Liana M Sega
We prove that if M, N are finite modules over a Gorenstein local ring R of codimension at most 4, then the vanishing of Ext^n_R(M,N) for n\gg 0 is equivalent to the vanishing of Ext^n_R(N,M) for n\gg 0. Furthermore, if the completion of $R$ has no embedded deformation, then such vanishing occurs if and only if M or N has finite projective dimension.
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Nearly Gorenstein projective monomial curves of small codimension
本文研究小余维数情形下的近Gorenstein射影单项曲线，完整刻画了余维数为2和3的这类曲线的结构与分类。
Structure theorems for Gorenstein ideals of codimension four with small number of generators
本文用表示论方法研究余维数为四的Gorenstein理想的极小自由消解，引入高阶结构映射族，并证明由六个元素极小生成的该类理想必为余维数为三的Gorenstein理想的超平面截面，同时提出七、八个生成元情形的猜想。
In this note, we calculate the Koszul homology of the codimension 3 Gorenstein ideals. We find filtrations for the Koszul homology in terms of modules with pure free resolutions and completely describe these resolutions. We also consider the Huneke-Ulrich deviation 2 ideals.
Log-concave Gorenstein sequences
该研究探讨不同余维数下相关序列的对数凹性，证明3维余Artinian Gorenstein序列对数凹、存在4维余非对数凹该类序列，且2维余level序列及任意余维compressed level希尔伯特函数均对数凹。
In this note we supply an elementary proof of the following well-known theorem of R. Stanley: the $h$-vectors of Gorenstein algebras of codimension 3 are SI-sequences, i.e. are symmetric and the first difference of their first half is an $O$-sequence.
The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree) of a Cohen-Macaulay standard graded algebra with certain extremal graded Betti numbers in its minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a stronger conjecture. We prove this conjecture for the case of codimension three graded Gorenstein algebras.
We study the weak Lefschetz property of a class of graded Artinian Gorenstein algebras of codimension three associated in a natural way to the Apéry set of a numerical semigroup generated by four natural numbers. We show that these algebras have the weak Lefschetz property whenever the initial degree of their defining ideal is small.
We study Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. We introduce rank matrices of linear forms for such algebras that represent the ranks of multiplication maps in various degrees. We show that there is a 1-1 correspondence between rank matrices and Jordan degree types. For Artinian Gorenstein algebras with codimension three we classify all rank matrices that occur for linear forms with vanishing third power. As a consequence, we show for such algebras that the possible Jordan types with parts of length at most four are uniquely determine…
The main goal of this paper is to characterize the Hilbert functions of all (artinian) codimension 4 Gorenstein algebras that have at least two independent relations of degree four. This includes all codimension 4 Gorenstein algebras whose initial relation is of degree at most 3. Our result shows that those Hilbert functions are exactly the so-called {\em SI-sequences} starting with (1,4,h_2,h_3,...), where $h_4 \leq 33$. In particular, these Hilbert functions are all unimodal. We also establish a more general unimodality result, which relies on the values of the Hilbert function not being too…
We analyze the structure of spinor coordinates on resolutions of Gorenstein ideals of codimension four. As an application we produce a family of such ideals with seven generators which are not specializations of Kustin-Miller model.
The famous structure theorem of Buchsbaum and Eisenbud gives a complete characterization of Gorenstein ideals of codimension 3 and their minimal free resolutions. We generalize the ideas of Buchsbaum and Eisenbud from Gorenstein ideals to Gorenstein algebras and present a description of Gorenstein algebras of any odd codimension. As an application we study the canonical ring of a numerical Godeaux surface.
We prove that the Hilbert functions of codimension four graded Gorenstein Artin algebras R/I are unimodal provided I has a minimal generator in degree less than five. It is an open question whether all Gorenstein h-vectors in codimension four are unimodal. In this paper, we prove that Hilbert functions of all artinian codimension four Gorenstein algebras starting with (1,4,10, 20, h_4,...), where h_4\leq 34 are unimodal. Combining this with the previously known results, we obtain that all Gorenstein h-vectors (1, h_1, h_2,h_3, h_4, ....) are unimodal if h_4\leq 3.
This paper applies G. Lyubeznik's notion of $F$-finite modules to describe in a very down-to-earth manner certain annihilator submodules of some top local cohomology modules over Gorenstein rings. As a consequence we obtain an explicit description of the test ideal of Gorenstein rings in terms of ideals in a regular ring.
We investigate symmetry in the vanishing of Tate cohomology for finitely generated modules over local Gorenstein rings. For finitely generated R-modules M and N over Gorenstein local ring R, it is shown that $\widehat{Ext}^i_R(M,N)=0$ for all $i\in\mathbb{Z}$ if and only if $\widehat{Ext}^i_R(N,M)=0$ for all $i\in\mathbb{Z}$.
We define a notion of Gorenstein flat dimension for unbounded complexes over left GF-closed rings. Over Gorenstein rings we introduce a notion of Gorenstein cohomology for complexes; we also define a generalized Tate cohomology for complexes over Gorenstein rings, and we show that there is a close connection between the absolute, the Gorenstein and the generalized Tate cohomology.
We determine new bounds on the entries of Gorenstein Hilbert functions, both in any fixed codimension and asymptotically. Our first main theorem is a lower bound for the degree $i+1$ entry of a Gorenstein $h$-vector, in terms of its entry in degree $i$. This result carries interesting applications concerning unimodality: indeed, an important consequence is that, given $r$ and $i$, all Gorenstein $h$-vectors of codimension $r$ and socle degree $e\geq e_0=e_0(r,i)$ (this function being explicitly computed) are unimodal up to degree $i+1$. This immediately gives a new proof of a theorem of Stanle…
We study Gorenstein ideals of codimension $4$ derived from generic doublings of almost complete intersection perfect ideals of codimension $3$. We also investigate spinor coordinates of such Gorenstein ideals with $8$ and $9$ generators. For an ideal $J$ of commutative ring $R$, the $R/J$ module $J/J^2$ is called conormal module and $R/J$-dual of $J/J^2$ is called normal module. We study properties of conormal and normal modules of almost complete intersection perfect ideals of codimension $3$.
The purpose of this paper is to give some equivalent conditions to the socle and Bass numbers' conjectures which raised by C. Huneke in (Problems on local cohomology, Free resolutions in commutative algebra and algebraic geometry, Res. Notes Math. 1992, pp. 93-108). In addition, some results about certain Gorenstein ideals are included.
Jordan degree type for codimension three Gorenstein algebras of small Sperner number
该研究将余维数2分次代数的Jordan次数型研究拓展至小Sperner数的余维数3分次Artinian Gorenstein代数，完全确定了s=3,4,5时对应Gorenstein序列的所有可能JDT，并划定了s=6时的潜在JDT范围。
Assume $R$ is a polynomial ring over a field and $I$ is a homogeneous Gorenstein ideal of codimension $g\ge3$ and initial degree $p\ge2$. We prove that the number of minimal generators $ν(I_p)$ of $I$ that are in degree $p$ is bounded above by $ν_0={p+g-1\choose g-1}-{p+g-3\choose g-1}$, which is the number of minimal generators of the defining ideal of the extremal Gorenstein algebra of codimension $g$ and initial degree $p$. Further, $I$ is itself extremal if $ν(I_p)=ν_0$.
We focus on the structure of a homogeneous Gorenstein ideal $I$ of codimension three in a standard polynomial ring $R=\kk[x_1,\ldots,x_n]$ over a field $\kk$, assuming that $I$ is generated in a fixed degree $d$. For such an ideal $I$ this degree comes along with the minimal number of generators of $I$ and the degree of the entries of the associated skew-symmetric matrix in a simple formula. We give an elementary characteristic-free argument to the effect that, for any such data linked by this formula, there exists a Gorenstein ideal $I$ of codimension three filling them. We conjecture that, f…
Codimension two Artinian algebras $A$ have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras - the most promising results so far have concerned the weak Lefschetz property for such algebras. We here show that every standard-graded codimension three Artinian Gorenstein algebra $A$ having low maximum value of the Hilbert function - at most six - has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is grea…
Annihilation of cohomology over one dimensional almost Gorenstein rings
本文研究一维几乎Gorenstein环的上同调湮灭理想，给出其与稳定范畴湮灭理想相等的充分条件，并证明其与导子理想一致。
On the weak Lefschetz property of Artinian Gorenstein algebras of codimension three in arbitrary characteristic
本文将特征零代数闭域上三维余维Artinian Gorenstein代数的弱Lefschetz性质结论推广到任意特征的无限域，同时证明了相关的公因子结论。
I describe the projective resolution of a codimension 4 Gorenstein ideal, aiming to extend Buchsbaum and Eisenbud's famous result in codimension 3. The main result is a structure theorem stating that the ideal is determined by its (k+1) x 2k matrix of first syzygies, viewed as a morphism from the ambient regular space to the Spin-Hom variety SpH_k in Mat(k+1,2k). This is a general result encapsulating some theoretical aspects of the problem, but, as it stands, is still some way from tractable applications.
We give counterexamples to the following conjecture of Auslander: given a finitely generated module $M$ over an Artin algebra $Λ$, there exists a positive integer $n_M$ such that for all finitely generated $Λ$-modules $N$, if $\Ext_Λ^i(M,N)=0$ for all $i\gg 0$, then $\Ext_Λ^i(M,N)=0$ for all $i\geq n_M$. Some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local Gorenstein rings.
A Gorenstein A-algebra R of codimension 2 is a perfect finite A-algebra such that R=Ext^2(R,A) holds as R-modules, A being a Cohen-Macaulay local ring with dim(A)-dim_A(R)=2. I prove a structure theorem for these algebras improving on an old theorem of M. Grassi. Special attention is paid to the question how the ring structure of R is encoded in its Hilbert resolution. It is shown that R is automatically a ring once one imposes a weak depth condition on a determinantal ideal derived from a presentation matrix of R over A. The interplay of Gorenstein algebras and Koszul modules as introduced by…
Higher Lorentzian Polynomials, Higher Hessians, and the Hodge-Riemann Property for Graded Oriented Artinian Gorenstein Algebras in Codimension Two
该研究引入$i$-洛伦兹多项式，证明其为余维2有向阿廷戈伦斯坦代数的Macaulay对偶生成元，建立其与全非负Toeplitz矩阵的一一对应，推导相关推论并给出新的Toeplitz矩阵惠特尼型定理。
New families of Artinian Gorenstein algebras with the weak Lefschetz property
本文构造了任意余维数且具有二项式 Macaulay 对偶生成元的 Artinian Gorenstein 代数新族，并证明其满足弱或强 Lefschetz 性质，是相关姊妹篇的补充。
Let $R$ be a Gorenstein local ring with maximal ideal $\mathfrak{m}$ satisfying $\mathfrak{m}^3=0\ne\mathfrak{m}^2$. Set $k=R/\mathfrak{m}$ and $e=\text{rank}_{k}(\mathfrak{m}/\mathfrak{m}^2)$. If $e>2$ and $M$, $N$ are finitely generated $R$-modules, we show that the formal power series $\sum_{i=0}^\infty\text{rank}_{k}\left(\text{Ext}^i_R(M,N)\otimes_R k \right)t^i$ and $\sum_{i=0}^\infty\text{rank}_{k}\left(\text{Tor}_i^R(M,N)\otimes_R k \right)t^i$ are rational, with denominator $1-et+t^2$.
Artinian Gorenstein algebras with binomial Macaulay dual generator
本文系统研究具有二项式Macaulay对偶生成元的Artinian Gorenstein代数，在余维数3证明其满足强Lefschetz性质并给出完全交刻画，同时为任意余维数建立最优的弱Lefschetz性质充分条件。
In this paper I investigate minimal surfaces of general type with p_g=5, q=0 for which the 1-canonical map is a birational morphism onto a surface in P^4 (so called canonical surfaces in P^4) via a structure theorem for the Hilbert resolutions of the canonical rings of the afore-mentioned surfaces, viewed as Gorenstein algebras of codimension 2 over the homogeneous coordinate ring of P^4. I discuss how the ring structure of such an algebra is encoded in its resolution. Among other things I show how this method can be applied to analyze the moduli space of canonical surfaces with p_g=5, q=0, K^…
The aim of this paper is to study the theory of cohomology annihilators over commutative Gorenstein rings. We adopt a triangulated category point of view and study the annihilation of stable category of maximal Cohen-Macaulay modules. We prove that in dimension one the cohomology annihilator ideal and the conductor ideal coincide under mild assumptions. We present a condition on a ring homomorphism between Gorenstein rings which allows us to carry the cohomology annihilator of the domain to that of the codomain. As an application, we generalize the Milnor-Jung formula for algebraic curves to t…
Let $\fa$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$-module. For an $R$-module $X$, we denote any of the quantities $\fd_R X$, $\Gfd_R X$ and $\GCfd_RX$ by $\T(X)$. Let $M$ be an $R$-module such that $\H_{\fa}^i(M)=0$ for all $i\neq n$. It is proved that if $\T(X)<\infty$, then $\T(\H_{\fa}^n(M))\leq\T(M)+n$ and the equality holds whenever $M$ is finitely generated. With the aid of these results, among other things, we characterize Cohen-Macaulay modules, dualizing modules and Gorenstein rings.
We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function (1,3,6,6,3,1), we give a complete answer in every characteristic by translating the problem to one of studying geometric aspects of certain morphisms from $\mathbb P^2$ to $\mathbb P^3$, and Hesse configurations in $\mathbb P^2$.
Let $R$ be a Gorenstein local ring, $\frak{a}$ an ideal in $R$, and $M$ an $R$-module. The local cohomology of $M$ supported at $\frak{a}$ can be computed by applying the $\frak{a}$-torsion functor to an injective resolution of $M$. Since $R$ is Gorenstein, $M$ has a complete injective resolution, so it is natural to ask what one gets by applying the $\frak{a}$-torsion functor to it. Following this lead, we define stable local cohomology for modules with complete injective resolutions. This gives a functor to the stable category of Gorenstein injective modules. We show that in many ways this b…
Let $(R,\fm)$ be a local ring and let $C$ be a semidualizing $R$--module. In this paper, we are concerned in $C$--injective and $G_{C}$--injective dimensions of certain local cohomology modules of $R$. Firstly, the injective dimension of $C$ and the above quantities of dimensions is compared. Then, as an application of the above comparisons, a characterization of a dualizing module of $R$ is given. Finally, it is shown that if $R$ is Cohen-Macaulay of dimension $d$ such that $\H_{\fm}^{d}(C)$ is $C$--injective, then $R$ is Gorernstein. This is an answer to the question which was recently prese…
We study the Rees algebra of a perfect Gorenstein ideal of codimension 3 in a hypersurface ring. We provide a minimal generating set of the defining ideal of these rings by introducing a modified Jacobian dual and applying a recursive algorithm. Once the defining equations are known, we explore properties of these Rees algebras such as Cohen-Macaulayness and Castelnuovo-Mumford regularity.
The weak Lefschetz property for artinian Gorenstein algebras of small Sperner number
本文证明了基座度数为d且Sperner数至多为d+1的artinian Gorenstein代数均满足弱Lefschetz性质，且与余维数无关，并指出该界限在一般情况下是严格的。
We study Gorenstein liaison of codimension two subschemes of an arithmetically Gorenstein scheme X. Our main result is a criterion for two such subschemes to be in the same Gorenstein liaison class, in terms of the category of ACM sheaves on X. As a consequence we obtain a criterion for X to have the property that every codimension 2 arithmetically Cohen-Macaulay subscheme is in the Gorenstein liaison class of a complete intersection. Using these tools we prove that every arithmetically Gorenstein subscheme of $\mathbb{P}^n$ is in the Gorenstein liaison class of a complete intesection and we a…
In this paper several quasi-Gorenstein counterparts to some known properties of Gorenstein rings are given. We, furthermore, give an explicit description of the attach prime ideals of certain local cohomology modules.
A projectively normal Calabi-Yau threefold $X \subseteq \mathbb{P}^n$ has an ideal $I_X$ which is arithmetically Gorenstein, of Castelnuovo-Mumford regularity four. Such ideals have been intensively studied when $I_X$ is a complete intersection, as well as in the case where $X$ is codimension three. In the latter case, the Buchsbaum-Eisenbud theorem shows that $I_X$ is given by the Pfaffians of a skew-symmetric matrix. A number of recent papers study the situation when $I_X$ has codimension four. We prove there are 16 possible betti tables for an arithmetically Gorenstein ideal $I$ with $\math…
We investigate relative cohomology functors on subcategories of abelian categories via Auslander-Buchweitz approximations and the resulting strict resolutions. We verify that certain comparison maps between these functors are isomorphisms and introduce a notion of perfection for this context. Our main theorem is a balance result for relative cohomology that simultaneously recovers theorems of Holm and the current authors as special cases.
We study the concept of liaison addition for codimension two subschemes of an arithmetically Gorenstein projective scheme. We show how it relates to liaison and biliaison classes of subschemes and use it to investigate the structure of Gorenstein liaison equivalence classes, extending the known theory for complete intersection liaison of codimension two subschemes. In particular, we show that on the non-singular quadric threefold in projective 4-space, every non-licci ACM curve can be obtained from a single line by successive liaison additions with lines and CI-biliaisons.
It has been conjectured that {\it all} graded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded algebras $A$ of the Apéry set of $M$-pure symmetric numerical semigroups generated by four natural numbers. In 2010, Bryant proved that these algebras are graded Artinian Gorenstein algebras of codimension three. In a recent article, Guerrieri showed that if $A$ is not a complete intersection, then $A$ is of form $A=R/I$ with $R=K[x,y,z]$ and \begin{align*} I…
A generalization of Grothendieck's non-vanishing theorem is proved for a module which is finite over a local homomorphism. It is also proved that the Gorenstein injective dimension of such a module, if finite, is bounded below by its Krull dimension and is equal to the supremum of the depths of the localizations of the ring over primes in the support of the module.
Free resolutions and Lefschetz properties of some Artin Gorenstein rings of codimension four
本文针对余维数4、正则度不超过6的Artinian Gorenstein环，研究其极小自由分解与弱/强Lefschetz性质及Jordan型之间的关系。
The Jordan type of an Artinian algebra is the Jordan block partition associated to multiplication by a generic element of the maximal ideal. We study the Jordan type for Artinian Gorenstein (AG) local algebras A, and the interaction of Jordan type with the symmetric decomposition of the Hilbert function H(A). We give examples of Gorenstein sequences H for which the family Gor(H) of AG algebras having Hilbert function H has several irreducible components, each corresponding to a symmetric decomposition of H. The component structure results from the intersection of two opposing filtrations of th…
We classify all possible $h$-vectors of graded artinian Gorenstein algebras in socle degree 4 and codimension $\leq 17$, and in socle degree 5 and codimension $\leq 25$. We obtain as a consequence that the least number of variables allowing the existence of a nonunimodal Gorenstein $h$-vector is 13 for socle degree 4, and 17 for socle degree 5. In particular, the smallest nonunimodal Gorenstein $h$-vector is $(1,13,12,13,1)$, which was constructed by Stanley in his 1978 seminal paper on level algebras. This solves a long-standing open question in this area. All of our results are characteristi…
For homomorphism K-->S of commutative rings, where K is Gorenstein and S is essentially of finite type and flat as a K-module, the property that all non-trivial fiber rings of K-->S are Gorenstein is characterized in terms of properties of the cohomology modules Ext_n^{S\otimes_KS}S{S\otimes_KS}.
Let $H=\langle n_1, \ldots ,n_4\rangle$ be a numerical semigroup generated by $4$ elements, which is symmetric and let $k[H]$ be the semigroup ring of $H$ over a field $k$. H. Bresinski proved that the defining ideal of $k[H]$ is minimally generated by $3$ or $5$ elements. We give a new short proof of Bresinski's Theorem using the structure theorem of Buchsbaum and Eisenbud on the minimal free resolution of Gorenstein rings of embedding codimension $3$.
Let $(R,\fm)$ be a commutative Noetherian local ring and let $M$ be an $R$-module which is a relative Cohen-Macaulay with respect to a proper ideal $\fa$ of $R$ and set $n:=\h_{M}\fa$. We prove that $\ind M<\infty$ if and only if $\ind\H^{n}_\fa(M)<\infty$ and that $\ind\H^{n}_\fa(M)=\ind M-n$. We also prove that if $R$ has a dualizing complex and $\Gid_{R} M<\infty$, then $\Gid_{R}\H^{n}_\fa(M)<\infty$ and $\Gid_{R}\H^{n}_\fa(M)=\Gid_{R} M-n$. Moreover if $R$ and $M$ are Cohen-Macaulay, then it is proved that $\Gid_{R} M<\infty$ whenever $\Gid_{R}\H^{n}_\fa(M)<\infty$. Next, for a finitely ge…
We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.
Let $X$ be a normal arithmetically Gorenstein scheme in ${\mathbb P}^n$. We give a criterion for all codimension two ACM subschemes of $X$ to be in the same Gorenstein biliaison class on $X$, in terms of the category of ACM sheaves on $X$. These are sheaves that correspond to the graded maximal Cohen--Macaulay modules on the homogeneous coordinate ring of $X$. Using known results on MCM modules, we are able to determine the Gorenstein biliaison classes of codimension two subschemes of certain varieties, including the nonsingular quadric surface in ${\mathbb P}^3$, and the cone over it in ${\ma…
Graded skew-commutative rings occur often in practice. Here are two examples: 1) The cohomology ring of a compact three-dimensional manifold. 2) The cohomology ring of the complement of a hyperplane arrangement (the Orlik-Solomon algebra). We present some applications of the homological theory of these graded skew-commutative rings. In particular we find compact oriented 3-manifolds without boundary for which the Hilbert series of the Yoneda Ext-algebra of the cohomology ring of the fundamental group is an explicit transcendental function. This is only possible for large first Betti numbers of…
On the Hilbert function of Artinian local complete intersections of codimension three
本文研究了余维数为三的二次 Artinian 局部完全交的 Hilbert 函数，刻画了其容许性条件，给出了有效构造，并证明其理想的对称分解由 Hilbert 函数决定。
Local Gorenstein duality in chromatic group cohomology
本文研究了紧 Lie 群分类空间上链谱的局部 Gorenstein 对偶性，证明其在多种环谱（如 Lubin-Tate 理论、拓扑 K-理论和拓扑模形式）中系统成立，并通过下降结果扩展了适用范围。
Let \fa be an ideal of a commutative Noetherian ring R and M and N two finitely generated R-modules. Let \cd_{\fa}(M,N) denote the supremum of the i's such that H^i_{\fa}(M,N)\neq 0. First, by using the theory of Gorenstein homological dimensions, we obtain several upper bounds for \cd_{\fa}(M,N). Next, over a Cohen-Macaulay local ring (R,\fm), we show that \cd_{\fm}(M,N)=\dim R-\grade(\Ann_RN,M), provided that either projective dimension of M or injective dimension of N is finite. Finally, over such rings, we establish an analogue of the Hartshorne-Lichtenbaum Vanishing Theorem in the context…
We propose a concept of module liaison that extends Gorenstein liaison of ideals and provides an equivalence relation among unmixed modules over a commutative Gorenstein ring. Analyzing the resulting equivalence classes we show that several results known for Gorenstein liaison are still true in the more general case of module liaison. In particular, we construct two maps from the set of even liaison classes of modules of fixed codimension into stable equivalence classes of certain reflexive modules. As a consequence, we show that the intermediate cohomology modules and properties like being pe…
The paper describes a duality phenomenon for cohomology theories with the character of Gorenstein rings. For a connective cohomology theory with the p-local integers in degree 0, and coefficient ring R_* Gorenstein of shift 0, this states that for X with R_*(X) torsion, we have R^*(X)=Σ^a Hom( R_*(X), Z/p^{\infty}). A corresponding statement for modules over a commutative Gorenstein ring spectrum is also proved. [Minor typographical and bibliographic changes to the last version.]
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