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.
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…
The Castelnuovo-Mumford regularity of a graded ring is an important invariant in computational commutative algebra, and there is increasing interest in multigraded generalizations. We study connections between two recent definitions of multigraded regularity with a view towards a better understanding of the multigraded Hilbert function of fat point schemes in P^{n_1} x ... x P^{n_k}.
$V$ is a complete intersection scheme in a multiprojective space if it can be defined by an ideal $I$ with as many generators as $\textrm{codim}(V)$. We investigate the multigraded regularity of complete intersections scheme in $\mathbb{P}^n\times \mathbb{P}^m$. We explicitly compute many values of the Hilbert functions of $0$-dimensional complete intersections. We show that these values only depend upon $n,m$, and the bidegrees of the generators of $I$. As a result, we provide a sharp upper bound for the multigraded regularity of $0$-dimensional complete intersections.
We develop a multigraded variant of Castelnuovo-Mumford regularity. Motivated by toric geometry, we work with modules over a polynomial ring graded by a finitely generated abelian group. As in the standard graded case, our definition of multigraded regularity involves the vanishing of graded components of local cohomology. We establish the key properties of regularity: its connection with the minimal generators of a module and its behavior in exact sequences. For an ideal sheaf on a simplicial toric variety X, we prove that its multigraded regularity bounds the equations that cut out the assoc…
In this paper, we consider $\mathbb{Z}^{r}-$graded modules on the $\mathrm{Cl}(X)$ $-$graded Cox ring $\mathbb{C}[x_{1},\dotsc,x_{r}]$ of a smooth complete toric variety $X$. Using the theory of Klyachko filtrations in the reflexive case, we construct a collection of lattice polytopes codifying the multigraded Hilbert function of the module. We apply this approach to reflexive $\mathbb{Z}^{s+r+2}$-graded modules over non-standard bigraded polynomial rings $\mathbb{C}[x_{0},\dotsc,x_{s},y_{0},\dotsc,y_{r}]$. In this case, we give sharp bounds for the multigraded regularity index of their multig…
We give a criterion for the section ring of an ample line bundle to be Koszul in terms of multigraded regularity. We discuss an application to polytopal semigroup rings.
Using multigraded Castelnuovo-Mumford regularity, we study the equations defining a projective embedding of a variety X. Given globally generated line bundles B_1, ..., B_k on X and integers m_1, ..., m_k, consider the line bundle L := B_1^m_1 \otimes ... \otimes B_k^m_k. We give conditions on the m_i which guarantee that the ideal of X in P(H^0(X,L)) is generated by quadrics and the first p syzygies are linear. This yields new results on the syzygies of toric varieties and the normality of polytopes.
Let S = k[x_1,...,x_n] be a Z^r-graded ring with deg (x_i) = a_i \in Z^r for each i and suppose that M is a finitely generated Z^r-graded S-module. In this paper we describe how to find finite subsets of Z^r containing the multidegrees of the minimal multigraded syzygies of M. To find such a set, we first coarsen the grading of M so that we can view M as a Z-graded S-module. We use a generalized notion of Castelnuovo-Mumford regularity, which was introduced by D. Maclagan and G. Smith, to associate to M a number which we call the regularity number of M. The minimal degrees of the multigraded m…
We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety X with a given multigraded Hilbert polynomial. To establish this bound, we introduce a new combinatorial tool, called a Stanley filtration, for studying monomial ideals in the homogeneous coordinate ring of X. As a special case, we obtain a new proof of Gotzmann's regularity theorem. We also discuss applications of this bound to the construction of multigraded Hilbert schemes.
This paper is concerned with the relationships between two concepts, vanishing of cohomology groups and the structure of free resolutions. In particular, we study the connection between vanishing theorems for the local cohomology of multigraded modules and the structure of their free multigraded resolutions.
We relate the set of complete reduction vectors of a $\mathbb{Z}^s$-graded admissible filtration of ideals $\mathcal{F}$ with the set of multigraded regularities of $G(\mathcal{F}).$ We prove reg$(G(\mathcal{F}))$=reg$(\mathcal{R}(\mathcal{F})).$ We establish a relation between the sets of complete reduction vectors of $\mathcal{F}$ and postulation vectors of $\mathcal{F}$ under some cohomological conditions.
In this article we extend a previous definition of Castelnuovo-Mumford regularity for modules over an algebra graded by a finitely generated abelian group. Our notion of regularity is based on Maclagan and Smith's definition, and is extended first by working over any commutative base ring, and second by considering local cohomology with support in an arbitrary finitely generated graded ideal $B$, obtaining, for each $B$, a $B$-regularity region. The first extension provides a natural approach for working with families of sheaves or of graded modules, while the second opens new applications. We…
Let $A$ be a Noetherian standard $\mathbb{N}$-graded algebra over an Artinian local ring $A_0$. Let $I_1,\ldots,I_t$ be homogeneous ideals of $A$ and $M$ a finitely generated $\mathbb{N}$-graded $A$-module. We prove that there exist two integers $k$ and $k'$ such that \[ \mathrm{reg}(I_1^{n_1} \cdots I_t^{n_t} M) \leq (n_1 + \cdots + n_t) k + k' \quad\mbox{for all }~n_1,\ldots,n_t \in \mathbb{N}. \]
We study the properties of F-rationality and F-regularity in multigraded rings and their diagonal subalgebras. The main focus is on diagonal subalgebras of bigraded rings: these constitute an interesting class of rings since they arise naturally as homogeneous coordinate rings of blow-ups of projective varieties. As a consequence of some of the results obtained here, it is shown that there exist standard bigraded hypersurfaces whose rings of invariants under torus actions have rational singularities, but are not of F-regular type. Another application is the construction of families of rings wi…
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…
In this paper we consider multi-graded extended Rees algebras of zero dimensional ideals which are Cohen-Macaulay (CM) with minimal multiplicity. We show that the minimal multiplicity property can occur only for the ordinary extended Rees algebra and the bigraded extended Rees algebra. For the bigraded extended Rees algebra we find necessary conditions for it to be CM with minimal multiplicity. We also produce bigraded Rees algebras which are Cohen-Macaulay with minimal multiplicity.
Maclagan and Smith \cite{MaclaganSmith} developed a multigraded version of Castelnuovo-Mumford regularity. Based on their definition we will prove in this paper that for a smooth curve $C\subseteq ¶^a\times¶^b$ $(a, b\geq 2)$ of bidegree $(d_1,d_2)$ with nondegenerate birational projections the ideal sheaf $\mathcal{I}_{C|¶^a\times¶^b}$ is $(d_2-b+1,d_1-a+1)$-regular. We also give an example showing that in some cases this bound is the best possible.
In this article we analyze the implicitization problem of the image of a rational map $ϕ: X --> P^n$, with $T$ a toric variety of dimension $n-1$ defined by its Cox ring $R$. Let $I:=(f_0,...,f_n)$ be $n+1$ homogeneous elements of $R$. We blow-up the base locus of $ϕ$, $V(I)$, and we approximate the Rees algebra $Rees_R(I)$ of this blow-up by the symmetric algebra $Sym_R(I)$. We provide under suitable assumptions, resolutions $\Z.$ for $Sym_R(I)$ graded by the torus-invariant divisor group of $X$, $Cl(X)$, such that the determinant of a graded strand, $\det((\Z.)_μ)$, gives a multiple of the i…
We study the multigraded Poincaré-Betti series of $A=S/\aaf$, where $S$ is the ring of polynomials in $n$ indeterminates divided by the monomial ideal $\aaf$. There is a conjecture about the multigraded Poincaré-Betti series by Charalambous and Reeves which they proved in the case, where the Taylor resolution is minimal. We introduce a conjecture about the minimal $A$-free resolution of the residue class field and show that this conjecture implies the conjecture of Charalambous and Reeves. We prove our conjecture for several classes of algebras. The conjecture implies that $A$ is Golod if and …
This paper concerns the study of a class of clutters called simplicial subclutters. Given a clutter $\mathcal{C}$ and its simplicial subclutter $\mathcal{D}$, we compare some algebraic properties and invariants of the ideals $I, J$ associated to these two clutters, respectively. We give a formula for computing the (multi)graded Betti numbers of $J$ in terms of those of $I$ and some combinatorial data about $\mathcal{D}$. As a result, we see that if $\mathcal{C}$ admits a simplicial subclutter, then there exists a monomial $u \notin I$ such that the (multi)graded Betti numbers of $I+(u)$ can be…
This dissertation focuses on the following topics: (1) asymptotic prime divisors over complete intersection rings, (2) asymptotic stability of complexities over complete intersection rings, (3) asymptotic linear bounds of Castelnuovo-Mumford regularity in multigraded modules, and (4) characterizations of regular local rings via syzygy modules of the residue field.
Let R be the quotient of a polynomial ring over a field k by an ideal generated by monomials. We derive a formula for the multigraded Poincare' series of R, i.e., the generating function for the ranks of the modules in a minimal multigraded free resolution of k over R. The formula can be expressed in terms of the homology of lower intervals in a certain lattice associated to the minimal set of generators for the ideal.
Notions of Castelnuovo-Mumford regularity and of $a^*$ invariant were extended from standard graded algebras to the toric setting. We here focus our attention on the standard multigraded case, which corresponds to a product of $k$ projective spaces. A natural notion for a $\mathbb Z^k$-graded module is its support: degrees in which it is not zero. A stabilized version of it is adding $-\mathbb N^k$, in order for the complement (vanishing region) to be stable by addition of $\mathbb N^k$. Cohomology of twists of a sheaf on a product of projective spaces, provided by a graded module, are given b…
We express the multigraded Betti numbers of an arbitrary monomial ideal in terms of the multigraded Betti numbers of two basic classes of ideals. This decompo- sition has multiple applications. In some concrete cases, we use it to construct minimal resolutions of classes of monomial ideals; in other cases, we use it to compute projective dimensions. To illustrate the effectiveness of the structural decomposition, we give a new proof of a classic theorem by Charalambous.
Let $R:= K[x_1,\ldots,x_{n}]$ be a polynomial ring over an infinite field $K$, and let $I \subset R$ be a homogeneous ideal with respect to a weight vector $ω= (ω_1,\ldots,ω_n) \in (\mathbb{Z}^+)^n$ such that $\dim(R/I) = d$. In this paper we study the minimal graded free resolution of $R/I$ as $A$-module, that we call the Noether resolution of $R/I$, whenever $A :=K[x_{n-d+1},\ldots,x_n]$ is a Noether normalization of $R/I$. When $d=2$ and $I$ is saturated, we give an algorithm for obtaining this resolution that involves the computation of a minimal Gröbner basis of $I$ with respect to the we…
Let $ρ_C$ be the regularity of the Hilbert function of a projective curve $C$ in $\mathbb P^n_K$ over an algebraically closed field $K$ and $α_1,...,α_{n-1}$ be minimal degrees for which there exists a complete intersection of type $(α_1,...,α_{n-1})$ containing the curve $C$. Then the Castelnuovo-Mumford regularity of $C$ is upper bounded by $\max\{ρ_C+1,α_1+...+α_{n-1}-(n-2)\}$. We study and, for space curves, refine the above bound providing several examples.
We investigate the minimal number of generators $μ$ and the depth of divisorial ideals over normal semigroup rings. Such ideals are defined by the inhomogeneous systems of linear inequalities associated with the support hyperplanes of the semigroup. The main result is that for every bound $C$ there exist, up to isomorphism, only finitely divisorial ideals $I$ such that $μ(I)\le C$. It follows that there exist only finitely many Cohen--Macaulay divisor classes. Moreover we determine the minimal depth of all divisorial ideals and the behaviour of $μ$ and depth in ``arithmetic progressions'' in t…
We introduce the class of lattice-linear monomial ideals and use the LCM-lattice to give an explicit construction for their minimal free resolution. The class of lattice-linear ideals includes (among others) the class of monomial ideals with linear free resolution and the class of Scarf monomial ideals. Our main tool is a new construction by Tchernev that produces from a map of posets $η:P\lra\mbb{N}^n$ a sequence of multigraded modules and maps.
The question we address in this paper is: which monomial ideals have minimal cellular resolutions, that is, minimal resolutions obtained from homogenizing the chain maps of CW-complexes? Velasco gave families of examples of monomial ideals that do not have minimal cellular resolutions, but those examples have large minimal generating sets. In this paper, we show that if a monomial ideal has at most four generators, then the ideal and its (monomial) Artinian reductions have minimal cellular resolutions. When the ideal is generated by two monomials, we can give a precise description of the CW-co…
We show that given a finitely generated standard-graded algebra of dimension $d$ over an infinite field, its graded Noether normalizations obey a certain kind of `generic exchange', allowing one to pass between any two of them in at most $d$ steps. We prove analogous generic exchange theorems for minimal reductions of an ideal, minimal complete reductions of a set of ideals, and minimal complete reductions of multigraded $k$-algebras. Finally, we unify all these results into a common axiomatic framework by introducing a new topological-combinatorial structure we call a generic matroid, which i…
In this short note we introduce a notion of extremality for Betti numbers of a minimal free resolution, which can be seen as a refinement of the notion of Mumford-Castelnuovo regularity. We show that extremal Betti numbers of an arbitrary submodule of a free S-module are preserved when taking the generic initial module. We relate extremal multigraded Betti numbers in the minimal resolution of a square free monomial ideal with those of the monomial ideal corresponding to the Alexander dual simplicial complex and generalize theorems of Eagon-Reiner and Terai. As an application we give easy (alte…
We investigate multi-graded Gorenstein semigroup algebras associated with an infinite family of reflexive lattice simplices. For each of these algebras, we prove that their multigraded Poincaré series is rational. Our method of proof is to produce for each algebra an explicit minimal free resolution of the ground field, in which the resolution reflects the recursive structure encoded in the denominator of the finely-graded Poincaré series. Using this resolution, we show that these algebras are not Koszul, and therefore rationality is non-trivial. Our results demonstrate how interactions betwee…
Consider an ideal $I\subset K[x_1,..., x_n]$, with $K$ an arbitrary field, generated by monomials of degree two. Assuming that $I$ does not have a linear resolution, we determine the step $s$ of the minimal graded free resolution of $I$ where nonlinear syzygies first appear, we show that at this step of the resolution nonlinear syzygies are concentrated in degree $s+3$, and we compute the corresponding graded Betti number $β_{s,s+3}$. The multidegrees of these nonlinear syzygies are also determined and the corresponding multigraded Betti numbers are shown to be all equal to 1.
Toward a partial classification of monomial ideals with $d$-linear resolution, in this paper, some classes of $d$-uniform clutters which do not have linear resolution, but every proper subclutter of them has a $d$-linear resolution, are introduced and the regularity and Betti numbers of circuit ideals of such clutters are computed. Also, it is proved that for given two $d$-uniform clutters $\mathcal{C}_1, \mathcal{C}_2$, the Castelnuovo-Mumford regularity of the ideal $I(\bar{\mathcal{C}_1 \cup \mathcal{C}_2})$ is equal to the maximum of regularities of $I(\bar{\C}_1)$ and $I(\bar{\C}_2)$, whe…
We focus in this paper on edge ideals associated to bipartite graphs and give a combinatorial characterization of those having regularity 3. When the regularity is strictly bigger than 3, we determine the first step $i$ in the minimal graded free resolution where there exists a minimal generator of degree $>i+3$, show that at this step the highest degree of a minimal generator is $i+4$, and determine the value of the corresponding graded Betti number $β_{i,i+4}$ in terms of the combinatorics of the associated bipartite graph. The results can then be easily extended to the non-squarefree case t…
In this paper we study the Castelnuovo-Mumford regularity of the path ideals of finite simple graphs. We find new upper bounds for various path ideals of gap free graphs. In particular we prove that the path ideals of gap free and claw graphs have linear minimal free resolutions.
Let A be the twisted commutative algebra freely generated by d indeterminates of degree 1. We show that the regularity of an A-module can be bounded from the first floor(d^2/4) + 2 terms of its minimal free resolution. This extends results of Church and Ellenberg from the d=1 case.
