We show that the Koszul homology algebra of a quotient by the edge ideal of a forest is generated by the lowest linear strand. This provides a large class of Koszul algebras whose Koszul homology algebras satisfy this property. We obtain this result by constructing the minimal graded free resolution of a quotient by such an edge ideal via the so called iterated mapping cone construction and using the explicit bases of Koszul homology given by Herzog and Maleki. Using these methods we also recover a result of Roth and Van Tuyl on the graded Betti numbers of quotients of edge ideals of trees.
We investigate various module-theoretic properties of Koszul homology under mild conditions. These include their depth, $S_2$-property and their Bass numbers
The Koszul homology algebra of a commutative local (or graded) ring $R$ tends to reflect important information about the ring $R$ and its properties. In fact, certain classes of rings are characterized by the algebra structure on their Koszul homologies. In this paper we survey some classical results on the Koszul homology algebras of such rings and highlight some applications. We report on recent progress on the Koszul homology algebras of Koszul algebras and examine some open questions on the topic.
In this note, we calculate the Koszul homology of the codimension 3 Gorenstein ideals. We find filtrations for the Koszul homology in terms of modules with pure free resolutions and completely describe these resolutions. We also consider the Huneke-Ulrich deviation 2 ideals.
It gives a class of $p$-Borel principal ideals of a polynomial algebra over a field $K$ for which the graded Betti numbers do not depend on the characteristic of $K$ and the Koszul homology modules have monomial cyclic basis. Also it shows that all principal $p$-Borel ideals have binomial cycle basis on Koszul homology modules.
Given a Serre class $\mathcal{S}$ of modules, we compare the containment of the Koszul homology, Ext modules, Tor modules, local homology, and local cohomology in $\mathcal{S}$ up to a given bound $s \geq 0$. As some applications, we give a full characterization of noetherian local homology modules. Further, we establish a comprehensive vanishing result which readily leads to the formerly known descriptions of the numerical invariants width and depth in terms of Koszul homology, local homology, and local cohomology. Also, we immediately recover a few renowned vanishing criteria scattered about…
We show that the Koszul homology algebra of the second Veronese subalgebra of a polynomial ring over a field of characteristic zero is generated, as an algebra, by the homology classes corresponding to the syzygies of the lowest linear strand.
Let $R$ be a standard graded commutative algebra over a field $k$, let $K$ be its Koszul complex viewed as a differential graded $k$-algebra, and let $H$ be the homology algebra of $K$. This paper studies the interplay between homological properties of the three algebras $R$, $K$, and $H$. In particular, we introduce two definitions of Koszulness that extend the familiar property originally introduced by Priddy: one which applies to $K$ (and, more generally, to any connected differential graded $k$-algebra) and the other, called strand-Koszulness, which applies to $H$. The main theoretical res…
We study Koszul homology over Gorenstein rings. If an ideal is strongly Cohen-Macaulay, the Koszul homology algebra satisfies Poincaré duality. We prove a version of this duality which holds for all ideals and allows us to give two criteria for an ideal to be strongly Cohen-Macaulay. The first can be compared to a result of Hartshorne and Ogus; the second is a generalization of a result of Herzog, Simis, and Vasconcelos using sliding depth.
With a particular focus on explicit computations and applications of the Koszul homology and Betti numbers of monomial ideals, the main goals of this thesis are the following: Analyze the Koszul homology of monomial ideals and apply it to describe the structure of monomial ideals. Describe algorithms to perform efficient computations of the homological invariants of monomial ideals. Apply the theory and computations on monomial ideals to problems inside and outside mathematics The thesis introduces as a main tool Mayer-Vietoris trees of monomial ideals.
Let $k$ be a field and $R$ a standard graded $k$-algebra. We denote by $\operatorname{H}^R$ the homology algebra of the Koszul complex on a minimal set of generators of the irrelevant ideal of $R$. We discuss the relationship between the multiplicative structure of $\operatorname{H}^R$ and the property that $R$ is a Koszul algebra. More generally, we work in the setting of local rings and we show that certain conditions on the multiplicative structure of Koszul homology imply strong homological properties, such as existence of certain Golod homomorphisms, leading to explicit computations of Po…
Differential modules are natural generalizations of complexes. In this paper, we study differential modules with complete intersection homology, comparing and contrasting the theory of these differential modules with that of the Koszul complex. We construct a Koszul differential module that directly generalizes the classical Koszul complex and investigate which properties of the Koszul complex can be generalized to this setting.
We discuss a class of modules, which we call $\underline{g}$-weak complete intersection modules, inspired by the weak complete intersection ideals studied by Rahmati, Striuli, and Yang and we present explicit formulas for the generators of Koszul homology with coefficients in a $\underline{g}$-weak complete intersection module. This generalizes work of Herzog and of Corso, Goto, Huneke, Polini, and Ulrich. We use these explicit formulas to study connections between $\underline{g}$-weak complete intersection ideals and weak complete intersection ideals.
We study the conjecture that if $I,J$ are $μ-$primary in a regular local ring $(R,μ)$ with dim$(R)=n$, then $\frac{I \cap J}{IJ} \cong \Tor_1(R/I, R/J)$ needs at least $n$ generators, and a related conjecture about the number of generators of the first Koszul homology module of an Artinian local ring $(A,m)$. In this manuscript, we focus our attention on the complete intersection defect of the Artinian ring and its quotient by the Koszul elements. We prove that the number of generators of the first Koszul homology module of $x_1,...,x_n \in m$ on an Artinian local ring $(A,m)$ is at least $n+ …
In a polynomial ring $R$ with $n$ variables, for every homogeneous ideal $I$ and for every $p\leq n$ we consider the Koszul homology $H_i(p,R/I)$ with respect to a sequence of $p$ of generic linear forms and define the Koszul-Betti number $β_{ijp}(R/I)$ of $R/I$ to be the dimension of the degree $j$ part of $H_i(p,R/I)$. In characteristic 0, we show that the Koszul-Betti numbers of any ideal $I$ are bounded above by those of any gin of $I$ and also by those of the Lex-segment of $I$. We also investigate the set $Gins(I)$ of all the gin of $I$ and show that the Koszul-Betti numbers of any ideal…
Let $M$ be a finite module over a noetherian ring $R$ with a free resolution of length 1. We consider the generalized Koszul complexes $\mathcal{C}_{\barλ}(t)$ associated with a map $\barλ:M\to\mathcal{H}$ into a finite free $R$-module $\mathcal{H}$ (see [IV], section 3), and investigate the homology of $\mathcal{C}_{\barλ}(t)$ in the special setup when $\grade I_M=\rank M=\dim R$. ($I_M$ is the first non-vanishing Fitting ideal of $M$.) In this case the (interesting) homology of $\mathcal{C}_{\barλ}(t)$ has finite length, and we deduce some length formulas. As an application we give a short a…
Let $(R,\mathfrak{m})$ be a Noetherian local ring, and let $M$ be a finitely generated $R$-module of dimension $d$. We prove that the set $\left\{\frac{l(M/IM)}{e(I, M)} \right\}_{\sqrt{I}=\mathfrak{m}}$ is bounded below by ${1}/{d!e(\overline{R})}$ where $\overline{R}=R/Ann(M)$. Moreover, when $\widehat{M}$ is equidimensional, this set is bounded above by a finite constant depending only on $M$. The lower bound extends a classical inequality of Lech, and the upper bound answers a question of Stückrad--Vogel in the affirmative. As an application, we obtain results on uniform behavior of the le…
The Koszul homology of modules of the polynomial ring $R$ is a central object in commutative algebra.It is strongly related with the minimal free resolution of these modules, and thus with regularity, Hilbert functions, etc. Here we consider the case of modules of the form $R/I$ where $I$ is a monomial ideal. So far, some good algorithms have been given in the literature and implemented in different Computer Algebra Systems (e.g. CoCoa, Singular, Macaulay), which compute minimal free resolutions of modules of the form $R/I$ with $I$ an ideal in $R$, which include the case of $I$ being a monomi…
This paper introduces the study of LG-quadratic quotients of exterior algebras, showing that they are Koszul, as in the commutative case. We construct an example of an LG-quadratic algebra that is not G-quadratic and another example that is Koszul but not LG-quadratic. This is only the second known Koszul algebra that is not LG-quadratic and the first that is noncommutative.
Koszul homology of monomial ideals provides a description of the structure of such ideals, not only from a homological point of view (free resolutions, Betti numbers, Hilbert series) but also from an algebraic viewpoint. In this paper we show that, in particular, the homology at degree (n-1), with n the number of indeterminates of the ring, plays an important role for this algebraic description in terms of Stanley and irreducible decompositions.
We extend to one dimensional quotients the result of A. Conca and S. Murai on the convexity of the regularity of Koszul cycles. By providing a relation between the regularity of Koszul cycles and Koszul homologies we prove a sharp regularity bound for the Koszul homologies of a homogeneous ideal in a polynomial ring under the same conditions.
Let R be a commutative noetherian ring. Let M be a finitely generated R-module. In this paper, we reconstruct M from its Koszul homology with respect to a suitable sequence of elements of R by taking direct summands, syzygies and extensions, and count the number of those operations. Using this result, we consider generation and classification of certain subcategories of the category of finitely generated R-modules, its bounded derived category and the singularity category of R.
Let C be a semidualizing complex over a noetherian local ring A. If there exists a local homomorphism with source A satisfying some homological properties, then C is dualizing.
For a system of non-homogeneous polynomials it was constructed explicit complex morphism of a dual complex to the Koszul complex into the Koszul complex. If the ideal of these polynomials is 0-dimensional, then this mapping is a homotopic equivalence, thus in this case it was obtained explicit duality of the Koszul complex.
Let $S$ be the power series ring or the polynomial ring over a field $K$ in the variables $x_1,\ldots,x_n$, and let $R=S/I$, where $I$ is proper ideal which we assume to be graded if $S$ is the polynomial ring. We give an explicit description of the cycles of the Koszul complex whose homology classes generate the Koszul homology of $R=S/I$ with respect to $x_1,\ldots,x_n$. The description is given in terms of the data of the free $S$-resolution of $R$. The result is used to determine classes of Golod ideals, among them proper ordinary powers and proper symbolic powers of monomial ideals. Our t…
We show that the graded maximal ideal of a graded $K$-algebra $R$ has linear quotients for a suitable choice and order of its generators if the defining ideal of $R$ has a quadratic Gröbner basis with respect to the reverse lexicographic order, and show that this linear quotient property for algebras defined by binomial edge ideals characterizes closed graphs. Furthermore, for algebras defined by binomial edge ideals attached to a closed graph and for join-meet rings attached to a finite distributive lattice we present explicit Koszul filtrations.
Several spectral sequence techniques are used in order to derive information about the structure of finite free resolutions of graded modules. These results cover estimates of the minimal number of generators of defining ideals of projective varieties. There are also investigations about the shifts and the dimension of Betti numbers.
Let $R$ be a local commutative noetherian ring and $HKR$ the homology ring of the corresponding Koszul complex. We study the homological properties of $HKR$ in particular the so-called Avramov spectral sequence. When the embedding dimension of $R$ is four and when $R$ can be presented with quadratic relations we have found 102 cases where this spectral sequence degenerates and only three cases where it does not degenerate. We also determine completely the Hilbert series of the bigraded Tor of these $HKR$ in tables A-D of section 5. We also study some higher embedding dimensions. Among the meth…
This work concerns the Koszul complex $K$ of a commutative noetherian local ring $R$, with its natural structure as differential graded $R$-algebra. It is proved that under diverse conditions, involving the multiplicative structure of $H(K)$, any dg $R$-algebra automorphism of $K$ induces the identity map on $H(K)$. In such cases, it is possible to define an action of the automorphism group of $R$ on $H(K)$. On the other hand, numerous rings are described for which $K$ has automorphisms that do not induce the identity on $H(K)$. For any $R$, it is shown that the group of automorphisms of $H(K)…
Let R be a commutative ring and I an ideal in R which is locally generated by a regular sequence of length d. Then, each projective R/I-module V has an R-projective resolution P. of length d. In this paper, we compute the homology of the n-th Koszul complex associated with the homomorphism P_1 --> P_0 for all n, if d = 1. This computation yields a new proof of the classical Adams-Riemann-Roch formula for regular closed immersions which does not use the deformation to the normal cone any longer. Furthermore, if d = 2, we compute the homology of the complex N Sym^2 K(P.) where K and N denote the…
We present a monomial ideal $\mathfrak{a} \subset S$ such that $S/\mathfrak{a}$ is not Golod, even though the product on its Koszul homology is trivial. This constitutes a counterexample to a well-known result by Berglund and Jöllenbeck (the error can be traced to a mistake in an earlier article by Jöllenbeck). On the positive side, we show that if $R$ is a monomial ring such that the $r$-ary Massey product vanish for all $r \leq \max(2, \mathrm{reg} R-2)$, then $R$ is Golod. In particular, if $R$ is the Stanley-Reisner ring of a simplicial complex of dimension at most $3$, then $R$ is Golod i…
A graded K-algebra R has property N_p if it is generated in degree 1, has relations in degree 2 and the syzygies of order less or equal to p on the relations are linear. The Green-Lazarsfeld index of R is the largest p such that it satisfies the property N_p. Our main results assert that (under a mild assumption on the base field K) the c-th Veronese subring of a polynomial ring has Green-Lazarsfeld index greater or equal to c+1. The same conclusion also holds for an arbitrary standard graded algebra, provided c is sufficiently large.
We will describe how we can identify the structure of the Koszul algebra for trivariate monomial ideals from minimal free resolutions. We use recent work of L. Avramov, where he classifies the behavior of Bass numbers of embedding codepth 3 commutative local rings. His classification relies on a corresponding classification of their respective Koszul algebras, which is comprised of 5 categories. Using Avramov's classification of the Koszul algebra, along with their respective Bass series we will learn how to identify the Koszul algebra structure by inspecting the minimal free resolution of the…
This article outgrew from an effort to understand our basic question: Are the annihilators of the non-zero Koszul homology modules $H_i$ of an unmixed ideal $I$ contained in the integral closure $\bar{I}$ of $I$? We also obtain some variations on a result of Burch, which continue the theme of the paper in that they deal with annihilators of homology and integrally closed ideals.
In this paper we extend the well-known iterated mapping cone procedure to monomial ideals in strongly Koszul algebras. We study properties of ideals generated by monomials in commutative Koszul algebras and show that the linear strand of ideals generated by linear forms is obtained as a subcomplex of the Priddy complex. In the case of strongly Koszul algebras, this shows that the minimal free resolution of a monomial ideal admitting linear quotients is obtained as an iterated mapping cone, immediately extending results for such ideals in polynomial rings to strongly Koszul algebras. We then co…
Conditions on the Koszul complex of a noetherian local ring $R$ guarantee that $\mathrm{Tor}^{R}_{i}(M,N)$ is non-zero for infinitely many $i$, when $M$ and $N$ are finitely generated $R$-modules of infinite projective dimension. These conditions are obtained from results concerning Tor of differential graded modules over certain trivial extensions of commutative differential graded algebras.
All powers of lexsegment ideals with linear resolution (equivalently, with linear quotients) have linear quotients with respect to suitable orders of the minimal monomial generators. For a large subclass of the lexsegment ideals the corresponding Rees algebra has a quadratic Gröbner basis, thus it is Koszul. We also find other classes of monomial ideals with linear quotients whose powers have linear quotients too.
Let $S$ be a regular local ring or a polynomial ring over a field and $I$ be an ideal of $S$. Motivated by a recent result of Herzog and Huneke, we study the natural question of whether $I^m$ is a Golod ideal for all $m\geq 2$. We observe that the Golod property of an ideal can be detected through the vanishing of certain maps induced in homology. This observation leads us to generalize some known results from the graded case to local rings and obtain new classes of Golod ideals.
We study the Koszul property of a standard graded $K$-algebra $R$ defined by the binomial edge ideal of a pair of graphs $(G_1,G_2)$. We show that the following statements are equivalent: (i) $R$ is Koszul; (ii) the defining ideal $J_{G_1,G_2}$ of $R$ has a quadratic Gröbner basis; (iii) the graded maximal ideal of $R$ has linear quotients with respect to a suitable order of its generators
Estimates are obtained for the degrees of minimal syzygies of quotient algebras of polynomial rings. For a class that includes Koszul algebra in almost all characteristics, these degrees are shown to increase by at most 2 from one syzygy module to the next one. Even slower growth is proved if, in addition, the algebra satisfies Green and Lazarsfeld's condition N_q with q > 1.
Two ideals $I$ and $J$ are called transverse if $I \cap J = IJ$. We show that the obstructions defined by Avramov for classes of (sequentially) transverse ideals in regular local rings are always $0$. In particular, we compute an explicit free resolution and Koszul homology for all such ideals. Moreover, we construct an explicit trivial Massey operation on the associated Koszul complex and hence (by Golod's construction) a minimal free resolution of the residue field over the quotient defined by the product of transverse ideals. We conclude with questions about the existence of associative mul…
We conjecture that the stable Khovanov homology of torus knots can be described as the Koszul homology of an explicit non-regular sequence of quadratic polynomials. The corresponding Poincare series turns out to be related to the Rogers-Ramanujan identity.
Trivariate Splines on Fans of Hyperplane Arrangements and Koszul Homology
University of Copenhagen（哥本哈根大学）；New Mexico State University（新墨西哥州立大学）；CUNEF Universidad（CUNEF大学）；University of Dayton（代顿大学）；Hue University（顺化大学）；University of Missouri-Kansas City（密苏里大学堪萨斯城分校）；Western University（韦仕敦大学）；McMaster University（麦克马斯特大学）；Swansea University（斯旺西大学）
We construct a free resolution of $R/I^s$ over $R$ where $I\ideal R$ is generated by a (finite or infinite) regular sequence. This generalizes the Koszul complex for the case $s=1$. For $s>1$, we easily deduce that the algebra structure of $\Tor^R_*(R/I,R/I^s)$ is trivial and the reduction map $R/I^s\lra R/I^{s-1}$ induces the trivial map of algebras.
The algebra of basic covers of a graph G, denoted by \A(G), was introduced by Juergen Herzog as a suitable quotient of the vertex cover algebra. In this paper we show that if the graph is bipartite then \A(G) is a homogeneous algebra with straightening laws and thus is Koszul. Furthermore, we compute the Krull dimension of \A(G) in terms of the combinatorics of G. As a consequence we get new upper bounds on the arithmetical rank of monomial ideals of pure codimension 2. Finally, we characterize the Cohen-Macaulay property and the Castelnuovo-Mumford regularity of the edge ideal of a certain cl…
In a paper in 1962, Golod proved that the Betti sequence of the residue field of a local ring attains an upper bound given by Serre if and only if the homology algebra of the Koszul complex of the ring has trivial multiplications and trivial Massey operations. This is the origin of the notion of Golod ring. Using the Koszul complex components he also constructed a minimal free resolution of the residue field. In this article, we extend this construction up to degree five for any local ring. We describe how the multiplicative structure and the triple Massey products of the homology of the Koszu…
Cohen-Macaulayness, unmixedness, the structure of the canonical module and the stability of the Hilbert function of algebraic residual intersections are studied in this paper. Some conjectures about these properties are established for large classes of residual intersections without restricting local number of generators of the ideals involved. A family of approximation complexes for residual intersections is constructed to determine the above properties. Moreover some general properties of the symmetric powers of quotient ideals are determined which were not known even for special ideals with…
Let $G$ be a simple graph on the vertex set $V(G) = [n] = \{1,...,n\}$ and edge ideal $E(G)$. We consider the class of closed graphs. A closed graph is a simple graph satisfying the following property: for all edges $\{i, j\}$ and $\{k, \ell\}$ with $i < j$ and $k < \ell$ one has $\{j, \ell\}\in E(G)$ if $i = k$, and $\{i, k\}\in E(G)$ if $j = \ell$. We state some criteria for the closedness of a graph $G$ that do not depend necessarily from the labelling of its vertex set. Consequently, if $S = K[x_1,..., x_n, y_1,..., y_n]$ is a polynomial ring in $2n$ variables with coefficients in a field …
