopf algebra for A. We show that each of the semisimple Hopf algebras H2n2 of Pansera, and A4m and B4m of Masuoka is a reflection Hopf algebra for an AS regular algebra of dimension 2 or 3.

Abstract: Publication date: Available online 7 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Harry Gulliver This is the second of two papers on the injective spectrum of a right noetherian ring. In [4], we defined the injective spectrum as a topological space associated to a ring (or, more generally, a Grothendieck category), which generalises the Zariski spectrum. We established some results about the topology and its links with Krull dimension, and computed a number of examples.In the present paper, which can largely be read independently of the first, we extend these results by defining a sheaf of rings on the injective spectrum and considering sheaves of modules over this structure sheaf and their relation to modules over the original ring. We then explore links with the spectrum of prime torsion theories developed by Golan [2] and use this torsion-theoretic viewpoint to prove further results about the topology.

Abstract: Publication date: Available online 7 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Niels Feld We generalize Rost's theory of cycle modules [20] using the Milnor-Witt K-theory instead of the classical Milnor K-theory. We obtain a (quadratic) setting to study general cycle complexes and their (co)homology groups. The standard constructions are developed: proper pushfoward, (essentially) smooth pullback, long exact sequences, spectral sequences and products, as well as the homotopy invariance property; in addition, Gysin morphisms for lci morphisms are constructed. We prove an adjunction theorem linking our theory to Rost's. This work extends Schmid's thesis [22].

Abstract: Publication date: Available online 7 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Rosa M. Miró-Roig, Quang Hoa Tran We study the weak Lefschetz property of a class of graded Artinian Gorenstein algebras of codimension three associated in a natural way to the Apéry set of a numerical semigroup generated by four natural numbers. We show that these algebras have the weak Lefschetz property whenever the initial degree of their defining ideal is small.

Abstract: Publication date: Available online 7 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Jeffrey Bergen, Piotr Grzeszczuk Let X be a set of noncommuting variables, and G={σx}x∈X, D={δx}x∈X be sequences of automorphisms and skew derivations of a ring R. It is proved that if the ring R is semiprime Goldie, then the free skew extension R[X;G,D] is semiprimitive.

Abstract: Publication date: Available online 7 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Emmanuel Briand, Samuel A. Lopes, Mercedes Rosas We investigate the combinatorics of the general formulas for the powers of the operator h∂d, where ∂ is a differential operator on an arbitrary noncommutative ring in which h is central. New formulas for the generalized Stirling numbers are obtained, as well as results on the divisibility by primes of the coefficients which occur in the normally ordered form of h∂d. All of the above applies to the theory of formal differential operator rings.

Abstract: Publication date: Available online 7 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Gianluca Occhetta, Luis E. Solá Conde In the framework of the problem of characterizing complete flag manifolds by their contractions, the complete flags of type F4 and G2 satisfy the property that any possible tower of Bott–Samelson varieties dominating them birationally deforms in a nontrivial moduli. In this paper we illustrate the fact that, at least in some cases, these deformations can be explained in terms of automorphisms of Schubert varieties, providing variations of certain isotropic structures on them. As a corollary, we provide a unified and completely algebraic proof of the characterization of complete flag manifolds in terms of their contractions.

Abstract: Publication date: Available online 7 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): M. Cárdenas, F.F. Lasheras, A. Quintero, R. Roy In this paper, we consider an equivalence relation within the class of finitely presented discrete groups attending to their asymptotic topology rather than their asymptotic geometry. More precisely, we say that two finitely presented groups G and H are “proper 2-equivalent” if there exist (equivalently, for all) finite 2-dimensional CW-complexes X and Y, with π1(X)≅G and π1(Y)≅H, so that their universal covers X˜ and Y˜ are proper 2-equivalent. It follows that this relation is coarser than the quasi-isometry relation. We point out that finitely presented groups which are 1-ended and semistable at infinity are classified, up to proper 2-equivalence, by their fundamental pro-group, and we study the behaviour of this relation with respect to some of the main constructions in combinatorial group theory. A (finer) similar equivalence relation may also be considered for groups of type Fn,n≥3, which captures more of the large-scale topology of the group. Finally, we pay special attention to the class of those groups G which admit a finite 2-dimensional CW-complex X with π1(X)≅G and whose universal cover X˜ has the proper homotopy type of a 3-manifold. We show that if such a group G is 1-ended and semistable at infinity then it is proper 2-equivalent to either Z×Z×Z, Z×Z or F2×Z (here, F2 is the free group on two generators). As it turns out, this applies in particular to any group G fitting as the middle term of a short exact sequence of infinite finitely presented groups, thus classifying such group extensions up to proper 2-equivalence.

Abstract: Publication date: Available online 6 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Juan García Escudero We make a correction to the Euler number of a quintic threefold considered in the paper Calabi–Yau threefolds with small Hodge numbers associated with a one-parameter family of polynomials and compute the Hodge numbers.

Abstract: Publication date: Available online 12 December 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Karim Johannes Becher Let E be a field of characteristic different from 2 which is the centre of a quaternion division algebra and which is not euclidean. Then there exists a biquaternion division algebra over the rational function field E(t) which does not contain any quaternion algebra defined over E. The proof is based on the study of Bezoutian forms developed in [1].

Abstract: Publication date: Available online 9 December 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Meng-Kiat Chuah, Rita Fioresi We study the equal rank real forms of affine non-twisted Kac-Moody Lie superalgebras by Cartan automorphisms and Vogan diagrams. We introduce admissible positive root systems and Hermitian real forms, then show that a real form has admissible positive root system if and only if it is Hermitian. As a result, we use the Vogan diagrams to classify the Hermitian real forms.

Abstract: Publication date: Available online 6 December 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Anuj Jakhar, Sudesh K. Khanduja Let K=Q(θ) be an algebraic number field with θ in the ring AK of algebraic integers of K having minimal polynomial f(x) over Q. For a prime number p, let ip(f) denote the highest power of p dividing the index [AK:Z[θ]]. Let f¯(x)=ϕ¯1(x)e1⋯ϕ¯r(x)er be the factorization of f(x) modulo p into a product of powers of distinct irreducible polynomials over Z/pZ with ϕi(x)∈Z[x] monic. Let the integer l≥1 and the polynomial N(x)∈Z[x] be defined by f(x)=∏i=1rϕi(x)ei+plN(x),N‾(x)≠0¯. In this paper, we prove that ip(f)≥∑i=1ruidegϕi(x), where

Abstract: Publication date: Available online 4 December 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Juan Elias, Roser Homs, Bernard Mourrain We analyze and present an effective solution to the minimal Gorenstein cover problem: given a local Artin k-algebra A=k〚x1,…,xn〛/I, compute an Artin Gorenstein k-algebra G=k〚x1,…,xn〛/J such that ℓ(G)−ℓ(A) is minimal. We approach the problem by using Macaulay's inverse systems and a modification of the integration method for inverse systems to compute Gorenstein covers. We propose new characterizations of the minimal Gorenstein cover and present a new algorithm for the effective computation of the variety of all minimal Gorenstein covers of A for low Gorenstein colength. Experimentation illustrates the practical behavior of the method.

Abstract: Publication date: Available online 2 December 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Sagnik Chakraborty, Rajendra V. Gurjar, Dibyendu Mondal In this paper, we prove some results about inclusions of complete (or analytic) local rings. The main result says that if A⊆B are equicharacteristic zero complete (or analytic) local domains with the same uncountable residue field, such that the map Spec B→Spec A is surjective, then the integral closure of A in B is a finite A-module. In particular, if A⊆B is a pure extension of complete (or analytic) local domains, then the integral closure of A in B is a finite A-module.

Abstract: Publication date: Available online 30 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Diego Lobos Maturana, Steen Ryom-Hansen We give a concrete construction of a graded cellular basis for the generalized blob algebra Bn introduced by Martin and Woodcock. The construction uses the isomorphism between KLR-algebras and cyclotomic Hecke algebras, proved by Brundan-Kleshchev and Rouquier. It gives rise to a family of Jucys-Murphy elements for Bn.

Abstract: Publication date: Available online 29 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): A.A. Ambily, Ravi A. Rao We consider the Dickson–Siegel–Eichler–Roy's (DSER) subgroup of the orthogonal group OR(Q⊥H(R)m) of a quadratic space with a hyperbolic summand over a commutative ring in which 2 is invertible, rankQ=n≥1 and m≥1. We show that it is a normal subgroup of the orthogonal group OR(Q⊥H(R)m), for m≠2. In particular, when Q≅H(R)r for r≥1 and m≥2, the DSER elementary orthogonal group EOR(Q,H(R)m) coincides with the usual elementary orthogonal group EO2(r+m)(R) and it is a normal subgroup in OR(H(R)r+m). We also prove that the DSER elementary orthogonal group EOR(Q,H(P)) is a normal subgroup of OR(Q⊥H(P)), where Q is a quadratic space and H(P) is the hyperbolic space of the finitely generated projective module over a commutative ring R, with P a finitely generated projective module with rank(P)≥2 and rank(Q)≥1.

Abstract: Publication date: Available online 28 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Pascal Koiran, Mateusz Skomra Let F(x,y)∈C[x,y] be a polynomial of degree d and let G(x,y)∈C[x,y] be a polynomial with t monomials. We want to estimate the maximal multiplicity of a solution of the system F(x,y)=G(x,y)=0. Our main result is that the multiplicity of any isolated solution (a,b)∈C2 with nonzero coordinates is no greater than 52d2t2. We ask whether this intersection multiplicity can be polynomially bounded in the number of monomials of F and G, and we briefly review some connections between sparse polynomials and algebraic complexity theory.

Abstract: Publication date: Available online 28 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Yuanlin Li, Qinghai Zhong A ring R is said to be clean if each element of R can be written as the sum of a unit and an idempotent. In a recent article (J. Algebra, 405 (2014), 168-178), Immormino and McGoven characterized when the group ring Z(p)[Cn] is clean, where Z(p) is the localization of the integers at the prime p. In this paper, we consider a more general setting. Let K be an algebraic number field, OK be its ring of integers, and R be a localization of OK at some prime ideal. We investigate when R[G] is clean, where G is a finite abelian group, and obtain a complete characterization for such a group ring to be clean for the case when K=Q(ζn) is a cyclotomic field or K=Q(d) is a quadratic field.