Matlis Duality Research Articles

Artinian Gorenstein algebras that are free extensions over k[t]/(tn), and Macaulay duality

T. Harima and J. Watanabe studied the Lefschetz properties of free extension Artinian algebras C over a base A with fiber B. The free extensions are deformations of the usual tensor product; when C is also Gorenstein, so are A and B, and it is natural to ask for the relation among the Macaulay dual generators for the algebras. Writing a dual generator F for C as a homogeneous “polynomial” in T and the dual variables for B, and given the dual generator for B, we give sufficient conditions on F that ensure that C is a free extension of A=k[t]∕(tn) with fiber B. We give examples exploring the sharpness of the statements. We also consider a special set of coinvariant algebras C which are free extensions of A, but which do not satisfy the sufficient conditions of our main result.

Flat cotorsion modules over Noether algebras

For a module-finite algebra over a commutative noetherian ring, we give a complete description of flat cotorsion modules in terms of prime ideals of the algebra, as a generalization of Enochs' result for a commutative noetherian ring. As a consequence, we show that pointwise Matlis duality gives a bijective correspondence between the isoclasses of indecomposable injective left modules and the isoclasses of indecomposable flat cotorsion right modules. This correspondence is an explicit realization of Herzog's homeomorphism induced from elementary duality of Ziegler spectra.

On flat generators and Matlis duality for quasicoherent sheaves

We show that for a quasicompact quasiseparated scheme $X$, the following assertions are equivalent: (1) the category $\operatorname{QCoh}(X)$ of all quasicoherent sheaves on $X$ has a flat generator; (2) for every injective object $\mathcal E$ of $\operatorname{QCoh}(X)$, the internal hom functor into $\mathcal E$ is exact; (3) the scheme $X$ is semiseparated.

On top local cohomology modules, Matlis duality and tensor products

Let [Formula: see text] be an ideal of a local ring [Formula: see text] with [Formula: see text] the cohomological dimension of [Formula: see text] in [Formula: see text]. In the case that [Formula: see text], we first give a bound for [Formula: see text], where [Formula: see text] and [Formula: see text] is complete. Later, [Formula: see text], [Formula: see text] and [Formula: see text] are examined. In the case [Formula: see text], the set [Formula: see text] is considered.

Minimaxness and finiteness properties of local homology and local cohomology modules

We prove some results concerning minimaxness and finiteness of local homology modules and by Matlis duality we extend some results for the minimaxness and finiteness of local cohomology modules. We introduce the concept of C-minimax R-modules, and we discuss the maximum and minimum integers such that local homology and local cohomology modules are C-minimax. As a consequence, we find minimum integers such that local homology and local cohomology modules are of finite length.

On Switala's Matlis duality for [formula omitted]-modules

We show that N. Switala's Matlis duality for D-modules yields a generalization of a result of R. Hartshorne and C. Polini.

Gorenstein homological dimensions and some duality results

Let [Formula: see text] be a local ring and [Formula: see text] denote the Matlis duality functor. Assume that [Formula: see text] possesses a normalized dualizing complex [Formula: see text] and [Formula: see text] and [Formula: see text] are two homologically bounded complexes of [Formula: see text]-modules with finitely generated homology modules. We will show that if G-dimension of [Formula: see text] and injective dimension of [Formula: see text] are finite, then [Formula: see text] Also, we prove that if Gorenstein injective dimension of [Formula: see text] and projective dimension of [Formula: see text] are finite, then [Formula: see text] These results provide some generalizations of Suzuki’s Duality Theorem and the Herzog–Zamani Duality Theorem.

On the de Rham homology and cohomology of a complete local ring in equicharacteristic zero

Let $A$ be a complete local ring with a coefficient field $k$ of characteristic zero, and let $Y$ be its spectrum. The de Rham homology and cohomology of $Y$ have been defined by R. Hartshorne using a choice of surjection $R\rightarrow A$ where $R$ is a complete regular local $k$-algebra: the resulting objects are independent of the chosen surjection. We prove that the Hodge–de Rham spectral sequences abutting to the de Rham homology and cohomology of $Y$, beginning with their $E_{2}$-terms, are independent of the chosen surjection (up to a degree shift in the homology case) and consist of finite-dimensional $k$-spaces. These $E_{2}$-terms therefore provide invariants of $A$ analogous to the Lyubeznik numbers. As part of our proofs we develop a theory of Matlis duality in relation to ${\mathcal{D}}$-modules that is of independent interest. Some of the highlights of this theory are that if $R$ is a complete regular local ring containing $k$ and ${\mathcal{D}}={\mathcal{D}}(R,k)$ is the ring of $k$-linear differential operators on $R$, then the Matlis dual $D(M)$ of any left ${\mathcal{D}}$-module $M$ can again be given a structure of left ${\mathcal{D}}$-module, and if $M$ is a holonomic ${\mathcal{D}}$-module, then the de Rham cohomology spaces of $D(M)$ are $k$-dual to those of $M$.

Volume Polynomials and Duality Algebras of Multi-Fans

We introduce a theory of volume polynomials and corresponding duality algebras of multi-fans. Any complete simplicial multi-fan $\Delta$ determines a volume polynomial $V_\Delta$ whose values are the volumes of multi-polytopes based on $\Delta$. This homogeneous polynomial is further used to construct a Poincare duality algebra $\mathcal{A}^*(\Delta)$. We study the structure and properties of $V_\Delta$ and $\mathcal{A}^*(\Delta)$ and give applications and connections to other subjects, such as Macaulay duality, Novik--Swartz theory of face rings of simplicial manifolds, generalizations of Minkowski's theorem on convex polytopes, cohomology of torus manifolds, computations of volumes, and linear relations on the powers of linear forms. In particular, we prove that the analogue of the $g$-theorem does not hold for multi-polytopes.

Grothendieck–Neeman duality and the Wirthmüller isomorphism

We clarify the relationship between Grothendieck duality à la Neeman and the Wirthmüller isomorphism à la Fausk–Hu–May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensor-triangulated categories, which leads to a surprising trichotomy: there exist either exactly three adjoints, exactly five, or infinitely many. We highlight the importance of so-called relative dualizing objects and explain how they give rise to dualities on canonical subcategories. This yields a duality theory rich enough to capture the main features of Grothendieck duality in algebraic geometry, of generalized Pontryagin–Matlis duality à la Dwyer–Greenless–Iyengar in the theory of ring spectra, and of Brown–Comenetz duality à la Neeman in stable homotopy theory.

J.W. Milnor’s diagonal element in Poincaré Duality Algebras, Macaulay Duality, the formulae of Thom and Wu, and a new characterization of Artin–Gorenstein Algebras

The purpose of this note is to provide a purely algebraic interpretation for the cohomology class J. W. Milnor calls the diagonal cohomology class in [11] Chapter 11 and is used by him in his discussion of Poincare duality and characteristic classes of manifolds. As defined there it is an element of the relative cohomology algebra \(H^n(M \times M ,M \times M \setminus \Delta (M))\) where \(\Delta : M \mathop {\hookrightarrow }\limits M \times M\) is the diagonal embedding and provides a replacement for the Thom class (see e.g., [3] and [10] page 56 et seq) of the tangent bundle of M. Here M should be a closed smooth oriented manifold of dimension n. We use one of Milnor’s Theorems to define an analogous element for any Poincare duality algebra over a field and show that it is a Macaulay dual for the kernel of the multiplication map of that algebra. Using a formula of R. Thom then allows us to define Stiefel–Whitney classes if the Poincare duality algebra has the structure of an unstable algebra over the Steenrod algebra. In such a case the algebra also has Wu classes and we show that the formulae of Thom and Wu relating these characteristic classes in the topological case hold in the more general algebraic context thereby recovering and extending an old result of J.F. Adams in a new way.

On a class of power ideals

In this paper we study the class of power ideals generated by the kn forms (x0+ξg1x1+…+ξgnxn)(k−1)d where ξ is a fixed primitive kth-root of unity and 0≤gj≤k−1 for all j. For k=2, by using a Zkn+1-grading on C[x0,…,xn], we compute the Hilbert series of the associated quotient rings via a simple numerical algorithm. We also conjecture the extension for k>2. Via Macaulay duality, those power ideals are related to schemes of fat points with support on the kn points [1:ξg1:…:ξgn] in Pn. We compute Hilbert series, Betti numbers and Gröbner basis for these 0-dimensional schemes. This explicitly determines the Hilbert series of the power ideal for all k: that this agrees with our conjecture for k>2 is supported by several computer experiments.

Local cohomology with support in ideals of maximal minors and sub-maximal Pfaffians

We compute the GL-equivariant description of the local cohomology modules with support in the ideal of maximal minors of a generic matrix, as well as of those with support in the ideal of 2n×2n Pfaffians of a (2n+1)×(2n+1) generic skew-symmetric matrix. As an application, we characterize the Cohen–Macaulay modules of covariants for the action of the special linear group SL(G) on G⊕m. The main tool we develop is a method for computing certain Ext modules based on the geometric technique for computing syzygies and on Matlis duality.

Some results on local homology and local cohomology modules

In this paper, we obtain some results about the local homology modules of Artinian modules, and by Matlis duality we obtain some results about the local cohomology modules of finitely generated modules.

On the top local cohomology modules

Let ( R , m ) be a Noetherian local ring and I an ideal of R. Let M be a finitely generated R-module with dim M = d . It is clear by Matlis duality that if R is complete then H I d ( M ) satisfies the following property: (⁎) Ann R ( 0 : H I d ( M ) p ) = p for all prime ideals p ⊇ Ann R H I d ( M ) . However, H I d ( M ) does not satisfy the property (⁎) in general. In this paper we characterize the property (⁎) of H I d ( M ) in order to study the catenarity of the ring R / Ann R H I d ( M ) , the set of attached primes Att R H I d ( M ) , the co-support Cos R ( H I d ( M ) ) , and the multiplicity of H I d ( M ) . We also show that if H I d ( M ) satisfies the property (⁎) then H I d ( M ) ≅ H m d ( M / N ) for some submodule N of M.

Matlis Duality and Finiteness Properties of Generalized Local Cohomology Modules

Let [Formula: see text] be an ideal of a commutative Noetherian local ring [Formula: see text] and M, N be two finitely generated R-modules such that M is of finite projective dimension n. Let t be a positive integer. We show that if there exists a regular sequence [Formula: see text] with [Formula: see text] and the i-th local cohomology module [Formula: see text] of N with respect to [Formula: see text] is zero for all i > t, then [Formula: see text], where D(-):= Hom R(-,E). Also, we prove that if N is a Cohen-Macaulay R-module of dimension d, then the generalized local cohomology module [Formula: see text] is co-Cohen-Macaulay of Noetherian dimension d. Finally, with an elementary proof, we show that [Formula: see text] is finite.

Matlis duals of local cohomology modules and their endomorphism rings

Let \({(R, \mathfrak{m})}\) denote a local ring. Let \({I \subset R}\) be an ideal with c = grade I. Let D(·) denote the Matlis duality functor. In recent research there is an interest in the structure of the local cohomology module \({H^c_I := H^c_I(R)}\), in particular in the endomorphism ring of \({D(H^c_I)}\). Let ER(k) be the injective hull of the residue field \({R/\mathfrak{m}}\). By investigating the natural map \({H^c_I \otimes D(H^c_I) \to E_R(k)}\) we are able to prove that the endomorphism rings of \({D(H^c_I)}\) and of \({H^c_I}\) are naturally isomorphic. This natural homomorphism is related to a quasi-isomorphism of a certain complex. As applications we show results when the endomorphism ring of \({D(H^c_I)}\) is naturally isomorphic to R generalizing results known under the additional assumption of \({H^i_I(R) = 0}\) for \({i \not= c}\).

Equivariant Matlis and the local duality

Generalizing the known results on graded rings and modules, we formulate and prove the equivariant version of the local duality on schemes with a group action. We also prove an equivariant analogue of Matlis duality.

Reducible family of height three level algebras

Let R = k [ x 1 , … , x r ] be the polynomial ring in r variables over an infinite field k, and let M be the maximal ideal of R. Here a level algebra will be a graded Artinian quotient A of R having socle Soc ( A ) = 0 : M in a single degree j. The Hilbert function H ( A ) = ( h 0 , h 1 , … , h j ) gives the dimension h i = dim k A i of each degree- i graded piece of A for 0 ⩽ i ⩽ j . The embedding dimension of A is h 1 , and the type of A is dim k Soc ( A ) , here h j . The family LevAlg ( H ) of level algebra quotients of R having Hilbert function H forms an open subscheme of the family of graded algebras or, via Macaulay duality, of a Grassmannian. We show that for each of the Hilbert functions H 1 = ( 1 , 3 , 4 , 4 ) and H 2 = ( 1 , 3 , 6 , 8 , 9 , 3 ) the family LevAlg ( H ) has several irreducible components (Theorems 2.3(A), 2.4). We show also that these examples each lift to points. However, in the first example, an irreducible Betti stratum for Artinian algebras becomes reducible when lifted to points (Theorem 2.3(B)). We show that the second example is the first in an infinite sequence of examples of type three Hilbert functions H ( c ) in which also the number of components gets arbitrarily large (Theorem 2.10). The first case where the phenomenon of multiple components can occur (i.e. the lowest embedding dimension and then the lowest type) is that of dimension three and type two. Examples of this first case have been obtained by the authors (unpublished) and also by J.O. Kleppe.

Associated primes of local cohomology modules and Matlis duality

Let ( R , m ) be a commutative Noetherian local ring of dimension d and I an ideal of R. We show that the set of associated primes of the local cohomology module H I 2 ( R ) is finite whenever R is regular. Also, it is shown that if x 1 , … , x d is a system of parameters for R, then D ( H ( x 1 , … , x i ) i ( R ) ) has infinitely many associated prime ideals for all i ⩽ d − 1 , where D ( − ) : = Hom R ( − , E ) denotes the Matlis dual functor and E : = E R ( R / m ) is the injective hull of the residue field R / m . Finally, we explore a counterexample of Grothendieck's conjecture by showing that, if d ⩾ 3 , then the R-module Hom R ( R / I , H I d − 1 ( R ) ) is not finitely generated, where I = ( x 1 ) ∩ ( x 2 , … , x d ) .