Symbolic powers,Serre conditions and Cohen-Macaulay Rees algebras

Summary LetR be a Cohen-Macaulay ring andI an unmixed ideal of heightg which is generically a complete intersection and satisfiesI ⁽ⁿ⁾=Iⁿ for alln≥1. Under what conditions will the Rees algebra be Cohen-Macaulay or have good depth? A series of partial answers to this question is given, relating the Serre condition (S _r ) of the associated graded ring to the depth of the Rees algebra. A useful device in arguments of this nature is the canonical module of the Rees algebra. By making use of the technique of the fundamental divisor, it is shown that the canonical module has the expected form: ω _R[It] ≅(t(1−t) ^g−2). The third author was partially supported by the NSF This article was processed by the author using theLaTex style filecljour1 from Springer-Verlag.     

The multiplicity and the Cohen-Macaulayness of extended Rees algebras of equimultiple ideals

Let I be an equimultiple ideal of Noetherian local ring A. This paper gives some multiplicity formulas of the extended Rees algebras T=A[It,t^-1]. In the case A generalized Cohen-Macaulay, we determine when T is Cohen-Macaulay and as an immediate consequence we obtain e.g., some criteria for the Cohen-Macaulayness of Rees algebra R(I) over a Cohen-Macaulay ring in terms of reduction numbers and ideals.     

On the Comparison of Rees's Valuations Over a Quasi-Unmixed Local Ring

In this paper, we will show that if (R, 𝔪) is a quasi-unmixed local ring, I an 𝔪-primary ideal of R and ?𝒱(I) is the set of Rees valuations of I, then the number of minimal prime ideals in the 𝔪-adic completion of R equals exactly the number of equivalence classes on the set ?𝒱(I) under the equivalence relation ～defined by: ν₁ ～ ν₂ if there exist a constant c ≥ 1 such that for all x ∈ R, ν₁(x) ≤ cν₂(x) and ν₂(x) ≤ cν₁(x).     

On the Rees algebra of certain codimension two perfect ideals

 Suppose ? is a set of arbitrary number of smooth points in ℙ² its defining ideal. In this paper, we study the Rees algebras of the ideals generated by I _t , t ≥α. When the points of ? are general, we give a set of defining equations for the Rees algebra . When the points of ? are arbitrary, we show that for all t≫ 0, the Rees algebra is Cohen-Macaulay and its defining ideal is generated by quadratics. A cohomological characterization for arithmetic Cohen-Macaulayness of subvarieties of a product space is also given. Received 4 April 2001     

Pure ℓ-Ideals in Lattice-Ordered Rings#

An ?-ideal I of a commutative lattice-ordered ring R with positive identity element is called a pure ?-ideal if R  =  I  + ?( x ) for each x  ∈  I , where ?(x) is the ?-annihilator of x in R . In this article, we give some results on pure ?-ideals and study the ?-ideal structure of a commutative lattice-ordered ring with positive identity element by using pure ?-ideals.     

Asymptotic Sequences over a Module

Let (R, 𝔪) be a Noetherian local ring and M be a submodule of the free module F = R ^r with height(I _r (M)) > 0. Asymptotic sequences over M will be defined analogous to Rees’ definition of asymptotic sequences over an ideal. It is then shown that all maximal asymptotic sequences over M have the same length. This length gives a bound on the analytic spread of M. Namely, if s is the length of a maximal asymptotic sequence over M then l(M) ≤dim R + rank M ? 1 ? s. Equality holds if R is quasi-unmixed.     

THE STRUCTURE OF THE TANGENT CONE: AN INTERESTING BIJECTION#

ABSTRACT

Let X = Spec(R) be a reduced equidimensional algebraic variety over an algebraically closed field k. Let Y = Spec(R/𝔮) be a codimension one ordinary multiple subvariety, where 𝔮 is a prime ideal of height 1 of R. If U is a nonempty open subset of Y and 𝔪 a closed point of U, we denote by A ? R _𝔪 its local ring in X, by 𝔭 the extension of 𝔮 in A, and by K the algebraic closure of the residue field k(𝔭).

Then there exists a bijection γ^𝔪:Proj(G _𝔭(A) ?  _A/𝔭 k) → Proj(G(A _𝔭) ?  _k(𝔭)K) such that for every subset Σ of Proj(G _𝔭(A) ?  _A/𝔭 k), the Hilbert function of Σ coincides with the Hilbert function of γ^𝔪(Σ). We examine some applications. We study the structure of the tangent cone at a closed point of a codimension one ordinary multiple subvariety.     

A Lower Bound on the Second Fiber Coefficient of the Fiber Cones

Let (R, 𝔪) be a Cohen–Macaulay local ring of dimension d > 0, I an 𝔪-primary ideal of R and K an ideal containing I. When depth G(I) ≥ d ? 1 and r(I | K) < ∞, we present a lower bound on the second fiber coefficient of the fiber cones, and also provide a characterization, in terms of f ₂(I, K), of the condition depth F _K (I) ≥ d ? 1.     

On Duality in the Homology Algebra of a Koszul Complex

The homology algebra of the Koszul complex K(x ₁, ..., x _n ; R) of a Gorenstein local ring R has Poincare duality if the ideal I = (x ₁, ..., x _n ) of R is strongly Cohen-Macaulay (i.e., all homology modules of the Koszul complex are Cohen-Macaulay) and under the assumption that dim R - grade I ⩽ 4 the converse is also true.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 77–81, 2003.     

Fiber Coefficients and Depth of Fiber Cones

Let (R, 𝔪) be a Cohen–Macaulay local ring of dimension d > 0, I an 𝔪-primary ideal of R, and K an ideal containing I. When r(I | K)<∞, we give a lower bound and an upper bound for f ₁(I). Under the above assumption on r(I | K) and depth G(I) ≥ d ? 1, we also provide a characterization, in terms of f ₁(I), of the condition depth F _K (I) ≥ d ? 1.     

Quadratic forms with values in invertible modules

LetR be a commutative ring,I an invertibleR-module, and consider quadratic spaces with values inI. The Clifford algebra of such a quadratic space is an algebra over the generalized Rees ring associated toI. We discuss the relation between the Witt module of quadratic spaces with values inI and the graded Witt ring and the graded Brauer-Wall group of the generalized Rees ring. This leads to the introduction of three distinguished subgroups of the graded Brauer-Wall group of the generalized Rees ring. The image of the Clifford functor is a subgroup of one of these three subgroups (the type 1 subgroup).     

Multi-symbolic Rees algebras and strong F-regularity

Let I be a divisorial ideal of a strongly F-regular ring A. K.-i. Watanabe raised the question whether the symbolic Rees algebra is Cohen-Macaulay whenever it is Noetherian. We develop the notion of multi-symbolic Rees algebras and use this to show that is indeed Cohen-Macaulay whenever a certain auxiliary ring is finitely generated over A. Received August 10, 1998 / in final form October 18, 1999 / Published online July 20, 2000     

A Family of Complex Irreducible Characters Possessed by Unitary and Special Unitary Groups Defined over Local Rings

Abstract

Let 𝒪 be a discrete valuation ring whose residue field 𝒪/𝔭 is finite and has odd characteristic. Let l be a positive integer. Set R = 𝒪/𝔭 ^l and let R = R[θ] be the ring obtained by adjoining to R a square root of a non-square unit. Consider the involution σ of R that fixes R elementwise and sends θ to ? θ. Let V be a free R-module of rank n > 0 endowed with a non-degenerate hermitian form ( , ) relative to σ. Let U _n (R) be the subgroup of GL(V) that preserves ( , ). Let SU _n (R) be the subgroup of all g ∈ U _n (R) whose determinant is equal to one. Let Ψ be the Weil character of U _n (R).

All irreducible constituents of Ψ are determined. An explicit character formula is given for each of them. In particular, all character degrees are computed. For n > 2 the corresponding results are also obtained for the restriction of Ψ to SU _n (R).     

Sequentially Cohen-Macaulay Modules Under Base Change

ABSTRACT

Assume that ?:(R, ± 𝔪) → (S, ± 𝔫) is a local flat homomorphism between commutative Noetherian local rings R and S. Let M be a finitely generated R-module. We investigate the ascent and descent of sequentially Cohen-Macaulay properties between the R-module M and the S-module M ? _R  S.     

On a Generalization of Injective Rings

Abstract

A ring R is called left IP-injective if every homomorphism from a left ideal of R into R with principal image is given by right multiplication by an element of R. It is shown that R is left IP-injective if and only if R is left P-injective and left GIN (i.e., r(I ∩ K) = r(I) + r(K) for each pair of left ideals I and K of R with I principal). We prove that R is QF if and only if R is right noetherian and left IP-injective if and only if R is left perfect, left GIN and right simple-injective. We also show that, for a right CF left GIN-ring R, R is QF if and only if Soc(R _R ) ? Soc( _R R). Two examples are given to show that an IP-injective ring need not be self-injective and a right IP-injective ring is not necessarily left IP-injective respectively.     

On the FIP Property for Extensions of Commutative Rings

ABSTRACT

A (unital) extension R ? T of (commutative) rings is said to have FIP (respectively be a minimal extension) if there are only finitely many (respectively no) rings S such that R ? S ? T. Transfer results for the FIP property for extensions of Nagata rings are obtained, including the following fact: if R ? T is a (module-) finite minimal ring extension, then R(X)?T(X) also is a (module-) finite minimal ring extension. The assertion obtained by replacing “is a (module-) finite minimal ring extension” with “has FIP” is valid if R is an infinite field but invalid if R is a finite field. A generalization of the Primitive Element Theorem is obtained by characterizing, for any field (more generally, any artinian reduced ring) R, the ring extensions R ? T which have FIP; and, if R is any field K, by describing all possible structures of the (necessarily minimal) ring extensions appearing in any maximal chain of intermediate rings between K and any such T. Transfer of the FIP and “minimal extension” properties is given for certain pullbacks, with applications to constructions such as CPI-extensions. Various sufficient conditions are given for a ring extension of the form R ? R[u], with u a nilpotent element, to have or not have FIP. One such result states that if R is a residually finite integral domain that is not a field and u is a nilpotent element belonging to some ring extension of R, then R ? R[u] has FIP if and only if (0 : u) ≠ 0. The rings R having only finitely many unital subrings are studied, with complete characterizations being obtained in the following cases: char(R)>0; R an integral domain of characteristic 0; and R a (module-)finite extension of ? which is not an integral domain. In particular, a ring of the last-mentioned type has only finitely many unital subrings if and only if (?:R)≠0. Some results are also given for the residually FIP property.     

Properties of the Fiber Cone of Ideals in Local Rings

Abstract

For an ideal I of a Noetherian local ring (R, m ) we consider properties of I and its powers as reflected in the fiber cone F(I) of I. In particular,we examine behavior of the fiber cone under homomorphic image R → R/J = R′ as related to analytic spread and generators for the kernel of the induced map on fiber cones ψ _J  : F _R (I) → F _R′(IR′). We consider the structure of fiber cones F(I) for which ker ψ _J  ≠ 0 for each nonzero ideal J of R. If dim F(I) = d > 0,μ(I) = d + 1 and there exists a minimal reduction J of I generated by a regular sequence,we prove that if grade(G ₊(I)) ≥ d ? 1,then F(I) is Cohen-Macaulay and thus a hypersurface.     

Commuting Derivations and Automorphisms of Certain Nilpotent Lie Algebras Over Commutative Rings

Let L be a finite-dimensional complex simple Lie algebra, L _? be the ?-span of a Chevalley basis of L, and L _R  = R ?_? L _? be a Chevalley algebra of type L over a commutative ring R. Let 𝒩(R) be the nilpotent subalgebra of L _R spanned by the root vectors associated with positive roots. A map ? of 𝒩(R) is called commuting if [?(x), x] = 0 for all x ∈ 𝒩(R). In this article, we prove that under some conditions for R, if Φ is not of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a central derivation (resp., automorphism), and if Φ is of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a sum (resp., a product) of a graded diagonal derivation (resp., automorphism) and a central derivation (resp., automorphism).     

The Cohen–Macaulay and Gorenstein Properties of Rings Associated to Filtrations

Let (R, m) be a Cohen–Macaulay local ring, and let ? = {F _i } _i∈? be an F ₁-good filtration of ideals in R. If F ₁ is m-primary we obtain sufficient conditions in order that the associated graded ring G(?) be Cohen–Macaulay. In the case where R is Gorenstein, we use the Cohen–Macaulay result to establish necessary and sufficient conditions for G(?) to be Gorenstein. We apply this result to the integral closure filtration ? associated to a monomial parameter ideal of a polynomial ring to give necessary and sufficient conditions for G(?) to be Gorenstein. Let (R, m) be a Gorenstein local ring, and let F ₁ be an ideal with ht(F ₁) = g > 0. If there exists a reduction J of ? with μ(J) = g and reduction number u: = r _J (?), we prove that the extended Rees algebra R′(?) is quasi-Gorenstein with a-invariant b if and only if J ⁿ : F _u  = F _n+b?u+g?1 for every n ∈ ?. Furthermore, if G(?) is Cohen–Macaulay, then the maximal degree of a homogeneous minimal generator of the canonical module ω _G(?) is at most g and that of the canonical module ω _R′(?) is at most g ? 1; moreover, R′(?) is Gorenstein if and only if J ^u : F _u  = F _u . We illustrate with various examples cases where G(?) is or is not Gorenstein.