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We extend the concept of Grobner bases to relative Grobner bases for ideals in and modules over quotient rings of a polynomial ring over a field. We develop a “relative” variant of both Buchberger’s criteria for avoiding reductions to zero and Schreyer’s theorem for a Grobner basis of the syzygy module. As main contribution, we then introduce the novel notion of relative involutive bases and present an algorithm for their explicit construction. Finally, we define the new notion of relatively quasi-stable ideals and exploit it for the algorithmic determination of coordinates in which finite relative Pommaret bases exist.

A construction due to Knorrer shows that if N is a maximal Cohen–Macaulay module over a hypersurface defined by f + y 2 , then the first syzygy of N / y N decomposes as the direct sum of N and its own first syzygy. This was extended by Herzog–Popescu to hypersurfaces f + y n , replacing N / y N by N / y n − 1 N . We show, in the same setting as Herzog–Popescu, that the first syzygy of N / y k N is always an extension of N by its first syzygy, and moreover that this extension has useful approximation properties. We give two applications. First, we construct a ring Λ over which every finitely generated module has an eventually 2-periodic projective resolution, prompting us to call it a “non-commutative hypersurface ring”. Second, we give upper bounds on the dimension of the stable module category (a.k.a. the singularity category) of a hypersurface defined by a polynomial of the form x 1 a 1 + … + x d a d .

Let S be a deeply embedded, equicharacteristic, Artinian Gorenstein local ring. We prove that if R is a non-Gorenstein quotient of S of small colength, then every totally reflexive R-module is free. Indeed, the second syzygy of the canonical module of R has a direct summand T which is a test module for freeness over R in the sense that if Tor + R ( T , N ) = 0 , for some finitely generated R-module N, then N is free.

Let p be an odd prime and Λ = Z [ D 2 p ] the integral group ring of the dihedral group D 2 p of order 2p. The syzygies Ω r ( Z ) are the stable classes of the intermediate modules in a free Λ-reso...

We introduce a relativization of the secant sheaves used by Ein, Green and Lazarsfeld and apply this construction to the study of syzygies of canonical curves. As a first application, we give a simpler proof of Voisin's Theorem for general canonical curves. This completely determines the terms of the minimal free resolution of the coordinate ring of such curves. Secondly, in the case of curves of even genus, we enhance Voisin's Theorem by providing a structure theorem for the last syzygy space, resolving the Geometric Syzygy Conjecture in even genus., Comment: Final version, to appear in Inventiones Math. The statements concerning positive characteristic have been removed and will be published separately. This submission supersedes arXiv:1910.06200 and arXiv:1907.07553

Construction of multivariate tight framelets is known to be a challenging problem because it is linked to the difficult problem on sum of squares of multivariate polynomials in real algebraic geometry. Multivariate dual framelets with vanishing moments generalize tight framelets and are not easy to be constructed either, since their construction is related to syzygy modules and factorization of multivariate polynomials. On the other hand, compactly supported multivariate framelets with directionality or high vanishing moments are of interest and importance in both theory and applications. In this paper we introduce the notion of a quasi-tight framelet, which is a dual framelet, but behaves almost like a tight framelet. Let ϕ ∈ L 2 ( R d ) be an arbitrary compactly supported real-valued M -refinable function with a general dilation matrix M and ϕ ˆ ( 0 ) = 1 such that its underlying real-valued low-pass filter satisfies the basic sum rule. We first constructively prove by a step-by-step algorithm that we can always easily derive from the arbitrary M -refinable function ϕ a directional compactly supported real-valued quasi-tight M -framelet in L 2 ( R d ) associated with a directional quasi-tight M -framelet filter bank, each of whose high-pass filters has one vanishing moment and only two nonzero coefficients. If in addition all the coefficients of its low-pass filter are nonnegative, then such a quasi-tight M -framelet becomes a directional tight M -framelet in L 2 ( R d ) . Furthermore, we show by a constructive algorithm that we can always derive from the arbitrary M -refinable function ϕ a compactly supported quasi-tight M -framelet in L 2 ( R d ) with the highest possible order of vanishing moments. We shall also present a result on quasi-tight framelets whose associated high-pass filters are purely differencing filters with the highest order of vanishing moments. Several examples will be provided to illustrate our main theoretical results and algorithms in this paper.

Let $$\varPi _n$$Πn be the set of bivariate polynomials of degree not greater than n. A $$\varPi _n$$Πn-correct set of nodes is a set such that the Lagrange interpolation problem with respect to these nodes has a unique solution. A maximal line of a $$\varPi _n$$Πn-correct set is any line containing exactly $$n+1$$n+1 nodes. Syzygy matrices can be used to find linear factors of the fundamental polynomials and detect maximal lines. We suggest to use matrix relations in order to generate syzygies, identify linear factors of fundamental polynomials and detect maximal lines. We interpret our results in the important case of GC sets trying to shed some light on the Gasca-Maeztu conjecture.

Kehrein and Kreuzer (J Pure Appl Algebra 205(2):279–295, 2006) presented an algorithm, referred to as the BorderBasis algorithm, for computing border bases. The main objective of this paper is to improve this computation by applying new structures from the theory of Grobner bases. In doing so, based on the method developed by Gao et al. (Math Comput 85(297):449–465, 2016), a new variant of the BorderBasis algorithm is proposed to compute simultaneously a border basis and also a Grobner basis for the syzygy module of the input polynomials (this pair of bases will be called a coupled border basis). The new algorithm, and also the original form of the BorderBasis algorithm, have been implemented in the computer algebra systems Maple and ApCoCoA. We compare the performance of our implementation of these algorithms via a set of benchmark polynomials. Our experiments show that our algorithm performs more efficiently than the original BorderBasis algorithm.

© 2019 Elsevier B.V. In a first part of this paper, we prove constancy of the canonical graded Betti table among the smooth curves in linear systems on Gorenstein weak Fano toric surfaces. In a second part, we show that Green's canonical syzygy conjecture holds for all smooth curves of genus at most 32 or Clifford index at most 6 on arbitrary toric surfaces. Conversely we use known results on Green's conjecture (due to Lelli-Chiesa)to obtain new facts about graded Betti tables of projectively embedded toric surfaces. ispartof: JOURNAL OF PURE AND APPLIED ALGEBRA vol:224 issue:2 pages:507-527 status: published

Let $$\mathfrak {g}$$ be a finite dimensional nilpotent p-restricted Lie algebra over a field k of characteristic p. For $$p\geqslant 5$$, we show that every endotrivial $$\mathfrak {g}$$-module is a direct sum of a syzygy of the trivial module and a projective module. The proof includes a theorem that the intersection of the maximal linear subspaces of the null cone of a nilpotent restricted p-Lie algebra for $$p \geqslant 5$$ has dimension at least two. We give an example to show that the statement about endotrivial modules is false in characteristic two. In characteristic three, another example shows that our proof fails, and we do not know a characterization of the endotrivial modules in this case.

A closed subscheme of codimension two T ⊂ P 2 is a quasi complete intersection (q.c.i.) of type ( a , b , c ) if there exists a surjective morphism O ( − a ) ⊕ O ( − b ) ⊕ O ( − c ) → I T . We give bounds on deg ⁡ ( T ) in function of a , b , c and r, the least degree of a syzygy between the three polynomials defining the q.c.i. (see Theorem 6 ). As a by-product we recover a theorem of du Plessis-Wall on the global Tjurina number of plane curves (see Theorem 20 ) and some other related results.

We start the study of reduced complex projective plane curves, whose Jacobian syzygy module has 3 generators. Among these curves one finds the nearly free curves introduced by the authors, and the plus-one generated line arrangements introduced by Takuro Abe. All the Thom–Sebastiani type plane curves, and more generally, any curve whose global Tjurina number is equal to a lower bound given by A. du Plessis and C.T.C. Wall, are 3-syzygy curves. Rational plane curves which are nearly cuspidal, i.e. which have only cusps except one singularity with two branches, are also related to this class of curves.

For an abelian category A , we define the category PEx( A ) of pullback diagrams of short exact sequences in A , as a subcategory of the functor category Fun( Δ , A ) for a fixed diagram category Δ. For any object M in PEx ( A ) , we prove the existence of a short exact sequence 0 → K → P → M → 0 of functors, where the objects are in PEx( A ) and P ( i ) ∈ Proj ( A ) for any i ∈ Δ . As an application, we prove that if ( C , D , E ) is a triple of syzygy finite classes of objects in mod Λ satisfying some special conditions, then Λ is an Igusa–Todorov algebra. Finally, we study lower triangular matrix Artin algebras and determine in terms of their components, under reasonable hypothesis, when these algebras are syzygy finite or Igusa–Todorov.

International audience; We provide constructive versions of Hilbert's syzygy theorem for Z and Z/nZ following Schreyer's method. Moreover, we extend these results to arbitrary coherent strict Bézout rings with a divisibility test for the case of finitely generated modules whose module of leading terms is finitely generated.

International audience; We introduce an improved Gröbner basis completion algorithm for operads. To this end, we define operadic rewriting systems as a machinery to rewrite in operads, whose rewriting rules do not necessarily depend on an ambient monomial order. A Gröbner basis of an operadic ideal can be seen as a confluent and terminating operadic rewriting system; thus, the completion of a Gröbner basis is equivalent to the completion of a rewriting system. We improve the completion algorithm by filtering out redundant S-polynomials and testing only essential ones. Finally, we show how the notion of essential S-polynomials can be used to compute Gröbner bases for syzygy bimodules. This work is motivated by the computation of minimal models of associative algebras and symmetric operads. In this direction, we show how our completion algorithm extends to the case of shuffle operads. CCS Concepts • Computing methodologies → Combinatorial algorithms; Algebraic algorithms.

Deciding the presence of the weak Lefschetz property often is a challenging problem. Continuing studies of Brenner and Kaid (2007), Cook II and Nagel (2011) and Migliore, Miro-Roig, Murai and Nagel (2013) we carry out an in-depth study of Artinian monomial ideals with four generators in three variables. We use a connection to lozenge tilings to describe semistability of the syzygy bundle of such an ideal, to determine its generic splitting type, and to decide the presence of the weak Lefschetz property. We provide results in both characteristic zero and positive characteristic.

In this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study $3$-syzygy arrangements and we present examples that admit unexpected curves., Comment: 19 pages, one figure, version after referee's remarks

We review computations of joint invariants on a linear symplectic space, discuss variations for an extension of group and space and relate this to other equivalence problems and approaches, most importantly to differential invariants., Comment: In this revision we added missing references, and essentially changed the presentation into a review. We corrected small errors, reduced the material on algebraic part, and extended it on geometric part. Thus we elaborate on known results from the classical invariant theory, discuss some extensions and draw relations to the differential invariants theory via symplectic invariant discretizations

We provide an expression for the conductor $c(\Delta)$ of a semimodule $\Delta$ of a numerical semigroup $\Gamma$ with two generators in terms of the syzygy module of $\Delta$ and the generators of the semigroup. In particular, we deduce that the difference between the conductor of the semimodule and the conductor of the semigroup is an element of $\Gamma$, as well as a formula for $c(\Delta)$ in terms of the dual semimodule of $\Delta$., Comment: Proof of Theorem 3.1 has been clarified. Some typos corrected along the whole paper. To Appear in Semigroup Forum. arXiv admin note: text overlap with arXiv:2011.01690. text overlap with arXiv:2011.01690

We present the newest release of the subsystem of Singular called Letterplace which exists since 2009. It is devoted to computations with finitely presented associative algebras over fields and offers Grobner(-Shirshov) bases over free algebras via the Letterplace correspondence of La Scala and Levandovskyy. This allows to use highly tuned commutative data structures internally and to reuse parts of existing algorithms in the non-commutative situation. The present version has been deeply reengineered, based on the experience with earlier and experimental versions. We offer an unprecedented functionality, some of which for the first time in the history of computer algebra. In particular, we present tools for elimination theory (via truncated Grobner bases and via supporting several kinds of elimination orderings), dimension theory (Gel'fand-Kirillov and global dimension), and for homological algebra (such as syzygy bimodules and lifts for ideals and bimodules) to name a few. Another article in this issue is devoted to the extension of Grobner bases to the coefficients in principal ideal rings including Z, which is also a part of this release. We report on comparison with other systems and on some advances in the theory. Quite nontrivial examples illustrate the abilities of the system.

We prove that the kernel of the evaluation morphism of global sections - namely the syzygy bundle - of a sufficiently ample line bundle on an abelian variety is stable. This settles a conjecture of Ein-Lazarsfeld-Mustopa, in the case of abelian varieties., Final version. To appear in Bull. Lond. Math. Soc

One way to compute a GCD of a pair of multivariate polynomials is by finding a certain syzygy. We can weaken this to create an “approximate syzygy”, for the purpose of computing an approximate GCD. The primary tools are Grobner bases and optimization. Depending on specifics of the formulation, one might use quadratic programming, linear programming, unconstrained with quadratic main term and quartic penalty, or a penalty-free sum-of-squares optimization. There are relative strengths and weaknesses to all four approaches, trade-offs in terms of speed vs. quality of result, size of problem that can be handled, and the like. Once a syzygy is found, there is a polynomial quotient to form, in order to get an approximation to an exact quotient. This step too can be tricky and requires careful handling. We will show what seem to be reasonable formulations for the optimization and quotient steps. We illustrate with several examples from the literature.

An explicit construction is given of a minimal free resolution of the ideal generated by all squarefree monomials of a given degree. The construction relies upon and exhibits the natural action of the symmetric group on the syzygy modules. The resolution is obtained over an arbitrary coefficient ring; in particular, it is characteristic free. Two applications are given: an equivariant resolution of De Concini-Procesi rings indexed by hook partitions, and a resolution of FI-modules., Comment: Updated references

A translational surface is a rational tensor product surface generated from two rational space curves by translating one curve along the other curve. Translational surfaces are invariant under rigid motions: translating and rotating the two generating curves translates and rotates the translational surface by the same amount. We construct three special syzygies for a translational surface from a μ-basis of one of the generating space curves, and we show how to compute the implicit equation of a translational surface from these three special syzygies. Examples are provided to illustrate our theorems and flesh out our algorithms.

The main objective of this paper is to present an improved algorithm to compute H-bases by relying on techniques from linear algebra. An H-basis is a specific generating set for a polynomial ideal ...

Hilbert famously showed that polynomials in n variables are not too complicated, in various senses. For example, the Hilbert Syzygy Theorem shows that the process of resolving a module by free modules terminates in finitely many (in fact, at most n) steps, while the Hilbert Basis Theorem shows that the process of finding generators for an ideal also terminates in finitely many steps. These results laid the foundations for the modern algebraic study of polynomials. Hilbert's results are not uniform in n: unsurprisingly, polynomials in n variables will exhibit greater complexity as n increases. However, an array of recent work has shown that in a certain regime---namely, that where the number of polynomials and their degrees are fixed---the complexity of polynomials (in various senses) remains bounded even as the number of variables goes to infinity. We refer to this as Stillman uniformity, since Stillman's Conjecture provided the motivating example. The purpose of this paper is to give an exposition of Stillman uniformity, including: the circle of ideas initiated by Ananyan and Hochster in their proof of Stillman's Conjecture, the followup results that clarified and expanded on those ideas, and the implications for understanding polynomials in many variables., This expository paper was written in conjunction with Craig Huneke's talk on Stillman's Conjecture at the 2018 JMM Current Events Bulletin

The geometric characterization, introduced by Chung and Yao, identifies node sets for total degree interpolation such that the Lagrange fundamental polynomials are products of linear factors. Sets satisfying the geometric characterization are usually called GC sets. Gasca and Maeztu conjectured that planar GC sets of degree n contain n + 1 collinear points. It has been shown that the conjecture holds for degrees not greater than 5 but it is still unsolved for general degree. One promising approach consists of studying the syzygies of the ideal of polynomials vanishing at the nodes. In order to describe syzygy matrices of GC sets, we analyze the extension of a GC set of degree n to a GC set of degree n + 1, by adding a n + 2 nodes on a line.

A finite-dimensional algebra $A$ over an algebraically closed field $K$ is called periodic if it is periodic under the action of the syzygy operator in the category of $A-A-$ bimodules. The periodic algebras are self-injective and occur naturally in the study of tame blocks of group algebras, actions of finite groups on spheres, hypersurface singularities of finite Cohen-Macaulay type, and Jacobian algebras of quivers with potentials. Recently, the tame periodic algebras of polynomial growth have been classified and it is natural to attempt to classify all tame periodic algebras. We introduce the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular tetrahedral algebras, are symmetric tame periodic algebras of period $4$. Moreover, we describe the socle deformations of the weighted surface algebras and prove that all these algebras are symmetric tame periodic algebras of period $4$. The main results of the paper form an important step towards a classification of all periodic symmetric tame algebras of non-polynomial growth, and lead to a complete description of all algebras of generalized quaternion type. Further, the orbit closures of the weighted surface algebras (and their socle deformations) in the affine varieties of associative $K$-algebra structures contain wide classes of tame symmetric algebras related to algebras of dihedral and semidihedral types, which occur in the study of blocks of group algebras with dihedral and semidihedral defect groups.

This paper is a survey of recent work on the asymptotic behavior of the syzygies of a smooth complex projective variety as the positivity of the embedding line bundle grows. After a quick overview of results from the 1980s and 1990s concerning the linearity of the first few terms of a resolution, we discuss a non-vanishing theorem to the effect that from an asymptotic viewpoint, essentially all of the syzygy modules that could be non-zero are in fact non-zero. We explain the quick new proof of this result in the case of Veronese varieties due to Erman and authors, and we explore some results and conjectures about the asymptotics of Betti numbers. Finally we discuss the case of syzygies of weight one, and the gonality conjecture on the syzygies of curves of large degree. The exposition also discusses numerous open questions and conjectures.

Let $${\mathrm{Syz}}_1(\mathfrak {m})$$ be the first syzygy of the graded maximal ideal $$\mathfrak {m}$$ of a polynomial ring $$K[x_1,\ldots ,x_n]$$ over a field K. The multiplicity and (Castelnuovo–Mumford) regularity of the symmetric algebra $${\mathrm{Sym}}({\mathrm{Syz}}_1(\mathfrak {m}))$$ are estimated by using the theory of s-sequences. It is proved that the multiplicity of $${\mathrm{Sym}}({\mathrm{Syz}}_1(\mathfrak {m}))$$ is 1 when $$n\ge 5$$ , and $$n-2$$ is an upper bound for its regularity. In virtue of Grobner bases, this bound is shown to be reached provided $$n\le 5$$ .

In this note, we give a slight improvement of a result of A. K\"uronya and V. Lozovanu about higher syzygies on abelian surfaces., Comment: 6 pages

μ -Bases are new representations for rational curves and surfaces which serve as a bridge between their parametric forms and implicit forms. Geometrically, μ -bases are represented by moving lines or moving planes, while their algebraic counterparts are special syzygies of the parametric equations of rational curves or surfaces. μ -bases have been proven to be significant in solving many important problems in geometric modeling, such as fast implicitization, singularity computation, reparametrization as well as providing easy inversion formulas for points. We review the state-of-the-art results in μ -bases theory and applications for rational curves and surfaces, and raise unsolved problems for future research.

Green’s canonical syzygy conjecture asserts a simple relationship between the Clifford index of a smooth projective curve and the shape of the minimal free resolution of its homogeneous ideal in the canonical embedding. We prove the analogue of this conjecture formulated by Bayer and Eisenbud for a class of non-reduced curves called ribbons. Our proof uses the results of Voisin and Hirschowitz–Ramanan on Green’s conjecture for general smooth curves.

Let ( A , m ) be a local complete intersection ring of dimension d and let I be an m -primary ideal. Let M be a maximal Cohen–Macaulay A -module. For i = 0 , 1 , ⋯ , d , let e i I ( M ) denote the i th Hilbert-coefficient of M with respect to I . We prove that for i = 0 , 1 , 2 , the function j ↦ e i I ( Syz j A ( M ) ) is of quasi-polynomial type with period 2. Let G I ( M ) be the associated graded module of M with respect to I . If G I ( A ) is Cohen–Macaulay and dim ⁡ A ≤ 2 we also prove that the functions j ↦ depth G I ( Syz 2 j + i A ( M ) ) are eventually constant for i = 0 , 1 . Let ξ I ( M ) = lim l → ∞ ⁡ depth G I l ( M ) . Finally we prove that if dim ⁡ A = 2 and G I ( A ) is Cohen–Macaulay then the functions j ↦ ξ I ( Syz 2 j + i A ( M ) ) are eventually constant for i = 0 , 1 .

To study syzygies of the Cox rings of del Pezzo surfaces, we calculate important syzygetic invariants such as the Hilbert functions, the Green-Lazarsfeld indices, the projective dimensions, and the Castelnuovo-Mumford regularities. Using these computations as well as the natural multigrading structures by the Picard groups of del Pezzo surfaces and Weyl group actions on Picard lattices, we determine the Betti diagrams of the Cox rings of del Pezzo surfaces of degree at most four., 15 pages, final version

We define canonical and $n$-canonical modules of a module-finite algebra over a Noether commutative ring and study their basic properties. Using $n$-canonical modules, we generalize a theorem on $(n,C)$-syzygy by Araya and Iima which generalize a well-known theorem on syzygies by Evans and Griffith. Among others, we prove a noncommutative version of Aoyama’s theorem which states that a canonical module descends with respect to a flat local homomorphism.

Let G be a finite simple graph on n vertices. Let J G &sub, K [ x 1 , &hellip, x n ] be the cover ideal of G. In this article, we obtain syzygies, Betti numbers, and Castelnuovo&ndash, Mumford regularity of J G s for all s &ge, 1 for certain classes of graphs G.

We introduce the notion of Burch ideals and Burch rings. They are easy to define, and can be viewed as generalization of many well-known concepts, for example integrally closed ideals of finite colength and Cohen--Macaulay rings of minimal multiplicity. We give several characterizations of these objects. We show that they satisfy many interesting and desirable properties: ideal-theoretic, homological, categorical. We relate them to other classes of ideals and rings in the literature., 23 pages, add Example 2.2, Prop 5.5 and Example 5.6

Let R be a commutative Noetherian ring. We consider the question of when n-syzygy modules over R are n-torsionfree in the sense of Auslander and Bridger. Our tools include Serre’s condition and certain conditions on the local Gorenstein property of R. Our main result implies the converse of a celebrated theorem of Evans and Griffith.

We compute the Hilbert–Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen–Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.

We characterize the syzygies and co-syzygies over 2-Calabi–Yau tilted algebras in terms of the Auslander–Reiten translation and the syzygy functor. We explore connections between the category of syzygies, the category of Cohen–Macaulay modules, the representation dimension of algebras and the Igusa–Todorov functions. In particular, we prove that the Igusa–Todorov dimensions of d -Gorenstein algebras are equal to d . For cluster-tilted algebras of Dynkin type D , we give a geometric description of the stable Cohen–Macaulay category in terms of tagged arcs in the punctured disc. We also describe the action of the syzygy functor in a geometric way. This description allows us to compute the Auslander–Reiten quiver of the stable Cohen–Macaulay category using tagged arcs and geometric moves.

In this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy o...

Let I be a monomial ideal in the polynomial ring S generated by elements of degree at most d. In this paper, it is shown that, if the i-th syzygy of I has no elements of degrees j,…,j+(d−1) (where j≥i+d), then (i+1)-th syzygy of I does not have any element of degree j+d. Then we give several applications of this result, including an alternative proof for Green–Lazarsfeld index of the edge ideals of graphs as well as an alternative proof for Froberg’s theorem on classification of square-free monomial ideals generated in degree 2 with linear resolution. Among all, we deduce a partial result on subadditivity of the syzygies for monomial ideals.

Let R0 be a Noetherian local ring and R a standard graded R0-algebra with maximal ideal 𝔪 and residue class field 𝕂 = R/𝔪. For a graded ideal I in R we show that for k ≫ 0: (1) the Artin-Rees number of the syzygy modules of Ik as submodules of the free modules from a free resolution is constant, and thereby the Artin-Rees number can be presented as a proper replacement of regularity in the local situation; and (2) R is a polynomial ring over the regular R0, the ring R/Ik is Golod, its Poincaré-Betti series is rational and the Betti numbers of the free resolution of 𝕂 over R/Ik are polynomials in k of a specific degree. The first result is an extension of the work by Swanson on the regularity of Ik for k ≫ 0 from the graded situation to the local situation. The polynomiality consequence of the second result is an analog of the work by Kodiyalam on the behaviour of Betti numbers of the minimal free resolution of R/Ik over R.

The purpose of this paper is to consider the$N$-center problem with collinear centers, to identify its syzygy sequences that can be realized by minimizers of the Lagrangian action functional and to count the number of such syzygy sequences. In particular, we show that the number of such realizable syzygy sequences of length$\ell$greater than or equal to two for the 3-center problem is at least$F_{\ell +2}-2$, where$\{F_{n}\}$is the Fibonacci sequence. Moreover, with fixed length$\ell$, the density of such realizable syzygy sequences of length$\ell$for the$N$-center problem approaches one as$N$increases to infinity. Using reflection symmetry, the minimizers that we found can be extended to periodic solutions.

We refine a well-known theorem of Auslander and Reiten about the extension closedness of n-th syzygies over noether algebras. Applying it, we obtain the converse of a celebrated theorem of Evans and Griffith on Serre's condition (S_n) and the local Gorensteiness of a commutative ring in height less than n. This especially extends a recent result of Araya and Iima concerning a Cohen-Macaulay local ring with canonical module to an arbitrary local ring., 9 pages

Let $R = K[X_1, ..., X_n]$ be a polynomial ring over some field $K$. In this paper, we prove that the $k$-th syzygy module of the residue class field $K$ of $R$ has Stanley depth $n-1$ for $\lfloor n/2 \rfloor \leq k < n$, as it had been conjectured by Bruns et. al. in 2010. In particular, this gives the Stanley depth for a whole family of modules whose graded components have dimension greater than $1$. So far, the Stanley depth is known only for a few examples of this type. Our proof consists in a close analysis of a matching in the Boolean algebra., minor corrections, 11 pages, 4 figures

Let G be a finite group with integral group ring Λ =Z[G]. The syzygies Ωr(Z) are the stable classes of the intermediate modules in a free Λ-resolution of the trivial module. They are of significance in the cohomology theory of G via the “co-represention theorem” Hr(G, N) = Hom𝒟er(Ωr(Z), N). We describe the Ωr(Z) explicitly for the dihedral groups D4n+2, so allowing the construction of free resolutions whose differentials are diagonal matrices over Λ.

In this paper we introduce an algorithm for computing the Stanley depth of a finitely generated multigraded module $M$ over the polynomial ring $\mathbb{K}[X_1, \ldots, X_n]$. As an application, we give an example of a module whose Stanley depth is strictly greater than the depth of its syzygy module. In particular, we obtain complete answers for two open questions raised by Herzog. Moreover, we show that the question whether $M$ has Stanley depth at least $r$ can be reduced to the question whether a certain combinatorially defined polytope $\mathscr{P}$ contains a $\mathbb{Z}^n$-lattice point., 19 pages. Section 5.3 of the first version has been withdrawn. Several typos have been also corrected in this second version. Accepted for publication in Mathematics of Computation

We introduce general weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular disc, triangle, tetrahedral and spherical algebras, are symmetric tame periodic algebras of period 4., Comment: arXiv admin note: text overlap with arXiv:1703.02346