$V$ 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.
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 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.
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}.
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.
Let $X$ be a complete $n$-dimensional simplicial toric variety with homogeneous coordinate ring $S$. We study the multigraded Hilbert function $H_Y$ of reduced $0$-dimensional subschemes $Y$ in $X$. We provide explicit formulas and prove non-decreasing and stabilization properties of $H_Y$ when $Y$ is a $0$-dimensional complete intersection in $X$. We apply our results to computing the dimension of some evaluation codes on $0$-dimensional complete intersection in simplicial toric varieties.
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 \[ \mathrm{reg}(I_1^{n_1} \cdots I_t^{n_t} M) \leq (n_1 + \cdots + n_t) k + k' \quad\mbox{for all }~n_1,\ldots,n_t \in \mathbb{N}. \]
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.
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.
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…
In this article we study the structure of residual intersections via constructing a finite complex which is acyclic under some sliding depth conditions on the cycles of the Koszul complex. This complex provides information on an ideal which coincides with the residual intersection in the case of geometric residual intersection; and is closely related to it in general. A new success obtained through studying such a complex is to prove the Cohen-Macaulayness of residual intersections of a wide class of ideals. For example we show that, in a Cohen-Macaulay local ring, any geometric residual inter…
In this article we establish bounds for the Castelnuovo-Mumford regularity of projective schemes in terms of the degrees of their defining equations. The main new ingredient in our proof is to show that generic residual intersections of complete intersection rational singularities again have rational singularities. When applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals.
The second Veronese ideal $I_n$ contains a natural complete intersection $J_n$ generated by the principal $2$-minors of a symmetric $(n\times n)$-matrix. We determine subintersections of the primary decomposition of $J_n$ where one intersectand is omitted. If $I_n$ is omitted, the result is the other end of a complete intersection link as in liaison theory. These subintersections also yield interesting insights into binomial ideals and multigraded algebra. For example, if $n$ is even, $I_n$ is a Gorenstein ideal and the intersection of the remaining primary components of $J_n$ equals $J_n+\lan…
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…
This paper proves that the Castelnuovo-Mumford regularities of the product and sum of two monomial complete intersection ideals are at most the sum of the regularities of the two ideals, and provides examples showing that these inequalities do not hold for general complete intersections.
Let I be a complete intersection in a polynomial ring over a field, the Castelnuovo-Mumford regularity of I^n is given by using an induction method. When I, J and K are three pure power complete intersections, it is proved that reg(IJK) is not more than reg(I)+reg(J)+reg(K).
Among reduced Noetherian prime characteristic commutative rings, we prove that a regular ring is precisely one where finite intersection of ideals commutes with taking bracket powers. However, reducedness is essential for this equivalence. Connections are made with Ohm-Rush content theory, intersection-flatness of the Frobenius map, and various flatness criteria.
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.
The study of regularity and complete intersection of a tensor product of commutative algebras possessing the same property started with Grothendieck in 1965 and has continued until today. Surprisingly, the homology theory of Andre and Quillen, developed by these authors in 1967, has never been used for this study. With the help of this theory, we can (slightly) generalize the results known up to now. But more important, we hope to convince the reader that this homology theory is the adequate tool to handle these problems: the proofs are very short and (assuming some flatness hypothesis) it all…
In this paper, we provide exact formulas for the Castelnuovo-Mumford regularity of powers of an almost complete intersection ideal $I$ which is generated by a homogeneous $d$-sequence. As applications, when $I$ is an almost complete intersection, taking the form of the (parity) binomial edge ideal of a connected graph, we can describe explicitly formulas for $\reg(I^t)$ for $t\ge 2$. The only exception is when $I$ is the parity binomial edge ideal of a graph which is obtained by adding an edge between two disjoint odd cycles.
Set $ A := Q/({\bf z}) $, where $ Q $ is a polynomial ring over a field, and $ {\bf z} = z_1,\ldots,z_c $ is a homogeneous $ Q $-regular sequence. Let $ M $ and $ N $ be finitely generated graded $ A $-modules, and $ I $ be a homogeneous ideal of $ A $. We show that (1) $ \mathrm{reg}\left( \mathrm{Ext}_A^{i}(M, I^nN) \right) \le ρ_N(I) \cdot n - f \cdot \left\lfloor \frac{i}{2} \right\rfloor + b \mbox{ for all } i, n \ge 0 $, (2) $ \mathrm{reg}\left( \mathrm{Ext}_A^{i}(M,N/I^nN) \right) \le ρ_N(I) \cdot n - f \cdot \left\lfloor \frac{i}{2} \right\rfloor + b' \mbox{ for all } i, n \ge 0 $, whe…
For a simplicial complex X and a field K, let h_i(X)=\dim \tilde{H}_i(X;K). It is shown that if X,Y are complexes on the same vertex set, then for all k h_{k-1}(X\cap Y) \leq \sum_{σ\in Y} \sum_{i+j=k} h_{i-1}(X[σ])\cdot h_{j-1}(\lk(Y,σ)) . A simplicial complex X is d-Leray over K, if h_i(Y)=0 for all induced subcomplexes Y \subset X and i \geq d. Let L_K(X) denote the minimal d such that X is d-Leray over K. The above theorem implies that if X,Y are simplicial complexes on the same vertex set then L_K(X \cap Y) \leq L_K(X) +L_K(Y). Reformulating this inequality in commutative algebra terms, w…
In this survey article we give a brief history of symbolic powers and its connection with the interesting problem of set-theoretic complete intersection. We also state a few problems and conjectures. Recently, in connection to symbolic powers is the containment problem. We list a few interesting results and related problems on the resurgence, Waldschmidt constant and Castelnuovo-Mumford regularity.
This paper main goal is to measure the defect of Cohen-Macaulayness, Gorensteiness, complete intersection and regularity for the tensor product of algebras over a ring. For this sake, we determine the homological invariants which are inherent to these notions, such as the Krull dimension, depth, injective dimension, type and embedding dimension of the tensor product constructions in terms of those of their components.
Regularity, complete intersection and Gorenstein properties of a local ring can be characterized by homological conditions on the canonical homomorphism into its residue field (Serre, Avramov, Auslander). It is also known that in positive characteristic, the Frobenius endomorphism can also be used for these characterizations (Kunz, ...), and more generally any contracting endomorphism. We introduce here a class of local homomorphisms, in some sense larger than all above, for which these characterizations still hold, providing an unified treatment for this class of homomorphisms.
We study the footprint function, with respect to a monomial order, of complete intersection graded ideals in a polynomial ring with coefficients in a field. For graded ideals of dimension one, whose initial ideal is a complete intersection, we give a formula for the footprint function and a sharp lower bound for the corresponding minimum distance function. This allows us to recover a formula for the minimum distance of an affine cartesian code and the fact that in this case the minimum distance and the footprint functions coincide. Then we present an extension of a result of Alon and Füredi, a…
Let $K$ be a field and $I$ a monomial ideal of the polynomial ring $S=K[x_1,..., x_n]$ generated by monomials $u_1,u_2,..., u_t$. We show that $S/I$ is pretty clean if either: 1) $u_1,u_2,..., u_t$ is a filter-regular sequence, 2) $u_1,u_2,..., u_t$ is a $d$-sequence; or 3) $I$ is almost complete intersection. In particular, in each of these cases, $S/I$ is sequentially Cohen-Macaulay and both Stanley's and $h$-regularity conjectures, on Stanley decompositions, hold for $S/I$. Also, we prove that if $I$ is the Stanley-Reisner ideal of a locally complete intersection simplicial complex on $[n]$…
