Azucena Tochimani, Rafael H. Villarreal
In this paper we give a classification of complete intersection vanishing ideals on parameterized sets of clutter type over finite fields.
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Complete intersection vanishing ideals on degenerate tori over finite fields
研究有限域上退化环面消失理想的完全交性质，建立其与数值半群环面理想的对应，并给出次数和正则指数的公式。
Let K be a finite field and let X be a subset of a projective space, over the field K, which is parameterized by monomials arising from the edges of a clutter. We show some estimates for the degree-complexity, with respect to the revlex order, of the vanishing ideal I(X) of X. If the clutter is uniform, we classify the complete intersection property of I(X) using linear algebra. We show an upper bound for the minimum distance of certain parameterized linear codes along with certain estimates for the algebraic invariants of I(X).
Let R be a complete intersection ring and let M and N be R-modules. It is shown that the vanishing of Ext^i_R(M,N) for a certain number of consecutive values of i starting at n forces the complete intersection dimension of M to be at most n-1. We also estimate the complete intersection dimension of the dual of M, in terms of vanishing of the cohomology modules, Ext^i_R(M,N).
We prove rigidity type results on the vanishing of stable (co)homology for modules of finite complete intersection dimension, results which generalize and improve upon known results. We also introduce a notion of pre-rigidity, which generalizes phenomena for modules of finite complete intersection dimension and complexity one. Using this concept, we prove results on length and vanishing of homology modules.
Let $I$ be ideal of an $n$-dimensional local Gorenstein ring $R$. In this paper we will describe several necessary and sufficient conditions such that the ideal $I$ becomes cohomologically complete intersections. In fact, as a technical tool, it will be shown that the vanishing $H^i_{I}(R)= 0$ for all $i\neq c= \grade (I)$ is equivalent to the vanishing of the Betti numbers of $H^c_{I}(R)$. This gives a new characterization to check the cohomologically complete intersections property with the homological properties of the vanishing of Tor modules of $H^c_{I}(R)$.
In this paper we are concerned with the vanishing of $\textnormal{Tor}$ over complete intersection rings. Building on results of C. Huneke, D. Jorgensen and R. Wiegand, and, more recently, H. Dao, we obtain new results showing that good depth properties on the $R$-modules $M$, $N$ and $M\otimes_RN$ force the vanishing of $\textnormal{Tor}^{R}_{i}(M,N)$ for all $i\geq 1$.
In this paper we exploit properties of Dao's eta-pairing as well as techniques of Huneke, Jorgensen, and Wiegand to study the vanishing of Tor_i(M,N) of finitely generated modules M, N over complete intersections. We prove vanishing of Tor_i(M, N) for all positive integers i under depth conditions on M, N, and their tensor product. Our arguments improve a result of Dao and establish a new connection between the vanishing of Tor and the depth of tensor products.
Let X* be a subset of an affine space A^s, over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x --> [x] and x --> [(x,1)] respectively, where [x] and [(x,1)] are points in the projective spaces P^{s-1} and P^s respectively. For certain clutters and for connected graphs, we were able to relate the algebraic invariants and properties of the vanishing ideals I(X) and I(Y). In a number of interesting cases, we compute its degree and regularity. For Hamiltonian bipartite graphs, we show the Eisenbud-Goto regularity conjecture. We …
We study two properties of modules over a local hypersurface $R$: decency and rigidity. We show that the vanishing of Hochster's function $θ^R(M,N)$, known to imply decent intersection, also implies rigidity. We investigate the vanishing of $θ^R(M,N)$ to obtain new results about decency and rigidity over hypersurfaces. We employ a mixture of techniques from Commutative Algebra and Intersection Theory of algebraic cycles.
We study H. Dao's invariant $η_c^R$ of pairs of modules defined over a complete intersection ring $R$ of codimension $c$ having an isolated singularity. Our main result is that $η_c^R$ vanishes for all pairs of modules when $R$ is a {\em graded} complete intersection ring of codimension $c > 1$ having an isolated singularity. A consequence of this result is that all pairs of modules over such a ring are $c$-$\Tor$-rigid.
We study the relation between the vanishing of André-Quillen homology and complete intersection flat dimension and we extend some of the existing results in the literature.
Classical definitions of locally complete intersection (l.c.i.) homomorphisms of commutative rings are limited to maps that are essentially of finite type, or flat. The concept introduced in this paper is meaningful for homomorphisms phi : R \longrightarrow S of commutative noetherian rings. It is defined in terms of the structure of phi in a formal neighborhood of each point of Spec S. We characterize the l.c.i. property by different conditions on the vanishing of the André-Quillen homology of the R-algebra S. One of these descriptions establishes a very general form of a conjecture of Quille…
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…
We show that every thick subcategory of the singularity category of a complete intersection ring is self dual. We also prove the analogous statement for thick subcategories of the bounded derived category and give applications to the symmetry of vanishing of cohomology. These results are also proved for certain complete intersection schemes.
Let $M$ be a finitely generated module over a local complete intersection $R$ of characteristic $p>0$. The property that $M$ has finite projective dimension can be characterized by the vanishing of $\ext_R^i({}^{f^n} R,M)$ for some $i>0$ and for some $n>0$.
Considering modules of finite complete intersection dimension over commutative Noetherian local rings, we prove (co)homology vanishing results in which we assume the vanishing of nonconsecutive (co)homology groups. In fact, the (co)homology groups assumed to vanish may be arbitrarily far apart from each other.
Let X be a subset of a projective space, over a finite field K, which is parameterized by the monomials arising from the edges of a clutter. Let I(X) be the vanishing ideal of X. It is shown that I(X) is a complete intersection if and only if X is a projective torus. In this case we determine the minimum distance of any parameterized linear code arising from X.
Let $ e_1, ..., e_c $ be positive integers and let $ Y \subseteq \mathbb{P}^n$ be the monomial complete intersection defined by the vanishing of $x_1^{e_1}, ..., x_c^{e_c}$. In this paper we study sharp upper bounds on the number of equations and syzygies of subschemes parametrized by the Hilbert scheme of points $Hilb^d(Y)$, and discuss applications to the Hilbert scheme of points $Hilb^d(X)$ of arbitrary complete intersections $X \subseteq \mathbb{P}^n$.
It is proved that if one of the finite modules M and N, over a local ring R, has reducible complexity and has finite Gorenstein dimension then the depth formula holds, provided TorR_i(M,N) = 0 for i>>0. We also study the vanishing of cohomology of a module of finite complete intersection dimension.
An ideal $I$ of a local Gorenstein ring $(R, \mathfrak m)$ is called cohomologically complete intersection whenever $H^i_I(R) = 0$ for all $i \not= \height I.$ Here $H^i_I(R), i \in \mathbb Z,$ denotes the local cohomology of $R$ with respect to $I.$ For instance, a set-theoretic complete intersection is a cohomologically complete intersection. Here we study cohomologically complete intersections from various homological points of view, in particular in terms of their Bass numbers of $H^c_I(R), c = \height I.$ As a main result it is shown that the vanishing $H^i_I(R) = 0$ for all $i \not= c$ i…
We study the syzygies of a codimension two ideal I = <f_1,f_2,f_3> in k[x,y,z]. Our main result is that the module of syzygies vanishing (scheme-theoretically) at the zero locus Z = V(I) is generated by the Koszul syzygies iff Z is a local complete intersection. The proof uses a characterization of complete intersections due to Herzog. When I is saturated, we relate our theorem to results of Weyman and of Simis and Vasconcelos. We conclude with an example of how our theorem fails for four generated local complete intersections in k[x,y,z] and we discuss generalizations to higher dimensions.
A local ring R is said to be a Roberts ring if tau_R([R]) = [Spec R]_dim R, where tau_R is the Riemann-Roch map for Spec R. Such rings satisfy a vanishing theorem for the Serre intersection multiplicity, as was established by Paul Roberts in his proof of the Serre vanishing conjecture. It is known that complete intersections are Roberts rings, and the first author proved that a determinantal ring is a Roberts ring precisely if it is complete intersection. Let A_d(n) denote the affine cone of the Grassmannian G_d(n) under the Plucker embedding. We determine which of the rings A_d(n) are Roberts…
The arithmetic rank of the residual intersections of a complete intersection ideal
研究完全交理想残差交的算术秩，确定其在任意特征下的上界，并给出其集合理论生成元的显式描述。
Let $(A,\mathfrak{m})$ be a complete intersection with $k = A/\mathfrak{m}$ algebraically closed. Let CMS(A) be the stable category of maximal CM $A$-modules. For a large class of thick subcategories $\mathcal{S}$ of CMS(A) we show that there is a theory of support varieties for the Verdier quotient $\mathcal{T} = $ CMS(A)$/\mathcal{S}$. As an application we show that the analogous version of Auslander-Reiten conjecture, Murthys result, Avramov-Buchweitz result on symmetry of vanishing of cohomology holds for $\mathcal{T}$.
Let $\mathscr{I}$ be an ideal sheaf on $P^n$. In the first part of this paper, we bound the asymptotic regularity of powers of $\mathscr{I}$ as $ps-3\leq \reg \mathscr{I}^p\leq ps+e$, where $e$ is a constant and $s$ is the $s$-invariant of $\mathscr{I}$. We also give the same upper bound for the asymptotic regularity of symbolic powers of $\mathscr{I}$ under some conditions. In the second part, by using multiplier ideal sheaves, we give a vanishing theorem of powers of $\mathscr{I}$ when it defines a local complete intersection subvariety with log canonical singularities.
In this paper, we focus on the initial degree and the vanishing of the Valabrega-Valla module of a pair of monomials ideals $J\subseteq I$ in a polynomials ring over a field $\mathbb{K}$. We prove that the initial degree of this module is bounded above by the maximum degree of a minimal generators of $J$. For edge ideals of graphs, a complete characterization of the vanishing of the Valabrega-Valla module is given. For higher degree ideals, we find classes which the Valabrega-Valla module vanishes. For the case that $J$ is the facet ideal of a clutter $\mathcal{C}$ and $I$ is the defining idea…
Let $R$ be a local complete intersection and $M,N$ are $R$-modules such that $\ell(\Tor_i^R(M,N))<\infty$ for $i\gg 0$. Imitating an approach by Avramov and Buchweitz, we investigate the asymptotic behavior of $\ell(\Tor_i^R(M,N))$ using Eisenbud operators and show that they have well-behaved growth. We define and study a function $η^R(M,N)$ which generalizes Serre's intersection multiplicity $χ^R(M,N)$ over regular local rings and Hochster's function $θ^R(M,N)$ over local hypersurfaces. We use good properties of $η^R(M,N)$ to obtain various results on complexities of $\Tor$ and $\Ext$, vanish…
Avramov and Buchweitz proved that for finitely generated modules $M$ and $N$ over a complete intersection local ring $R$, $\Ext^i_R(M,N)=0$ for all $i\gg 0$ implies $\Ext^i_R(N,M)=0$ for all $i\gg 0$. In this note we give some generalizations of this result. Indeed we prove the above mentioned result when (1) $M$ is finitely generated and $N$ is arbitrary, (2) $M$ is arbitrary and $N$ has finite length and (3) $M$ is complete and $N$ is finitely generated.
An ideal I of a local Cohen-Macaulay ring R is called a cohomologically complete intersection if H^i_I(R) = 0 for all i \neq c = height(I). Here H^i_I(R), i \in Z denotes the local cohomology of R with respect to I. For instance, a set-theoretic complete intersection is a cohomologically complete intersection. Here we study cohomologically complete intersections from various homological points of view. As a main result it is shown that the vanishing H^iI_(M) = 0 for all i \neq c is completely encoded in homological properties of H^cI_(M). These results extend those of Hellus and Schenzel (see …
Let $X$ be a closed equidimensional local complete intersection subscheme of a smooth projective scheme $Y$ over a field, and let $X_t$ denote the $t$-th thickening of $X$ in $Y$. Fix an ample line bundle $\mathcal{O}_Y(1)$ on $Y$. We prove the following asymptotic formulation of the Kodaira vanishing theorem: there exists an integer $c$, such that for all integers $t \geqslant 1$, the cohomology group $H^k(X_t,\mathcal{O}_{X_t}(j))$ vanishes for $k < \dim X$ and $j < -ct$. Note that there are no restrictions on the characteristic of the field, or on the singular locus of $X$. We also construc…
In this paper, motivated by a work of Luk and Yau, and Huneke and Wiegand, we study various aspects of the cohomological rigidity property of tensor product of modules over commutative Noetherian rings. We determine conditions under which the vanishing of a single local cohomology module of a tensor product implies the vanishing of all the lower ones, and obtain new connections between the local cohomology modules of tensor products and the Tate homology. Our argument yields bounds for the depth of tensor products of modules, as well as criteria for freeness of modules over complete intersecti…
Let $R$ be a local ring, and let $M$ and $N$ be finitely generated $R$-modules such that $M$ has finite complete intersection dimension. In this paper we define and study, under certain conditions, a pairing using the modules $\Ext_R^i(M,N)$ which generalizes Buchweitz's notion of the Herbrand diference. We exploit this pairing to examine the number of consecutive vanishing of $\Ext_R^i(M,N)$ needed to ensure that $\Ext_R^i(M,N)=0$ for all $i\gg 0$. Our results recover and improve on most of the known bounds in the literature, especially when $R$ has dimension at most two.
Indicator functions, v-numbers and Gorenstein rings in the theory of projective Reed-Muller-type codes
本文研究射影Reed-Muller型码的对偶性，利用v-数、Hilbert函数和指示函数给出全局对偶判据，推广了Gorenstein消失理想上的对偶定理，并分类了自对偶码及计算广义Hamming权函数的正则度指标。
In this paper we study a long-standing conjecture of Huneke and Wiegand which is concerned with the torsion submodule of certain tensor products of modules over one-dimensional local domains. We utilize Hochster's theta invariant and show that the conjecture is true for two periodic modules. We also make use of a result of Orlov and formulate a new condition which, if true over hypersurface rings, forces the conjecture of Huneke and Wiegand to be true over complete intersection rings of arbitrary codimension. Along the way we investigate the interaction between the vanishing of Tate (co)homolo…
Finiteness of homological dimensions of RHom complexes
本文研究了RHom复形和Ext模块的有限完备交集维度，证明了相关稳定性结果和标准，改进了Ghosh和Samanta的结果，并证明了Auslander-Reiten猜想。
Derived complete intersections and polynomial growth of Betti numbers over dg-algebras
本文研究了dg-代数上Betti数的多项式增长性质，推导出Gulliksen定理的导出版本，并建立了dg-代数模块的结构定理，同时扩展了Halperin关于局部环偏差消失的定理，从而恢复了Gulliksen定理。
Generalized symmetry in the vanishing of Ext
研究具有对偶复形的诺特环上同调有限复形 Ext 消失的广义对称性，通过额外同调条件扩展并统一局部戈伦斯坦环上相关对称结果，揭示对偶复形在其中的作用。
Let R be a (not necessarily local) Noetherian ring and M a finitely generated R-module of finite dimension d. Let \fa be an ideal of R and \fM denote the intersection of all prime ideals \fp in Supp_RH^d_{\fa}(M). It is shown that H^d_{\fa}(M)\simeq H^d_{\fM}(M)/\displaystyle{\sum_{n\in \mathbb{N}}}<\fM>(0:_{H^d_{\fM}(M)}\fa^n), where for an Artinian R-module A we put <\fM>A=\cap_{n\in \mathbb{N}} \fM^nA. As a consequence, it is proved that for all ideals \fa of R, there are only finitely many non-isomorphic top local cohomology modules H^d_{\fa}(M) having the same support. In addition, we est…
In this paper we study algebraic sets of pairs of matrices defined by the vanishing of either the diagonal of their commutator matrix or its anti-diagonal. We find a system of parameters for the coordinate rings of these two sets and their intersection and show that they are complete intersections. Moreover, we prove that these algebraic sets are $F$-pure over a field of positive prime characteristic and the algebraic set of pairs of matrices with the zero diagonal commutator is $F$-regular.
A quasi-complete intersection (q.c.i.) ideal of a local ring is an ideal with "free exterior Koszul homology"; the definition can also be understood in terms of vanishing of André-Quillen homology functors. Principal q.c.i. ideals are well understood, but few constructions are known to produce q.c.i. ideals of grade zero that are not principal. This paper examines the structure of q.c.i. ideals. We exhibit conditions on a ring $R$ which guarantee that every q.c.i. ideal of $R$ is principal. On the other hand, we give an example of a minimal q.c.i. deal $I$ which does not contain any principal …
The vanishing ideal I of a subspace arrangement is an intersection of linear ideals. We give a formula for the Hilbert polynomial of I if the subspaces meet transversally. We also give a formula for the Hilbert series of a product J of the linear ideals without any assumptions on the subspace arrangement. It turns out that the Hilbert series of J is a combinatorial invariant of the subspace arrangement: it only depends on the intersection lattice and the dimension function. The graded Betti numbers of J are determined by the Hilbert series, so they are combinatorial invariants as well. The res…
On the Ext Analog of the Euler Characteristic
本文研究欧拉特征的Ext类比——欧拉形式，并利用它解决Jorgensen关于完全交上模投射维数的问题，同时得到高阶Herbrand差消失的新结果。
Let $R$ be a standard graded algebra over a field $k$, with irrelevant maximal ideal $\fm$, and $I$ a homogeneous $R$-ideal. We study the asymptotic vanishing behavior of the graded components of the local cohomology modules $\{\HH{i}{\fm}{R/I^n}\}_{n\in \NN}$ for $i<\dim R/I$. We show that, when $\chara k= 0$, $R/I$ is Cohen-Macaulay, and $I$ is a complete intersection locally on $\Spec R \setminus\{\fm\}$, the lowest degrees of the modules $\{\HH{i}{\fm}{R/I^n}\}_{n\in \NN}$ are bounded by a linear function whose slope is controlled by the generating degrees of the dual of $I/I^2$. Our resul…
We show that a ring $R$ is regular if $Tor_{i}^{R}(R^{+},k) = 0$ for some $i\geq 1$ assuming further that $R$ is a $\mathbb{N}$-graded ring of dimension $2$ finitely generated over an equi-characteristic zero field $k$. This answers a question of Bhatt, Iyengar, and Ma. We use almost mathematics over $R^{+}$ to deduce properties of the noetherian ring $R$ and rational surface singularities. Moreover we show that $R^{+}$ in equi-characteristic zero is $m$-adically ideal(wise) separated, a condition which appears in the proof of local criterion for flatness. In dimension $2$ it is Ohm-Rush and i…
A higher algebraic approach to liftings of modules over derived quotients
本文用高阶代数方法，在自Ext^2消没条件下证明导出商上模提升的存在性，并借助Auslander零因子定理，得出该条件等价于局部完全交环。
We provide a formula (see Theorem 1.5) for the Matlis dual of the injective hull of $R/\mathfrak{p}$ where $\mathfrak p$ is a one dimensional prime ideal in a local complete Gorenstein domain $(R,\mathfrak{m})$. This is related to results of Enochs and Xu (see [4] and [3]). We prove a certain 'dual' version of the Hartshorne-Lichtenbaum vanishing (see Theorem 2.2). There is a generalization of local duality to cohomologically complete intersection ideals $I$ in the sense that for $I=\mathfrak{m}$ we get back the classical Local Duality Theorem. We determine the exact class of modules to which …
Let $R$ be a local complete intersection ring and let $M$ and $N$ be nonzero finitely generated $R$-modules. We employ Auslander's transpose in the study of the vanishing of Tor and obtain useful bounds for the depth of the tensor product $M\otimes_{R}N$. An application of our main argument shows that, if $M$ is locally free on the the punctured spectrum of $R$, then either $\depth(M\otimes_{R}N)\geq \depth(M)+\depth(N)-\depth(R)$, or $\depth(M\otimes_{R}N)\leq \cod(R)$. Along the way we generalize an important theorem of D. A. Jorgensen and determine the number of consecutive vanishing of $\T…
The generators of a colon ideal with an application to the weak Lefschetz property for monomial almost complete intersections in three variables
本文通过研究列理想生成元，探讨了三项式近完全交集的弱Lefschetz性质，并验证了一个猜想在特定情况下的正确性。
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