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 $i \not= c$ is completely encoded in homological properties of $H^c_I(R),$ in particular in its Bass numbers.
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 $I$ 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 $H^c_{I}(R)$.
It is a well-known result that, in projective space over a field, every set-theoretical complete intersection of positive dimension in connected in codimension one (Hartshorne [H1,3.4.6] or [H2, Theorem 1.3]). Another important connectedness result is that a local ring with disconnected punctured sprectrum has depth at most $1$ ([H1, Proposition 2.1]). The two results are related, Hartshorne calls the latter "the keystone to the proof" of the former (loc. cit). In this short note we show that the latter result generalizes smoothly from set-theoretical to cohomologically complete intersections,…
Let $(R,\fm)$ be a local ring and $\fa$ be an ideal of $R$. The inequalities $$\begin{array}{ll} \ \Ht(\fa) \leq \cd(\fa,R) \leq \ara(\fa) \leq l(\fa) \leq μ(\fa) \end{array}$$ are known. It is an interesting and long-standing problem to find out the cases giving equality. Thanks to the formal grade we give conditions in which the above inequalities become equalities.
Let A be a commutative noetherian ring, and \a an ideal in it. In this paper we continue the study, begun in [PSY1], of the derived \a-adic completion and the derived \a-torsion functors. Here are our results: (1) a structural characterization of bounded above cohomologically complete complexes; (2) the Cohomologically Complete Nakayama Theorem; and (3) a characterization of cohomologically cofinite complexes.
We prove two theorems on cohomologically complete complexes. These theorems are inspired by, and yield an alternative proof of, a recent theorem of P. Schenzel on complete modules.
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 I be an ideal of a Complete Cohen-Macaulay local ring R of dimension n. We wil show that the natural homomorphism Rto HomR(HcI(KR), HcI(KR)) is an isomorphism provided that I is a cohomologically compltete intersection ideal of grade c where KR (resp. HiI(.)) denote the canonical module (resp. i-th local cohomology with respect to the ideal I) of R. The same result is true for the Matlis dual of HcI(KR).
In this paper we completely characterize lattice ideals that are complete intersections or equivalently complete intersections finitely generated semigroups of $\bz^n\oplus T$ with no invertible elements, where $T$ is a finite abelian group. We also characterize the lattice ideals that are set-theoretic complete intersections on binomials.
Using the connections among almost complete intersection schemes, arithmetically Gorenstein schemes and schemes that are union of complete intersections we give a structure theorem for arithmetically Cohen-Macaulay union of two complete intersections of codimension $2.$ We apply the results for computing Hilbert functions and graded Betti numbers for such schemes.
We study the complete intersection property of monomial curves in the family $Γ_{å+ \jj} = {(t^{a_0 + j}, t^{a_1+j},..., t^{a_n + j}) ~ | ~ j \geq 0, ~ a_0 < a_1 <...< a_n}$. We prove that if $Γ_{å+\jj}$ is a complete intersection for $j \gg0$, then $Γ_{å+\jj+\underline{a_n}}$ is a complete intersection for $j \gg 0$. This proves a conjecture of Herzog and Srinivasan on eventual periodicity of Betti numbers of semigroup rings under translations for complete intersections. We also show that if $Γ_{å+\jj}$ is a complete intersection for $j \gg 0$, then $Γ_å$ is a complete intersection. We also c…
In this paper, we investigate the multiplicities and the log canonical thresholds of abelian quotient complete intersection singularities in term of the special datum. Moreover we give bounds of the multiplicity of abelian quotient complete intersection singularities.
We ask when certain complete intersections of codimension $r$ can lie on a generic hypersurface in $\PP^n$. We give a complete answer to this question when $2r \leq n+2$ in terms of the degrees of the hypersurfaces and of the degrees of the generators of the complete intersection.
$V$ 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.
It is proved in this paper that a locally complete intersection curve in a smooth affine C-algebra with trival conormal bundle is a set theoretic complete intersection if its corresponding class in the Grothendieck Group is torsion.
Waring rank of certain complete intersection forms
Nanyang Technological University（南洋理工大学）；Politecnico Di Torino（都灵理工大学）；Indian Institute of Technology Dharwad（达尔瓦德印度理工学院）；Indian Institute of Technology Bombay（孟买印度理工学院）
In this paper we describe all possible reduced complete intersection sets of points on Veronese surfaces. We formulate a conjecture for the general case of complete intersection subvarieties of any dimension and we prove it in the case of the quadratic Veronese threefold. Our main tool is an effective characterization of all possible Hilbert functions of reduced subvarieties of Veronese surfaces.
We show that every complete intersection of Laurent polynomials in an algebraic torus is isomorphic to a complete intersection of master functions in the complement of a hyperplane arrangement, and vice versa. We call this association Gale duality because the exponents of the monomials in the polynomials annihilate the weights of the master functions. We use Gale duality to give a Kouchnirenko theorem for the number of solutions to a system of master functions and to compute some topological invariants of generic master function complete intersections.
We provide a structure theorem for all almost complete intersection ideals of depth three in any Noetherian local ring. In particular, we find that the minimal generators are the pfaffians of suitable submatrices of an alternating matrix. From the graded version of the previous result, we characterize the graded Betti numbers of all 3-codimensional almost complete intersection schemes of P^r.
Valuation rings as limits of complete intersection rings
Simion Stoilow Institute of Mathematics of the Romanian Academy（罗马尼亚科学院西奥多·斯托伊洛数学研究所）；University of Bucharest, Faculty of Mathematics and Computer Science（布加勒斯特大学数学与计算机科学系）
In this note we build on our previous work with Takehiko Yasuda to prove a precise version of inversion of adjunction for varieties which are local complete intersections.
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.
We characterize the graphs $G$ for which their toric ideals $I_G$ are complete intersections. In particular we prove that for a connected graph $G$ such that $I_G$ is complete intersection all of its blocks are bipartite except of at most two. We prove that toric ideals of graphs which are complete intersections are circuit ideals. The generators of the toric ideal correspond to even cycles of $G$ except of at most one generator, which corresponds to two edge disjoint odd cycles joint at a vertex or with a path. We prove that the blocks of the graph satisfy the odd cycle condition. Finally 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.
A commutative local ring is generally defined to be a complete intersection if its completion is isomorphic to the quotient of a regular local ring by an ideal generated by a regular sequence. It has not previously been determined whether or not such a ring is necessarily itself the quotient of a regular ring by an ideal generated by a regular sequence. In this article, it is shown that if a complete intersection is a one dimensional integral domain, then it is such a quotient. However, an example is produced of a three dimensional complete intersection domain which is not a homomorphic image …
We extend the notions of complete intersection dimension and lower complete intersection dimension to the category of complexes with finite homology and verify basic properties analogous to those holding for modules. We also discuss the question of the behavior of complete intersection dimension with respect to short exact sequences.
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.
Let $(A,\mathfrak{m})$ be an abstract complete intersection and let $P$ be a prime ideal of $A$. In [1] Avramov proved that $A_P$ is an abstract complete intersection. In this paper we give an elementary proof of this result.
We establish the upper bound in the multiplicity conjecture of Herzog, Huneke and Srinivasan for the codimension three almost complete intersections. We also give some partial results in the case where I is the aci linked to a complete intersection in one step.
In this paper we classify the monomial complete intersection algebras, in two variables, and of positive characteristic, which has the strong Lef- schetz property. Together with known results, this gives a complete classi- fication of the monomial complete intersections with the strong Lefschetz property.
We compute the resultants for quadratic binomial complete intersections. As an application we show that any quadratic binomial complete intersection can have the set of square-free monomials as a vector space basis if the generators are put in a normal form.
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).
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).
We show that the Stanley-Reisner ideal of the one-dimensional simplicial complex whose diagram is an $n$-gon is always a set-theoretic complete intersection in any positive characteristic.
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.
We prove that any quadratic complete intersection with certain action of the symmetric group has the strong Lefschetz property over a field of characteristic zero. As a consequence of it we construct a new class of homogeneous complete intersections with generators of higher degrees which have the strong Lefschetz property.
Stanley proved that, in characteristic zero, all artinian monomial complete intersections have the strong Lefschetz property. We provide a positive characteristic complement to Stanley's result in the case of artinian monomial complete intersections generated by monomials all of the same degree, and also for arbitrary artinian monomial complete intersections in characteristic two. To establish these results, we first prove an a priori lower bound on the characteristics that guarantee the Lefschetz properties. We then use a variety of techniques to complete the classifications.
We show that, over a local complete intersection, every possible variety is realized as the cohomological support variety of some module. Moreover, we show that the projective variety of a complete indecomposable maximal Cohen-Macaulay module is connected.
We prove the strong Lefschetz property for certain complete intersections defined by products of linear forms, using a characterization of the strong Lefschetz property in terms of central simple modules.
In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen-Macaulay.
Let $A$ be a graded complete intersection over a field and $B$ the monomial complete intersection with the generators of the same degrees as $A$. The EGH conjecture says that if $I$ is a graded ideal in $A$, then there should be an ideal $J$ in $B$ such that $B/J$ and $A/I$ have the same Hilbert function. We show that if the EGH conjecture is true, then it can be used to prove that every graded complete intersection over any field has the Sperner property.
We prove that every quasi-complete intersection ideal is obtained from a pair of nested complete intersection ideals by way of a flat base change. As a by-product we establish a rigidity statement for the minimal two-step Tate complex associated to an ideal $I$ in a local ring $R$. Furthermore, we define a minimal two-step complete Tate complex $T$ for each ideal $I$ in a local ring $R$; and prove a rigidity result for it. The complex $T$ is exact if and only if $I$ is a quasi-complete intersection ideal; and in this case, $T$ is the minimal complete resolution of $R/I$ by free $R$-modules.
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.
In this paper, we prove that the Stanley--Reisner ideal of any connected simplicial complex of dimension $\ge 2$ that is locally complete intersection is a complete intersection ideal. As an application, we show that the Stanley--Reisner ideal whose powers are Buchsbaum is a complete intersection ideal.
Using an alternate description of support varieties of pairs of modules over a complete intersection, we give several new applications of such varieties, including results for support varieties of intermediate complete intersections. Simpler proofs of known results are also given.
