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Let A, B and C be artin algebras such that there is a recollement of D ( Mod A ) relative to D ( Mod B ) and D ( Mod C ) . We compare the algebras A, B and C with respect to syzygy-finite properties and Igusa-Todorov properties under suitable conditions and consider relevant results in the recollements of the bounded derived categories. Further, we characterize when the functor j ⁎ (resp., i ⁎ , i ! ) in a recollement ( D b ( mod B ) , D b ( mod A ) , D b ( mod C ) , i ⁎ , i ⁎ , i ! , j ! , j ⁎ , j ⁎ ) is an eventually homological isomorphism.

Let (R, $$\mathfrak{m}$$ ) be a standard graded K-algebra over a field K. Then R can be written as S/I, where I ⊆ (x1,…, xn)2 is a graded ideal of a polynomial ring S = K[x1,…, xn]. Assume that n ⩽ 3 and I is a strongly stable monomial ideal. We study the symmetric algebra SymR(Syz1( $$\mathfrak{m}$$ )) of the first syzygy module Syz1( $$\mathfrak{m}$$ ) of $$\mathfrak{m}$$ . When the minimal generators of I are all of degree 2, the dimension of SymR (Syz1( $$\mathfrak{m}$$ )) is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.

Let L be a globally generated line bundle over a smooth irreducible complex projective surface X. The syzygy bundle $$M_{L}$$ is the kernel of the evaluation map $$H^0(L)\otimes \mathcal O_X\rightarrow L$$ . We prove the L-stability of $$M_L$$ for Hirzebruch surfaces, del Pezzo surfaces and Enriques surfaces. The $$(-K_X)$$ -stability of syzygy bundles $$M_L$$ over del Pezzo surfaces is also obtained.

First we give a sharp upper bound for the cardinal m of a minimal set of generators for the module of Jacobian syzygies of a complex projective reduced plane curve C. Next we discuss the sharpness of an upper bound, given by A. du Plessis and C.T.C. Wall, for the global Tjurina number of such a curve C, in terms of its degree d and of the minimal degree $$r\le d-1$$ of a Jacobian syzygy. We give a homological characterization of the curves whose global Tjurina number equals the du Plessis-Wall upper bound, which implies in particular that for such curves the upper bound for m is also attained. A second characterization of these curves in terms of the 0-th Fitting ideal of their Jacobian module is also given. Finally we prove the existence of curves with maximal global Tjurina numbers for certain pairs (d, r). We conjecture that such curves exist for any pair (d, r), and that, in addition, they may be chosen to be line arrangements when $$r\le d-2$$ . This conjecture is proved for degrees $$d \le 11$$ .

We give another proof of the recent result of Ringel, which asserts equality between the finitistic dimension and delooping level of Nakayama algebras. The main tool is syzygy filtration method introduced in \cite{sen2019}. In particular, we give characterization of the finitistic dimension one Nakayama algebras., Comments are welcome!

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...

for each squarefree monomial ideal I ⊂ S = k[x1, …, xn], we associate a simple finite graph GI by using the first linear syzygies of I. The nodes of GI are the generators of I, and two vertices ui and uj are adjacent if there exist variables x, y such that xui = yuj. In the cases, where GI is a cycle or a tree, we show that I has a linear resolution if and only if I has linear quotients and if and only if I is variable-decomposable. In addition, with the same assumption on GI, we characterize all squarefree monomial ideals with a linear resolution. Using our results, we characterize all Cohen-Macaulay codimension 2 monomial ideals with a linear resolution. As another application of our results, we also characterize all Cohen-Macaulay simplicial complexes in the case, where $${G_{\rm{\Delta }}} \cong {G_{{I_{\rm{\Delta }}} \vee }}$$ is a cycle or a tree.

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

In computer algebra, it remains to be challenging to establish general computational theories for determining the equivalence of indexed polynomials. In previous work, the author solved the equivalence determination problem for Riemann tensor polynomials by extending Grobner basis theory. This paper extends the previous work to more general indexed polynomials that involve no eliminations of indices and functions, by the method of ST-restricted rings. A decomposed form of the Grobner basis of the defining syzygy set in each ST-restricted ring is provided, and then the canonical form of an indexed polynomial proves to be the normal form with respect to the Grobner basis in the ST-fundamental restricted ring.

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 $(R,\mathfrak m)$ be a local (Noetherian) ring of dimension $d$ and $M$ a finite length $R$-module with free resolution $G_\bullet$. De Stefani, Huneke, and Nunez-Betancourt explored two questions about the properties of resolutions of $M$. First, in characteristic $p>0$, what vanishing conditions on the Frobenius Betti numbers, $\beta_i^F(M, R) : = \lim_{e \to \infty} \lambda(H_i(F^e(G_\bullet)))/p^{ed}$, force pd$_R M < \infty$. Second, if pd$_R M = \infty $, does this force $d+2$nd or higher syzygies of $M$ to have infinite length. For the first question, they showed, under rather restrictive hypotheses, that $d+1$ consecutive vanishing Frobenius Betti numbers forces pd$_R M < \infty$. And when $d=1$ and $R$ is CM then one vanishing Frobenius Betti number suffices. Using properties of stably phantom homology, we show that these results hold in general, i.e., $d+1$ consecutive vanishing Frobenius Betti numbers force pd$_R M < \infty$, and, under the hypothesis that $R$ is CM, $d$ consecutive vanishing Frobenius Betti numbers suffice. For the second question, they obtain very interesting results when $d=1$. In particular, no third syzygy of $M$ can have finite length. Their main tool is, if $d=1$, to show, if the syzygy has a finite length, then it is an alternating sum of lengths of Tors. We are able to prove this fact for rings of arbitrary dimension, which allows us to show that if $d=2$, no third syzygy of $M$ can be finite length! We also are able to show that the question has a positive answer if the dimension of the socle of $H^0_{\mathfrak m}(R)$ is large relative to the rest of the module, generalizing the case of Buchsbaum rings.

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.

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.

Systems of polynomial equations are one of the most universal mathematical objects. Almost all problems of cryptographic analysis can be reduced to solving systems of polynomial equations. The corresponding direction of research is called algebraic cryptanalysis. In terms of computational complexity, systems of polynomial equations cover the entire range of possible variants, from the algorithmic insolubility of Diophantine equations to well-known efficient methods for solving linear systems. Buchberger’s method [5] brings the system of algebraic equations to a system of a special type defined by the Grobner original system of equations, which enables the elimination of dependent variables. The Grobner basis is determined based on an admissible ordering on a set of terms. The set of admissible orderings on the set of terms is infinite and even continual. The most time-consuming step in finding the Grobner basis by using Buchberger’s algorithm is to prove that all S-polynomials represent a system of generators of K[X]-module S-polynomials. Thus, a natural problem of finding this minimal system of generators arises. The existence of this system follows from Nakayama’s lemma. In this paper, we propose an algorithm for constructing this basis for any ordering.

Let C be a smooth complex irreducible projective curve of genus g, and let $$(L,H^0(L))$$ be a generated complete linear series of type $$(d,r+1)$$ over C. The syzygy bundle, denoted by $$M_L$$ , is the kernel of the evaluation map . The aim of this paper is twofold. Firstly, to give new examples of stable syzygy bundles admitting a theta divisor over Petri curves. We prove that if $$M_L$$ is strictly semistable then $$M_L$$ admits a theta divisor. Secondly, to study the cohomological semistability of $$M_L$$ , and in this direction we give another proof of cohomological semistability of $$M_L$$ when L induces a birational map. This proof gives us precise conditions for the cohomological semistability of $$M_L$$ where such conditions agree with the semistability conditions for $$M_L$$ . Finally, we relate these two properties by showing that under certain conditions on Petri curves the cohomological semistability of $$M_L$$ implies the existence of reducible theta divisor for $$M_L$$ .

Isaac ben Solomon Ibn al-Ḥadib (or al-Aḥdab) emigrated from Castile to Sicily no later than 1396. In astronomy, his most important work, written in Hebrew, is The paved way ( Oraḥ selula), a set of tables for the motions of the Sun and the Moon. Here, we focus attention on his unusual tables for finding the difference in time and the difference in longitude between mean and true syzygy, where syzygy refers to the conjunction and opposition of the Sun and the Moon. It is shown that he took into account the effect of Ptolemy’s second lunar model on the velocity of the Moon at syzygy, which was done by very few astronomers in the Middle Ages. It is also noteworthy that he took some parameters from the zij of al-Battānī and others from the Parisian Alfonsine Tables, using them inconsistently in these tables.

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.

This article presents the central paradigms of the understanding of love by early twentieth-century Russian philosophers. It shows that interpreting the meaning of erotic love as a way of overcomin...

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.

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.

We show that if R is a two dimensional standard graded domain (with the graded maximal ideal m) of characteristic p > 0 and I ⊂ R is a graded ideal with l ( R / I ) ∞ , then the F-threshold c I ( m ) can be expressed in terms of a strong HN (Harder-Narasimhan) slope of the canonical syzygy bundle on Proj R . Thus c I ( m ) is a rational number. This gives us a well defined notion, of the F-threshold c I ( m ) in characteristic 0, in terms of a HN slope of the syzygy bundle on Proj R . This generalizes our earlier result (in [13] ) where we have shown that if I has homogeneous generators of the same degree, then the F-threshold c I ( m ) is expressed in terms of the minimal strong HN slope (in char p) and in terms of the minimal HN slope (in char 0), respectively, of the canonical syzygy bundle on Proj R . In the present more general setting, the relevant slope may not be the minimal one. Here we also prove that, for a given pair ( R , I ) over a field of characteristic 0, if ( m p , I p ) is a reduction mod p of ( m , I ) then c I p ( m p ) ≠ c ∞ I ( m ) implies c I p ( m p ) has p in the denominator, for almost all p.

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.

A quaternion rational surface is a rational surface generated by two rational space curves via quaternion multiplication. In general, the structure of the graded minimal free resolution of a rational surface is unknown. The goal of this paper is to construct the graded minimal free resolution of a quaternion rational surface generated by two rational space curves. We will provide the explicit formulas for the maps of these graded minimal free resolutions. The approach we take is to utilize the information of the [math] -bases of the generating rational curves, and create the generating sets for the first and second syzygy modules in the graded minimal free resolutions. In addition, we show that the ideal generated by the first syzygy module expressed in terms of moving planes is exactly the same as the ideal generated by the parametrization in the affine ring.

There are some classical and elementary applications of Grobner theory that we will study in this chapter. We will consider the membership problem, and we will compute the syzygy module, free resolutions of modules, the intersection and quotient of ideals and submodules, the matrix presentation of a finitely presented module, and the kernel and the image of homomorphism between modules.

If A is an artin algebra, G\'elinas has introduced an interesting upper bound for the finitistic dimension of A, namely the delooping level del A. We assert that for any Nakayama algebra, its finitistic dimension is equal to del A. This yields also a new proof that the finitistic dimension of A and of its opposite algebra are equal, as shown recently by Sen. If S is a simple module, let e(S) be the minimum of the projective dimension of S and of its injective envelope (one of these numbers has to be finite); and e*(S) the minimum of the injective dimension of S and of its projective cover. Then the finitistic dimension of A is the maximum of the numbers e(S) as well as the maximum of the numbers e^*(S). Using suitable syzygy modules, we construct a permutation h of the simple modules S such that e*(h(S)) = e(S). In particular, this shows for any natural number z, that the number of simple modules S with e(S) = z is equal to the number of simple modules S' with e(S') = z., Comment: The paper has been slightly revised, some remarks (which hopefully are helpful) have been added

A city is an identity in manifold presentations or profiles, some of which are reviewed here. Why and how do cities and planning allow for so many differing and apparently incompatible analogies (and theories)? To ask, What is really going on? is perhaps to miss that manifold. For cities allow for regionalized analogies, overlapping and incompatible, and many may be valid, at least regionally, at the same time. More generally, we describe thinking in analogy.

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

We construct an explicit free resolution T for a maximal Cohen-Macaulay module M over a local complete intersection of codimension 2 with infinite residue field. The resolution is minimal when the module M is a sufficiently high syzygy. Our starting point is a layered free resolution L, described in [7], of length 2 over a regular local ring. We provide explicit formulas for the differential in T in terms of the differential and homotopies on the finite resolution L. One application of our construction is to describe Ulrich modules over a codimension 2 quadratic complete intersection.

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.

In this paper, we study commutative local rings whose residue field is a high syzygy. We are able to characterize such rings with embedding dimension at most 3 as those that are Artinian an...

For any given Feynman graph, the set of integrals with all possible powers of the propagators spans a vector space of finite dimension. We introduce the package {\sc Azurite} ({\bf A ZUR}ich-bred method for finding master {\bf I}n{\bf TE}grals), which efficiently finds a basis of this vector space. It constructs the needed integration-by-parts (IBP) identities on a set of generalized-unitarity cuts. It is based on syzygy computations and analyses of the symmetries of the involved Feynman diagrams and is powered by the computer algebra systems {\sc Singular} and {\sc Mathematica}. It can moreover analytically calculate the part of the IBP identities that is supported on the cuts., Comment: Version 1.1.0 of the package Azurite, with parallel computations. It can be downloaded from https://bitbucket.org/yzhphy/azurite/raw/master/release/Azurite_1.1.0.tar.gz

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.

Dedicated to Donald F. Sandner, MDFramed by the biblical tale of Moses and the golden calf and an archetypal spectrum of solar spirit and its earthly and bodily aspects described by Mircea Eliade, ...

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 .