We consider the minimal free resolution of a generic set of n+1 forms (not necessarily of the same degree) in a polynomial ring of n variables. The Hilbert function for such an ideal is known, thanks to a result of Stanley and of Watanabe. (For more than n+1 forms the Hilbert function is conjectured by Froberg.) However, in all but a few trivial cases, the resolution is not known. A conjecture of Iarrobino says that apart from Koszul syzygies, there should be no overlaps in the twists occurring in the free modules in the resolution. We are able to find the precise minimal free resolutions in many cases (depending on n and on the particular degrees) and give good bounds in many other cases. As a result, we see that Iarrobino's conjecture does not hold and we begin to have an understanding of what causes this failure. We conjecture that there are no such overlaps when all n+1 generators have the same degree. In the process of achieving these results, we extend a result of Boij by giving the graded Betti numbers of a generic Gorenstein algebra when n is even and the socle degree is large.
In this paper we prove that the generalized version of the Minimal Resolution Conjecture stated by Mustata holds for certain general sets of points on a smooth cubic surface $X \subset \mathbb{P}^3$. The main tool used is Gorenstein liaison theory and, more precisely, the relationship between the free resolutions of two linked schemes.
We investigate the standard generalized Gorenstein algebras of homological dimension three, giving a structure theorem for their resolutions. Moreover in many cases we are able to give a complete description of their graded Betti numbers.
In recent years, two different multigraded variants of Castelnuovo-Mumford regularity have been developed, namely multigraded regularity, defined by the vanishing of multigraded pieces of local cohomology modules, and the resolution regularity vector, defined by the multidegrees in a minimal free resolution. In this paper, we study the relationship between multigraded regularity and the resolution regularity vector. Our method is to investigate multigraded variants of the usual a*-invariant. This, in particular, provides an effective approach to examining the vanishing of multigraded pieces of…
Let Z be a fat point scheme in P^2 supported on general points. Here we prove that if the multiplicities are at most 3 and the length of Z is sufficiently high then the number of generators of the homogeneous ideal I_Z in each degree is as small as numerically possible. Since it is known that Z has maximal Hilbert function, this implies that Z has the expected minimal free resolution.
We formulate a conjecture on the behavior of the minimal free resolutions of sets of general points on arbitrary varieties embedded by complete linear series, in analogy with the well-known Minimal Resolution Conjecture for points in projective space. We then study in detail the case of curves. Our main result is that the Minimal Resolution Conjecture holds for general points on any canonical curve. On the other hand, for curves embedded with large degree, it always fails at a well-specified spot in the Betti diagram. The results are shown via techniques involving theta divisors associated to …
Let $A$ be a semigroup whose only invertible element is 0. For an $A$-homogeneous ideal we discuss the notions of simple $i$-syzygies and simple minimal free resolutions of $R/I$. When $I$ is a lattice ideal, the simple 0-syzygies of $R/I$ are the binomials in $I$. We show that for an appropriate choice of bases every $A$-homogeneous minimal free resolution of $R/I$ is simple. We introduce the gcd-complex $D_{gcd}(\bf b)$ for a degree $\mathbf{b}\in \A$. We show that the homology of $D_{gcd}(\bf b)$ determines the $i$-Betti numbers of degree $\bf b$. We discuss the notion of an indispensable c…
Let S be a standard N^k-graded polynomial ring over a field. Let I be a multigraded homogeneous ideal in S and let M be a finitely generated Z^k-graded S-module. We prove that the resolution regularity, a multigraded variant of Castelnuovo-Mumford regularity, of I^nM is asymptotically a linear function. This shows that the well known Z-graded phenomenon carries to multigraded situation.
Hochster established the existence of a commutative noetherian ring $\tilde C$ and a universal resolution $U$ of the form $0\to \tilde C^{e}\to \tilde C^{f}\to \tilde C^{g}\to 0$ such that for any commutative noetherian ring $S$ and any resolution $V$ equal to $0\to S^{e}\to S^{f}\to S^{g}\to 0$, there exists a unique ring homomorphism $\tilde C\to S$ with $V=U\otimes_{\tilde C} S$. In the present paper we assume that $f=e+g$ and we find the minimal resolution of ${\bf K}\otimes \tilde C$ by free $B$-modules, where $\bf K$ is a field of characteristic zero and $B$ is a polynomial ring over $\b…
We propose a notion of minimal free resolutions for differential modules, and we prove existence and uniqueness results for such resolutions. We also take the first steps toward studying the structure of minimal free resolutions of differential modules. Our main result in this direction explains a sense in which the minimal free resolution of a differential module is a deformation of the minimal free resolution of its homology; this leads to structural results that mirror classical theorems about minimal free resolutions of modules.
Minimal free resolutions of graded modules over a noetherian polynomial ring have been attractive objects of interest for more than a hundred years. We introduce and study two natural extensions in the setting of graded modules over a polynomial OI-algebra, namely minimal and width-wise minimal free resolutions. A minimal free resolution of an OI-module can be characterized by the fact that the free module in every fixed homological degree, say $i$, has minimal rank among all free resolutions of the module. We show that any finitely generated graded module over a noetherian polynomial OI-algeb…
We discuss the minimal free resolution of an irreducible projective subscheme X. If X is also reduced, we focus on the case when its degree equals two plus the codimension. The set of all possible graded Betti numbers is described if the codimension is two. In general, there are only partial results. Then we discuss Cohen-Macaulay structures on a linear subspace. For such curves of degree two, the classification has been obtained by Notari, Spreafico, and the author. In general, the classification problem seems rather difficult. Vatne has shown that such a classification of structures whose de…
We use the lcm-lattice of a monomial ideal to study its minimal free resolutions. A new concept called a Taylor basis of a minimal free resolution is introduced and then used throughout the paper. We give a method of constructing minimal free resolutions of a monomial ideal from its lcm-lattice, which is called the atomic lattice resolution theory. Some applications of this theory is given. As the main application, we rewrite the theory of poset resolutions, and we obtain an approximation formula for minimal free resolutions of all monomial ideals.
We consider powers of lexsegment ideals with a linear resolution (equivalently, with linear quotients) which are not completely lexsegment ideals. We give a complete description of their minimal graded free resolution.
A relatively compressed algebra with given socle degrees is an Artinian quotient $A$ of a given graded algebra $R/\fc$, whose Hilbert function is maximal among such quotients with the given socle degrees. For us $\fc$ is usually a ``general'' complete intersection and we usually require that $A$ be level. The precise value of the Hilbert function of a relatively compressed algebra is open, and we show that finding this value is equivalent to the Fröberg Conjecture. We then turn to the minimal free resolution of a level algebra relatively compressed with respect to a general complete intersecti…
A current research theme is to compare symbolic powers of an ideal $I$ with the regular powers of $I$. In this paper, we focus on the case that $I=I_X$ is an ideal defining an almost complete intersection (ACI) sets of points $X$ in $\mathbb{P}^1\times\mathbb{P}^1$. In particular, we describe a minimal free bigraded resolution of a non arithmetically Cohen-Macaulay (also non homogeneus) set of fat points $\mathcal Z$ whose support is an ACI. We call $\mathcal Z$ a fat ACI. We also show that its symbolic and ordinary powers are equal, i.e, $I_{\mathcal Z}^{(m)}=I_{\mathcal Z}^{m}$ for any $m\ge…
We show that the minimal resolution of the quotient of the polynomial algebra over a field by a cointerval edge ideal can be given the structure of a DG-algebra.
In this article, we give the explicit minimal free resolution of the associated graded ring of certain affine monomial curves in affine 4-space based on the standard basis theory. As a result, we give the minimal graded free resolution and compute the Hilbert function of the tangent cone of these families.
We study when Taylor resolutions of monomial ideals are minimal. We consider monomial ideals with linear quotients. In particular, we determine precisely the stable ideals and the monomial ideals with linear resolutions having the miminal Taylor resolutions.
We use the theory of poset resolutions to construct the minimal free resolution of an arbitrary stable monomial ideal in the polynomial ring whose coefficients are from a field. This resolution is recovered by utilizing a poset of Eliahou-Kervaire admissible symbols associated to a stable ideal. The structure of the poset under consideration is quite rich and in related analysis, we exhibit a regular CW complex which supports a minimal cellular resolution of a stable monomial ideal.
We give a necessary and sufficient condition on a homogeneous polynomial ideal for its Taylor complex to be exact. Then we give a combinatorial construction of a minimal resolution for ideals satisfying the above condition (in particular for monomial ideals).
We construct the minimal resolutions of three classes of monomial ideals: dominant, 1-semidominant, and 2-semidominant ideals. The families of dominant and 1-semidominant ideals extend those of complete and almost complete intersections. We show that dominant ideals give a precise characterization of when the Taylor resolution is minimal, 1-semidominant ideals are Scarf, and the minimal resolutions of 2-semidominant ideals can be obtained from their Taylor resolutions by eliminating faces and facets of equal multidegree, in arbitrary order. We study the combinatorial properties of these classe…
Numerical invariants of a minimal free resolution of a module $M$ over a regular local ring $(R,\n)$ can be studied by taking advantage of the rich literature on the graded case. The key is to fix suitable $\n$-stable filtrations ${\mathbb M} $ of $M $ and to compare the Betti numbers of $M$ with those of the associated graded module $ gr_{\mathbb M}(M). $ This approach has the advantage that the same module $M$ can be detected by using different filtrations on it. It provides interesting upper bounds for the Betti numbers and we study the modules for which the extremal values are attained. Am…
For a monomial ideal $I$, let $G(I)$ be its minimal set of monomial generators. If there is a total order on $G(I)$ such that the corresponding Lyubeznik resolution of $I$ is a minimal free resolution of $I$, then $I$ is called a Lyubeznik ideal. In this paper, we characterize the Lyubeznik ideals, and we discover some classes of Lyubeznik ideals.
Matrix factorizations of a hypersurface yield a description of the asymptotic structure of minimal free resolutions over the hypersurface. We introduce a new concept of matrix factorizations for complete intersections that allows us to describe the asymptotic structure of minimal free resolutions over complete intersections.
A tetrahedral curve is an unmixed, usually non-reduced, one-dimensional subscheme of projective 3-space whose homogeneous ideal is the intersection of powers of the ideals of the six coordinate lines. The second and third authors have shown that these curves have very nice combinatorial properties, and they have made a careful study of the even liaison classes of these curves. We build on this work by showing that they are "almost always" componentwise linear, i.e. their homogeneous ideals have the property that for any d, the degree d component of the ideal generates a new ideal whose minimal…
Mustaţă has given a conjecture for the graded Betti numbers in the minimal free resolution of the ideal of a general set of points on an irreducible projective algebraic variety. For surfaces in $\mathbb P^3$ this conjecture has been proven for points on quadric surfaces and on general cubic surfaces. In the latter case, Gorenstein liaison was the main tool. Here we prove the conjecture for general quartic surfaces. Gorenstein liaison continues to be a central tool, but to prove the existence of our links we make use of certain dimension computations. We also discuss the higher degree case, bu…
The Minimal Resolution Conjecture (MRC) for points on a projective variety X predicts that the Betti numbers of general sets of points in X are as small as the geometry (Hilbert function) of X allows. To a large extent, we settle this conjecture for a curve C with general moduli. We show that, independently of the genus, MRC holds for a general linear system of degree d and dimension r on C if and only if d>2r-1. We then proceed to find a full solution to the Ideal Generation Conjecture for curves with general moduli. In a different direction, we prove that K3 surfaces admit Ulrich bundles of …
Let $I$ be a graded ideal of $K[x_1,\ldots,x_n]$ generated by homogeneous polynomials of a same degree $d$, and assume that $I$ has linear quotients. In this note, we use Horseshoe Lemma to give a relatively direct inductive construction of a minimal free resolution of $I$, which is called a $d$-linear resolution.
We study the minimal free resolution of the tangent cone of Gorenstein monomial curves in affine 4-space. We give the explicit minimal free resolution of the tangent cone of non-complete intersection Gorenstein monomial curve whose tangent cone has five minimal generators and show that the possible Betti sequences are $(1,5,6,2)$ and $(1,5,5,1)$. Also, we compute the Hilbert function of the tangent cone of these families as a result.
We introduce the notion of a \emph{resolution supported on a poset}. When the poset is a CW-poset, i.e. the face poset of a regular CW-complex, we recover the notion of cellular resolution as introduced by Bayer and Sturmfels. Work of Reiner and Welker, and of Velasco, has shown that there are monomial ideals whose minimal free resolutions are not cellular, hence cannot be supported on any CW-poset. We show that for any monomial ideal there is a \emph{homology CW-poset} that supports a minimal free resolution of the ideal. This allows one to extend to every minimal resolution, essentially verb…
We construct a minimal free resolution of the semigroup ring k[C] in terms of minimal resolutions of k[A] and k[B] when <C> is a numerical semigroup obtained by gluing two numerical semigroups <A> and <B>. Using our explicit construction, we compute the Betti numbers, graded Betti numbers, regularity and Hilbert series of k[C], and prove that the minimal free resolution of k[C] has a differential graded algebra structure provided the resolutions of k[A] and k[B] possess them. We discuss the consequences of our results in small embedding dimensions. Finally, we give an extension of our main res…
We will explore some properties of minimal graded free resolutions of $R/I$, where $R$ is a trivariate polynomial ring over a field and $I$ is a monomial ideal. Our focus will be to consider a specific form of the resolutions when $I$ is primary to the homogeneous maximal ideal. We will identify certain characteristics of the last matrix of these resolutions, and observe differences in the resolutions for generic ideals in comparison to non-generic ideals. Finally, we learn how to identify whether $I$ is generic by knowing the structure of the last matrix in the minimal free resolution of $R/I…
Let Z be a finite set of double points in P^1 x P^1 and suppose further that X, the support of Z, is arithmetically Cohen-Macaulay (ACM). We present an algorithm, which depends only upon a combinatorial description of X, for the bigraded Betti numbers of I_Z, the defining ideal of Z. We then relate the total Betti numbers of I_Z to the shifts in the graded resolution, thus answering a special case of a question of T. Roemer.
We compute the linear strand of the minimal free resolution of the ideal generated by k x k sub-permanents of an n x n generic matrix and of the ideal generated by square-free monomials of degree k. The latter calculation gives the full minimal free resolution by work of Biagioli-Faridi-Rosas. Our motivation is to lay groundwork for the use of commutative algebra in algebraic complexity theory. We also compute several Hilbert functions relevant for complexity theory.
Given an arbitrary field k and an arithmetic sequence of positive integers m_0<...<m_n, we consider the affine monomial curve parameterized by X_0=t^{m_0},...,X_n=t^{m_n}. In this paper, we conjecture that the Betti numbers of its coordinate ring are completely determined by n and the value of m_0 modulo n. We first show that the defining ideal of the monomial curve can be written as a sum of two determinantal ideals. Using this fact, we describe the minimal free resolution of the coordinate ring in the following three cases: when m_0 is 1 modulo n (determinantal), when m_0 is n modulo n (almo…
With a view to study problems of smoothability, we construct a minimal free resolution for the coordinate ring of an algebroid monomial curve associated to an $AS$ numerical semigroup (i.e. generated by an arithmetic sequence), obtained independently of the result of \cite{gss2} and equipped with the explicit description of all the involved maps.
In this paper we study minimal free resolutions of some classes of monomial ideals. we first give a sufficient condition to check the minimality of the resolution obtained by the mapping cone. Using it, we obtain the Betti numbers of max-path ideals of rooted trees and ideals containing powers of variables. In particular, we discuss about resolutions of ideals of the form $J_{\mathcal{H}}+(x_{i_1}^2,\ldots, x_{i_m}^2)$ where $J_{\mathcal{H}}$ is the edge ideal of a hypergraph $\mathcal{H}$.
We construct free resolutions for quotient rings $R/\langle \mathcal{I}', \mathcal{I}\mathcal{J}, \mathcal{J}'\rangle$, give conditions for the quotient to be realized as a fiber product, and give criteria for the construction to be minimal. We then specialize this result to fiber products over a field $k$ and recover explicit formulas for Betti numbers, graded Betti numbers, and Poincaré series.
We give a comparison between the Poincare series of two monomial rings: $R=A/I$ and $R_q=A/I_q$ where $I_q$ is a monomial ideal generated by the $q$'th power of monomial generators of $I$. We compute the Poincare series for a class of monomial rings with minimal Taylor resolution. The paper was produced during Pragmatic 2011.
This paper is concerned with the combinatorial description of the graded minimal free resolution of certain monomial algebras which includes toric rings. Concretely, we explicitly describe how the graded minimal free resolution of those algebras is related to the combinatorics of some simplicial complexes. Our description may be interpreted as an algorithmic procedure to partially compute this resolution.
If r\geq 6, r\neq 9, we show that the Minimal Resolution Conjecture fails for a general set of m points in P^r for almost 1/2\sqrt r values of m. This strengthens the result of Eisenbud and Popescu [1999], who found a unique such m for each r in the given range. Our proof begins like a variation of that of Eisenbud and Popescu, but uses exterior algebra methods as explained by Eisenbud and Schreyer [2000] to avoid the degeneration arguments that were the most difficult part of the Eisenbud-Popescu proof. Analogous techniques show that the Minimal Resolution Conjecture fails for linearly normal…
We introduce a squarefree monomial ideal associated to the set of domino tilings of a $2\times n$ rectangle and proceed to study the associated minimal free resolution. In this paper, we use results of Dalili and Kummini to show that the Betti numbers of the ideal are independent of the underlying characteristic of the field, and apply a natural splitting to explicitly determine the projective dimension and Castelnuovo-Mumford regularity of the ideal.
We present an explicit construction of a minimal cellular resolution for the edge ideals of forests, based on discrete Morse theory. In particular, the generators of the free modules are subsets of the generators of the modules in the Lyubeznik resolution. This procedure allows to ease the computation of the graded Betti numbers and the projective dimension.
Based upon a previous work of Manjunath and Sturmfels for a finite, complete, undirected graph, and a refined algorithm by Eröcal, Motsak, Schreyer and Steenpaß for computing syzygies, we display a free resolution of the lattice ideal associated to a finite, strongly connected, weighted, directed graph. Moreover, the resolution is minimal precisely when the digraph is strongly complete.
We consider vector-spread Borel ideals. We show that these ideals have linear quotients and thereby we determine the graded Betti numbers and the bigraded Poincaré series. A characterization of the extremal Betti numbers of such a class of ideals is given. Finally, we classify all Cohen-Macaulay vector-spread Borel ideals.
A minimal monomial ideal is the combinatorially simplest monomial ideal whose lcm-lattice equals a given finite atomic lattice $\hat{L}$. The minimal ideal inherits many nice properties of any ideal $I$ whose lcm-lattice also equals $\hat{L}$, e.g. Cohen-Macaulayness and the dual property of having a linear resolution. Conversely, any ideal having a linear resolution is shown to be (essentially) minimal.
This is the second paper in a series of three papers. In the first paper of the series, "Artinian Gorenstein algebras with linear resolutions", (arXiv:1306.2523, J. of Algebra, to appear) we prove that it is possible to give the minimal resolution of the rings from the title in terms of the coefficients of the corresponding Macaulay inverse system. In this context, the word "give" means, "give in a polynomial manner". The first paper in the series proves, essentially, an existence theorem. The second and third papers in the series construct the explicit formulas for the resolution. The present…
Consider the affine space consisting of pairs of matrices $(A,B)$ of fixed size, and its closed subvariety given by the rank conditions $\operatorname{rank} A \leq a$, $\operatorname{rank} B \leq b$ and $\operatorname{rank} (A\cdot B) \leq c$, for three non-negative integers $a,b,c$. These varieties are precisely the orbit closures of representations for the equioriented $A_3$ quiver. In this paper we construct the (equivariant) minimal free resolutions of the defining ideals of such varieties. We show how this problem is equivalent to determining the cohomology groups of the tensor product of…
For a partition $λ$ of $n \in \mathbb{N}$, let $I^{\rm Sp}_λ$ be the ideal of $R=K[x_1,\ldots,x_n]$ generated by all Specht polynomials of shape $λ$. We assume that ${\rm char}(K)=0$. Then $R/I^{\rm Sp}_{(n-2,2)}$ is Gorenstein, and $R/I^{\rm Sp}_{(d,d,1)}$ is a Cohen-Macaulay ring with a linear free resolution. In this paper, we construct minimal free resolutions of these rings. Berkesch Zamaere, Griffeth, and Sam had already studied minimal free resolutions of $R/I^{\rm Sp}_{(n-d,d)}$, which are also Cohen-Macaulay, using heighly advanced technique of the representation theory. However we on…
