Abstract: Publication date: Available online 11 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Wolfgang RumpAbstractThe 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 KumarAbstractIn 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: June 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 6Author(s): A.-H. NokhodkarAbstractWe study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension of that field. Some generalized notions of isotropy, metabolicity and isometry are introduced and used to find a Witt decomposition for these forms. We then associate to every (skew) hermitian form over a division algebra with involution of the first kind a quadratic form defined on its underlying vector space. It is shown that this quadratic form, with its generalized notions of isotropy and isometry, can be used to determine the isotropy behaviour and the isometry class of (skew) hermitian forms.

Abstract: Publication date: June 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 6Author(s): Tomasz BrzezińskiAbstractTrusses, defined as sets with a suitable ternary and a binary operations, connected by the distributive laws, are studied from a ring and module theory point of view. The notions of ideals and paragons in trusses are introduced and several constructions of trusses are presented. A full classification of truss structures on the Abelian group of integers is given. Modules over trusses are defined and their basic properties and examples are analysed. In particular, the sufficient and necessary condition for a sub-heap of a module to induce a module structure on the quotient heap is established.

Abstract: Publication date: June 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 6Author(s): Bekir Danış, Müfit SezerAbstractWe study the generic initial ideals (gin) of certain ideals that arise in modular invariant theory. For all cases an explicit generating set is known we compute the generic initial ideal of the Hilbert ideal of a cyclic group of prime order for all monomial orders. We also consider the Klein four group and note that its Hilbert ideals are Borel fixed with certain orderings of the variables. In all situations we consider, it is possible to select a monomial order such that the gin of the Hilbert ideal is equal to its initial ideal. Along the way we show that gin respects a permutation of the variables in the monomial order.

Abstract: Publication date: June 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 6Author(s): Maarten SolleveldAbstractWe investigate Levi subgroups of a connected reductive algebraic group G, over a ground field K. We parametrize their conjugacy classes in terms of sets of simple roots and we prove that two Levi K-subgroups of G are rationally conjugate if and only if they are geometrically conjugate.These results are generalized to arbitrary connected linear algebraic K-groups. In that setting the appropriate analogue of a Levi subgroup is derived from the notion of a pseudo-parabolic subgroup.

Abstract: Publication date: June 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 6Author(s): Mi Young JangAbstractIn the present paper we prove that the Hilbert scheme of 0-dimensional subspaces on supercurves of dimension 1 1 exists and it is smooth. We also show that the Hilbert scheme is not projected in general.

Abstract: Publication date: June 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 6Author(s): Claus Michael Ringel, Pu ZhangAbstractLet k be a field and q a non-zero element of k. In Part I, we have exhibited a 6-dimensional k-algebra Λ=Λ(q) and we have shown that if q has infinite multiplicative order, then Λ has a 3-dimensional local module which is semi-Gorenstein-projective, but not torsionless, thus not Gorenstein-projective. This Part II is devoted to a detailed study of all the 3-dimensional local Λ-modules for this particular algebra Λ. If q has infinite multiplicative order, we will encounter a whole family of 3-dimensional local modules which are semi-Gorenstein-projective, but not torsionless.

Abstract: Publication date: June 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 6Author(s): Enrico Carlini, Maria Virginia Catalisano, Elena Guardo, Adam Van TuylAbstractIt remains an open problem to classify the Hilbert functions of double points in P2. Given a valid Hilbert function H of a zero-dimensional scheme in P2, we show how to construct a set of fat points Z⊆P2 of double and reduced points such that HZ, the Hilbert function of Z, is the same as H. In other words, we show that any valid Hilbert function H of a zero-dimensional scheme is the Hilbert function of a set a positive number of double points and some reduced points. For some families of valid Hilbert functions, we are also able to show that H is the Hilbert function of only double points. In addition, we give necessary and sufficient conditions for the Hilbert function of a scheme of a double points, or double points plus one additional reduced point, to be the Hilbert function of points with support on a star configuration of lines.

Abstract: Publication date: June 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 6Author(s): Lars Winther Christensen, Oana Veliche, Jerzy WeymanAbstractWhile every grade 2 perfect ideal in a regular local ring is linked to a complete intersection ideal, it is known not to be the case for ideals of grade 3. We soften the blow by proving that every grade 3 perfect ideal in a regular local ring is linked to a complete intersection or a Golod ideal. Our proof is indebted to a homological classification of Cohen–Macaulay local rings of codimension 3. That debt is swiftly repaid, as we use linkage to reveal some of the finer structures of this classification.

Abstract: Publication date: Available online 9 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): F. Ávila, G. Bezhanishvili, P.J. Morandi, A. ZaldívarAbstractFor 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 PereiroAbstractFor 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 TakahashiAbstractWe 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 WegenerAbstractWe 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 GonçalvesAbstractWe 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 WonAbstractLet 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 GulliverAbstractThis 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 FeldAbstractWe 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 TranAbstractWe 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 GrzeszczukAbstractLet 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 RosasAbstractWe 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: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): Leonardo M. Cabrer, Hilary A. PriestleyAbstractThis paper studies finitely generated quasivarieties of Sugihara algebras. These quasivarieties provide complete algebraic semantics for certain propositional logics associated with the relevant logic R-mingle. The motivation for the paper comes from the study of admissible rules. Recent earlier work by the present authors, jointly with Freisberg and Metcalfe, laid the theoretical foundations for a feasible approach to this problem for a range of logics—the Test Spaces Method. The method, based on natural duality theory, provides an algorithm to obtain the algebra of minimum size on which admissibility of sets of rules can be tested. (In the most general case a set of such algebras may be needed rather than just one.) The method enables us to identify this ‘admissibility algebra’ for each quasivariety of Sugihara algebras which is generated by an algebra whose underlying lattice is a finite chain. To achieve our goals, it was first necessary to develop a (strong) duality for each of these quasivarieties. The dualities promise also to provide a valuable new tool for studying the structure of Sugihara algebras more widely.

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): Bruno Deschamps, François LegrandRésuméDans cet article, nous montrons que le Problème Inverse de Galois sur un corps gauche H de dimension finie sur son centre k est équivalent à une variante du Problème Inverse de Galois sur k faisant intervenir une contrainte polynomiale. En application de ce résultat, nous montrons que, si k contient un corps ample, alors le Problème Inverse de Galois admet une réponse positive sur le corps H(t) des fractions rationnelles tordu à indéterminée centrale.AbstractIn this article, we show that the Inverse Galois Problem over a skew field H of finite dimension over its center k is equivalent to a variant of the Inverse Galois Problem over k involving a polynomial constraint. As an application, we show that if k contains an ample field, then the Inverse Galois Problem has a positive answer over the skew field H(t) of rational fractions with central indeterminate.

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): Yucai Su, Chunguang Xia, Lamei YuanAbstractWe classify extensions between finite irreducible conformal modules over a class of infinite Lie conformal algebras B(p) of Block type, where p is a nonzero complex number. We find that although certain finite irreducible conformal modules over B(p) are simply conformal modules over its Virasoro conformal subalgebra Vir, there exist more nontrivial extensions between these conformal B(p)-modules. For extensions between other conformal modules, the situation becomes rather different. As an application, we also solve the extension problem for a series of finite Lie conformal algebras b(n) for n≥1.

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): Paul J. TrumanAbstractLet K be a number field and let L/K be a tamely ramified radical extension of prime degree p. If K contains a primitive pth root of unity then L/K is a cyclic Kummer extension; in this case the group algebra K[G] (with G=Gal(L/K)) gives the unique Hopf-Galois structure on L/K, the ring of algebraic integers OL is locally free over OK[G] by Noether's theorem, and Gómez Ayala has determined a criterion for OL to be a free OK[G]-module. If K does not contain a primitive pth root of unity then L/K is a separable, but non-normal, extension, which again admits a unique Hopf-Galois structure. Under the assumption that p is unramified in K, we show that OL is locally free over its associated order in this Hopf-Galois structure and determine a criterion for it to be free. We find that the conditions that appear in this criterion are identical to those appearing in Gómez Ayala's criterion for the normal case.

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): Erik Insko, Julianna Tymoczko, Alexander WooAbstractHessenberg varieties are subvarieties of the flag variety parametrized by a linear operator X and a nondecreasing function h. The family of Hessenberg varieties for regular X is particularly important: they are used in quantum cohomology, in combinatorial and geometric representation theory, in Schubert calculus and affine Schubert calculus. We show that the classes of a regular Hessenberg variety in the cohomology and K-theory of the flag variety are given by making certain substitutions in the Schubert polynomial (respectively Grothendieck polynomial) for a permutation that depends only on h. Our formula and our methods are different from a recent result of Abe, Fujita, and Zeng that gives the class of a regular Hessenberg variety with more restrictions on h than here.

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): João Gouveia, Antonio Macchia, Rekha R. Thomas, Amy WiebeAbstractThe slack ideal of a polytope is a saturated determinantal ideal that gives rise to a new model for the realization space of the polytope. The simplest slack ideals are toric and have connections to projectively unique polytopes. We prove that if a projectively unique polytope has a toric slack ideal, then it is the toric ideal of the bipartite graph of vertex-facet non-incidences of the polytope. The slack ideal of a polytope is contained in this toric ideal if and only if the polytope is morally 2-level, a generalization of the 2-level property in polytopes. We show that polytopes that do not admit rational realizations cannot have toric slack ideals. A classical example of a projectively unique polytope with no rational realizations is due to Perles. We prove that the slack ideal of the Perles polytope is reducible, providing the first example of a slack ideal that is not prime.

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): A. Masuoka, A.N. ZubkovAbstractWe introduce the notion of Krull super-dimension of a super-commutative super-ring. This notion is used to describe regular super-rings and calculate Krull super-dimensions of completions of super-rings. Moreover, we use this notion to introduce the notion of super-dimension of any irreducible superscheme of finite type. Finally, we describe nonsingular superschemes in terms of sheaves of Kähler superdifferentials.

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): Yiqiang LiAbstractA canonical basis was constructed by Wang and the author in [6] inside Letzter's coideal subalgebra in quantum sl2. In this article, we provide an explicit description for the canonical bases and show that the bases coincide with the one defined algebraically by Bao-Wang in [1].

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): Vyacheslav Futorny, Luis Enrique Ramirez, Jian ZhangAbstractWe construct explicitly a large family of new simple modules for an arbitrary finite W-algebra of type A. A basis of these modules is given by the Gelfand-Tsetlin tableaux whose entries satisfy certain sets of relations. Characterization and an effective method of constructing such admissible relations are given. In particular we describe a family of simple infinite dimensional highest weight relation modules. We also prove a sufficient condition for the simplicity of tensor product of two highest weight relation modules and establish the simplicity of the tensor product any number of relation modules with generic highest weights. This extends the results of Molev to infinite dimensional highest weight modules.

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): Ngoc Phu HaAbstractIn a recent paper the authors Beliakova, Blanchet and Gainutdinov have shown that the modified trace on the category H-pmod of the projective modules corresponds to the symmetrised integral on the finite dimensional pivotal Hopf algebra H. We generalize this fact to the context of G-graded categories and Hopf G-coalgebra studied by Turaev-Virelizier. We show that the symmetrised G-integral on a finite type pivotal Hopf G-coalgebra induces a modified trace in the associated G-graded category.

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): Hans Franzen, Thorsten WeistAbstractFor an acyclic quiver with three vertices, we consider the canonical decomposition of a non-Schurian root and associate certain representations of a generalized Kronecker quiver. These representations correspond to points contained in the intersection of two subvarieties of a Grassmannian and give rise to representations of the original quiver, preserving indecomposability. We show that these subvarieties intersect using Schubert calculus. Provided that it contains a Schurian representation, the dimension of the intersection is what we expect by Kac's Theorem.

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): Viktor Bekkert, Hernán Giraldo, José A. Vélez-MarulandaAbstractLet k be a field of arbitrary characteristic, let Λ be a finite dimensional k-algebra, and let V be a finitely generated Λ-module. F. M. Bleher and the third author previously proved that V has a well-defined versal deformation ring R(Λ,V). If the stable endomorphism ring of V is isomorphic to k, they also proved under the additional assumption that Λ is self-injective that R(Λ,V) is universal. In this paper, we prove instead that if Λ is arbitrary but V is Gorenstein-projective then R(Λ,V) is also universal when the stable endomorphism ring of V is isomorphic to k. Moreover, we show that singular equivalences of Morita type (as introduced by X. W. Chen and L. G. Sun) preserve the isomorphism classes of versal deformation rings of finitely generated Gorenstein-projective modules over Gorenstein algebras. We also provide examples. In particular, if Λ is a monomial algebra in which there is no overlap (as introduced by X. W. Chen, D. Shen and G. Zhou) we prove that every finitely generated indecomposable Gorenstein-projective Λ-module has a universal deformation ring that is isomorphic to either k or to k〚t〛/(t2).

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): Daniel López N., Louis-François Préville-Ratelle, María RoncoAbstractWe introduce a simplicial object ({Dyckm}m≥0,Fi,Sj) in the category of non-symmetric algebraic operads, satisfying that Dyck0 is the operad of associative algebras and Dyck1 is J.-L. Loday’s operad of dendriform algebras. The dimensions of the operad Dyckm are given by the Fuss-Catalan numbers.Given a family of partially ordered sets P={Pn}n≥1 we show that, under certain conditions, the vector space spanned by the set of m-simpleces of P is a Dyckm algebra. This construction, applied to certain combinatorial Hopf algebras, whose associative product comes from a dendriform structure, provides examples of Dyckm algebras.

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): Cintya Wink de Oliveira Benedito, Carina Alves, Nelson Gomes Brasil Jr, Sueli Irene Rodrigues CostaAbstractIn this paper we propose a framework to construct algebraic lattices in dimensions 4n via ideals from maximal orders of a quaternion algebra whose center is a totally real number field. For n=1,2,3,4 and 6 it was possible to construct rotated versions of the densest lattices in their dimensions, D4,E8,K12,Λ16 and Λ24. We also present a family of lattices in dimension 2r from A=(−1,−1)Q(ζ2r+ζ2r−1) and a characterization of a maximal quaternion order of A by using the Chebyshev polynomials.

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): M. DomokosAbstractA minimal homogeneous generating system of the algebra of semi-invariants of tuples of two-by-two matrices over an infinite field of characteristic two or over the ring of integers is given. In an alternative interpretation this yields a minimal system of homogeneous generators for the vector invariants of the special orthogonal group of degree four over a field of characteristic two or over the ring of integers. An irredundant separating system of semi-invariants of tuples two-by-two matrices is also determined, it turns out to be independent of the characteristic.

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): Alexander Baranov, Hasan M. ShlakaAbstractWe study Jordan-Lie inner ideals of finite dimensional associative algebras and the corresponding Lie algebras and prove that they admit Levi decompositions. Moreover, we classify Jordan-Lie inner ideals satisfying a certain minimality condition and show that they are generated by pairs of idempotents.

Abstract: Publication date: May 2020Source: Journal of Pure and Applied Algebra, Volume 224, Issue 5Author(s): Francis N. Castro, Oscar Moreno, Ivelisse RubioAbstractWe improve a result of Carlitz about the number of variables needed for a system of polynomial equations with coefficients in Fq[X] to have non-trivial solutions by considering the p-weight degree of the polynomials. By providing infinite families of polynomials we illustrate that our improvement is significant and, in general, is tight.

Abstract: Publication date: Available online 7 January 2020Source: Journal of Pure and Applied AlgebraAuthor(s): Gianluca Occhetta, Luis E. Solá CondeAbstractIn 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. RoyAbstractIn 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 EscuderoAbstractWe 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 BecherAbstractLet 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 FioresiAbstractWe 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. KhandujaAbstractLet 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), ...

Abstract: Publication date: Available online 5 December 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Luiz Gustavo CordeiroAbstractOne important class of tools in the study of the connections between algebraic and topological structures are the “Banach–Stone type theorems”, which describe algebraic isomorphisms of algebras (or groups, lattices, etc.) of functions in terms of homeomorphisms between the underlying topological spaces. Several such theorems have been proven throughout the last century, however not all of them are comparable, and in particular no single one is the strongest. In this article, we describe a general framework which encompasses several of these results, and which allows for new applications related to groupoid algebras, and to groups of circle-valued functions. This is attained by a detailed study of “disjointness relations” on sets of functions, which play a central role (even if not explicitly) in previously-proven Banach–Stone type theorems.

Abstract: Publication date: Available online 4 December 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Juan Elias, Roser Homs, Bernard MourrainAbstractWe 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 MondalAbstractIn 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-HansenAbstractWe 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. RaoAbstractWe 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): Fanggui Wang, Hwankoo Kim, Tao XiongAbstractWe study the finististic weak dimensions of certain pullbacks. As an application, it is proved that the finististic weak dimension of a pseudo-valuation domain is 0, 1, or 2.

Abstract: Publication date: Available online 28 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Lixin MaoAbstractLet T=(A0UB) be a formal triangular matrix ring, where A and B are rings and U is a (B,A)-bimodule. Let C1 and C2 be two classes of left A-modules, D1 and D2 be two classes of left B-modules, we prove that: (1) If ToriA(U,C1)=0 for any i≥1, (C1,C2) and (D1,D2) are (resp. hereditary complete) cotorsion pairs, then (PD1C1,AD2C2) is a (resp. hereditary complete) cotorsion pair in T-Mod. (2) If ExtBi(U,D2)=0 for any i≥1, (C1,C2) and (D1,D2) are (resp. hereditary complete) cotorsion pairs, then (AD1C1,ID2C2) is a (resp. hereditary complete) cotorsion pair in T-Mod. In additio...

Abstract: Publication date: Available online 28 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Pascal Koiran, Mateusz SkomraAbstractLet 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 ZhongAbstractA 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.

Abstract: Publication date: Available online 28 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Alana Cavalcante, Mauricio Corrêa, Simone MarchesiAbstractThis paper is devoted to the study of holomorphic distributions of dimension and codimension one on smooth weighted projective complete intersection Fano manifolds X which is threedimensional and with Picard number equal to one. We study the relations between algebro-geometric properties of the singular set of singular holomorphic distributions and their associated sheaves. We characterize either distributions whose tangent sheaf or conormal sheaf are arithmetically Cohen Macaulay (aCM) on X. We also prove that a codimension one locally free distribution with trivial canonical bundle on any Fano threefold, with Picard number equal to one, has a tangent sheaf which either splits or it is stable.

Abstract: Publication date: Available online 28 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): M. Flores, D. GoundaroulisAbstractWe extend the Framization of the Temperley-Lieb algebra to Coxeter systems of type B. We first define a natural extension of the classical Temperley-Lieb algebra to Coxeter systems of type B and prove that such an extension supports a unique linear Markov trace function. We then introduce the Framization of the Temperley-Lieb algebra of type B as a quotient of the Yokonuma-Hecke algebra of type B. The main theorem provides necessary and sufficient conditions for the Markov trace defined on the Yokonuma-Hecke algebra of type B to pass to the quotient algebra. Using the main theorem, we construct invariants for framed links and classical links inside the solid torus.

Abstract: Publication date: Available online 28 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Stephen Lack, Giacomo TendasAbstractRegular and exact categories were first introduced by Michael Barr in 1971; since then, the theory has developed and found many applications in algebra, geometry, and logic. In particular, a small regular category determines a certain theory, in the sense of logic, whose models are the regular functors into Set. Barr further showed that each small and regular category can be embedded in a particular category of presheaves; then in 1990 Makkai gave a simple explicit characterization of the essential image of the embedding, in the case where the original regular category is moreover exact. More recently Prest and Rajani, in the additive context, and Kuber and Rosický, in the ordinary one, described a duality which connects an exact category with its (definable) category of models. Working over a suitable base for enrichment, we define an enriched notion of regularity and exactness, and prove a corresponding version of the theorems of Barr, of Makkai, and of Prest-Rajani/Kuber-Rosický.

Abstract: Publication date: Available online 27 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): S. El Baghdadi, L. Izelgue, A. TamoussitAbstractWe investigate some ring theoretic properties of almost Krull domains. By using the language of star operations, we shed new light on the work of Pirtle on almost Krull domains. That allows us to give a new characterization of these domains. On the other hand, we show that Int(D) issued from an almost Krull domain is D–locally free and we give a complete characterization of when Int(D) is an almost Krull domain.

Abstract: Publication date: Available online 27 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Ilaria CastellanoAbstractFor a topological flow (V,ϕ) - i.e., V is a linearly compact vector space and ϕ a continuous endomorphism of V - we gain a deep understanding of the relationship between (V,ϕ) and the Bernoulli shift: a topological flow (V,ϕ) is essentially a product of one-dimensional left Bernoulli shifts as many as ent⁎(V,ϕ) counts. This novel comprehension brings us to introduce a notion of corank for topological flows designed for coinciding with the value of the topological entropy of (V,ϕ). As an application, we provide an alternative proof of the so-called Bridge Theorem for locally linearly compact vector spaces connecting the topological entropy to the algebraic entropy by means of Lefschetz duality.

Abstract: Publication date: Available online 27 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Luca Barbieri-Viale, Mike PrestAbstractFollowing Nori's original idea we here provide certain motivic categories with a canonical tensor structure. These motivic categories are associated to a cohomological functor on a suitable base category and the tensor structure is induced by the cartesian tensor structure on the base category via a cohomological Künneth formula.

Abstract: Publication date: Available online 11 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): D. CulverAbstractIn previous work, the author analyzed the co-operations algebra for the second truncated Brown-Peterson spectrum at the prime p=2. The purpose of this paper is to carry out the necessary modifications to odd primes.

Abstract: Publication date: Available online 4 November 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Onofrio M. Di Vincenzo, Vincenzo NardozzaAbstractLet F be any field, G a finite abelian group and let A, B be F-algebras graded by subgroups of G. If M is a G-graded free (A,B)-bimodule, we describe the G-graded polynomial identities of the triangular algebra of M and, in case the field F has characteristic zero, we provide the description of its G-graded cocharacters by means of the graded cocharacters of A and B.

Abstract: Publication date: Available online 25 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Ido EfratAbstractFor a prime number p and a free profinite group S on the basis X, let S(n,p), n=1,2,…, be the lower p-central filtration of S. For p>n, we give a combinatorial description of H2(S/S(n,p),Z/p) in terms of the Shuffle algebra on X.

Abstract: Publication date: Available online 25 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Boštjan Gabrovšek, Eva HorvatAbstractWe present a reduced Burau-like representation for the mixed braid group on one strand representing links in lens spaces and show how to calculate the Alexander polynomial of a link directly from the mixed braid.

Abstract: Publication date: Available online 25 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Taku SuzukiAbstractA generalization of S. Mukai's conjecture says that if X is a Fano n-fold with Picard number ρX and pseudo-index iX, then ρX(iX−1)≤n, with equality if and only if X≅(PiX−1)ρX. In this paper, we prove that this conjecture holds if n=6 and either X admits an extremal contraction of fiber type or X admits no small extremal contractions.

Abstract: Publication date: Available online 24 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Daniel Panario, Lucas Reis, Qiang WangAbstractLet Fq be the finite field with q elements, where q is a power of a prime. We discuss recursive methods for constructing irreducible polynomials over Fq of high degree using rational transformations. In particular, given a divisor D>2 of q+1 and an irreducible polynomial f∈Fq[x] of degree n such that n is even or D≢2(mod4), we show how to obtain from f a sequence {fi}i≥0 of irreducible polynomials over Fq with deg(fi)=n⋅Di.

Abstract: Publication date: Available online 22 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Michael PerlmanAbstractWe study the structure of local cohomology with support in Pfaffian varieties as a module over the Weyl algebra DX of differential operators on the space of skew-symmetric matrices X=⋀2Cn. The simple composition factors of these modules are known by the work of Raicu-Weyman, and when n is odd, the general theory implies that the local cohomology modules are semi-simple. When n is even, we show that the local cohomology is a direct sum of indecomposable modules coming from the pole order filtration of the Pfaffian hypersurface. We then determine the Lyubeznik numbers for Pfaffian rings by computing local cohomology with support in the origin of the indecomposable summands referred to above.

Abstract: Publication date: Available online 22 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Gunnar Carlsson, Benjamin FilippenkoAbstractGiven finite metric spaces (X,dX) and (Y,dY), we investigate the persistent homology PH⁎(X×Y) of the Cartesian product X×Y equipped with the sum metric dX+dY. Interpreting persistent homology as a module over a polynomial ring, one might expect the usual Künneth short exact sequence to hold. We prove that it holds for PH0 and PH1, and we illustrate with the Hamming cube {0,1}k that it fails for PHn,n≥2. For n=2, the prediction for PH2(X×Y) from the expected Künneth short exact sequence has a natural surjection onto PH2(X×Y). We compute the nontrivial kernel of this surjection for the splitting of Hamming cubes {0,1}k={0,1}k−1×{0,1}. For all n≥0, the interleaving distance between the prediction for PHn(X×Y) and the true persistent homology is bounded above by the minimum of the diameters of X and Y. As preliminary results of independent interest, we establish an algebraic Künneth formula for simplicial modules over the ring κ[R+] of polynomials with coefficients in a field κ and exponents in R+=[0,∞), as well as a Künneth formula for the persistent homology of R+-filtered simplicial sets – both of these Künneth formulas hold in all homological dimensions

Abstract: Publication date: Available online 21 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Guillermo AlesandroniAbstractLet S be a polynomial ring in n variables, over an arbitrary field. Let M be the family of all monomial ideals in S. Using combinatorial methods, we give an explicit characterization of all M∈M, such that pd(S/M)=n. In addition, we give the total, graded, and multigraded Betti numbers of S/M in homological degree n, for all M∈M. Finally, we show that for each M∈M, with pd(S/M)=n, the sum of the total Betti numbers of S/M is at least 2n.

Abstract: Publication date: Available online 21 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Dimitra Kosta, Apostolos ThomaAbstractComputing the complexity of Markov bases is an extremely challenging problem; no formula is known in general and there are very few classes of toric ideals for which the Markov complexity has been computed. A monomial curve C in A3 has Markov complexity m(C) two or three. Two if the monomial curve is complete intersection and three otherwise. Our main result shows that there is no d∈N such that m(C)≤d for all monomial curves C in A4. The same result is true even if we restrict to complete intersections. We extend this result to all monomial curves in An, where n≥4.

Abstract: Publication date: Available online 21 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Isaac BirdAbstractOver a Cohen-Macaulay local ring we consider two extensions of the maximal Cohen-Macaulay modules from the viewpoint of definable subcategories, which are closed under direct limits, direct products and pure submodules. After presenting these categories, we compare them and consider which properties they inherit from the maximal Cohen-Macaulay modules. We then consider some further properties of these classes and how they interact with the entire module category.

Abstract: Publication date: Available online 21 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): A.S. SivatskiAbstractLet k be a field, chark≠2, let φ be an anisotropic quadratic form over k, dimφ≥2, V the underlying linear space of φ. As usual, denote by D(φ) the set of nonzero values of φ. Given a positive integer m, we say that φ is m-essential if there exists a nonzero polynomial p∈k[x1,…,xm] such that p∈D(φk(x1,…,xm)), but p∉D(ψk(x1,…,xm))) for any anisotropic form ψ over k with dimψ

Abstract: Publication date: Available online 18 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Rubén A. Hidalgo, Saúl QuispeAbstractLet Gn be the dicyclic group of order 4n. We observe that, up to isomorphisms, (i) for n≥2 even there is exactly one regular dessin d'enfant with automorphism group Gn, and (ii) for n≥3 odd there are exactly two of them. Each of them is produced on well known hyperelliptic Riemann surfaces. We obtain that the minimal genus over which Gn acts purely-non-free is σp(Gn)=n (this coincides with the strong symmetric genus of Gn when n is even). For each of the triangular conformal actions, every non-trivial subgroup of Gn has genus zero quotient, in particular, that the isotypical decomposition, induced by the action of Gn, of its jacobian variety has only one component. We also study conformal/anticonformal actions of Gn, on closed Riemann surfaces, with the property that Gn admits anticonformal elements. It is known that Gn always acts on a genus one Riemann surface with such a property. We observe that the next genus σhyp(Gn)≥2 over which Gn acts in that way is n+1 for n≥2 even, and 2n−2 for n≥3 odd. We also provide examples of pseudo-real Riemann surfaces admitting Gn as the full group of conformal/anticonformal automorphisms.

Abstract: Publication date: Available online 18 October 2019Source: Journal of Pure and Applied AlgebraAuthor(s): Jaiung Jun, Kalina Mincheva, Louis RowenAbstractWe develop the basic theory of projective modules and splitting over semirings, within the more general setting of systems. Systems provide a common language for most tropical algebraic approaches including supertropical algebra, hyperrings (specifically hyperfields), and fuzzy rings. This enables us to prove analogues of classical theorems for tropical and hyperring theory in a unified way. In this context we prove a Dual Basis Lemma and versions of Schanuel's Lemma.