Simplicial join via tensor product

Let be a field, and M and N two finitely generated graded modules over standard graded -algebras A and B, respectively. We will study generalized, sequentially, almost, and approximately Cohen–Macaulay as well as clean, and pretty clean properties of the -module through the corresponding properties of M and N. The behavior of these properties with respect to the simplicial join of two simplicial (multi)complexes will be revealed as corollaries.     

ON FLATNESS RELATIVE TO A TORSION THEORY

Abstract

For a torsion radical, σ, σ-flatness (as defined in [8]) is studied. Many of the properties of flat modules are shown to extend and σ-flat modules are characterized under certain restrictions on the associated filter, Lσ. We also define semi-σ-flat modules in a natural way and find conditions equivalent to every R-module being semi-σ-flat.     

Products of projective S-Systems

The tree main results of this paper deal with the generating vectors of simple infinite and finite dimesional modules arising as composition factors of the Kac modules K↓ G ₀ and K of the Lie superalgebra sl(m/n).     

On inverse-direct systems of modules

Inverse-direct systems of modules have been considered by Eklof and Mekler, see [P.C. Eklof, A.H. Mekler, Almost free modules, 2nd ed., North Holland, 2002]. The systems we are going to study are different: we do not assume the condition that certain composite maps are identity maps (this forces the direct summand property). In this paper inverse-direct systems will be considered where certain composite maps lie in the center of the respective endomorphism rings. We investigate how the limits are modified if the connecting maps are changed by automorphisms of the modules. It will also be shown that one can define a composition between the systems modified by these automorphisms such that those whose limits are non-isomorphic under the canonical maps form an abelian group. This group can be described in terms of the first derived functor of the inverse limit functor.We also study the relation to vanishing inverse limits: in certain cases, the maps can be modified in such a way that the inverse limit of the new system becomes 0. In the final section, we use self-idealizations in order to construct sets of non-isomorphic modules (over suitable uncountable rings) that are direct limits of the same collection of modules with different connecting maps.     

Vector fields and a family of linear type modules related to free divisors

This paper has three main goals. We start describing a method for computing the polynomial vector fields tangent to a given algebraic variety; this is of interest, for instance, in view of (effective) foliation theory. We then pass to furnishing a family of modules of linear type (that is, the Rees algebra equals the symmetric algebra), formed with vector fields related to suitable pairs of algebraic varieties, one of them being a free divisor in the sense of K. Saito. Finally, we derive freeness criteria for modules retaining a certain tangency feature, so that, in particular, well-known criteria for free divisors are recovered.     

Strongly Gorenstein projective, injective, and flat modules

In this paper, we study a particular case of Gorenstein projective, injective, and flat modules, which we call, respectively, strongly Gorenstein projective, injective, and flat modules. These last three classes of modules give us a new characterization of the first modules, and confirm that there is an analogy between the notion of “Gorenstein projective, injective, and flat modules” and the notion of the usual “projective, injective, and flat modules”.     

Prime filtrations and Stanley decompositions of squarefree modules and Alexander duality

In this paper we study how prime filtrations and squarefree Stanley decompositions of squarefree modules over the polynomial ring and over the exterior algebra behave with respect to Alexander duality. The results which we obtained suggest a lower bound for the regularity of a \mathbb Zⁿ{\mathbb {Z}^n}-graded module in terms of its Stanley decompositions. For squarefree modules this conjectured bound is a direct consequence of Stanley’s conjecture on Stanley decompositions. We show that for pretty clean rings of the form R/I, where I is a monomial ideal, and for monomial ideals with linear quotient our conjecture holds.     

Tilting modules over almost perfect domains

We provide a complete classification of all tilting modules and tilting classes over almost perfect domains, which generalizes the classifications of tilting modules and tilting classes over Dedekind and 1-Gorenstein domains. Assuming the APD is Noetherian, a complete classification of all cotilting modules is obtained (as duals of the tilting ones).     

Self-dual modules over local rings of curve singularities

Let C be a reduced curve singularity. C is called of finite self-dual type if there exist only finitely many isomorphism classes of indecomposable, self-dual, torsion-free modules over the local ring of C. In this paper it is shown that the singularities of finite self-dual type are those which dominate a simple plane singularity.     

On Modules Which are Isomorphic to Relatively Divisible or Pure Submodules of Each Other

Abstract

In this note, we provide a generalization of a well-known result of module theory which states that two injective modules are isomorphic when they are isomorphic to submodules of each other. More precisely, we show here that two RD-injective (respectively, pure-injective) modules over an integral domain are isomorphic if they are isomorphic to relatively divisible (respectively, pure) sub- modules of each other.     

Artinian level modules of embedding dimension two

We prove that a sequence of positive integers (h₀,h₁,…,h_c) is the Hilbert function of an artinian level module of embedding dimension two if and only if h_i−1−2h_i+h_i+1≤0 for all 0≤i≤c, where we assume that h₋₁=h_c+1=0. This generalizes a result already known for artinian level algebras. We provide two proofs, one using a deformation argument, the other a construction with monomial ideals. We also discuss liftings of artinian modules to modules of dimension one.     

Prime Filtrations and Primary Decompositions of Modules

We give a new characterization for prime filtrations of an R-module M in terms of primary decomposition of the zero submodule of M. Using this characterization we prove that some classes of monomial ideals like generic and cogeneric monomial ideals are clean, pretty clean, or almost clean.     

Correction to “Tensor products of modules and the rigidity of Tor”, Math. Annalen, 299 (1994), 449–476

This note makes a correction to the paper “Tensor products of modules and the ridigity of Tor”, a correction which is needed due to an incorrect convention for the depth of the zero module.     

Taylor complexes for Koszul boundaries

For all boundary modules of the Koszul complex of a monomial sequence we construct complexes, which we call Taylor complexes. For a monomial d-sequences these complexes provide free resolutions of the boundary modules. Let M be the ideal generated by a monomial d-sequence. We use the Taylor complexes to construct minimal free resolutions of the Rees algebra and the associated graded ring of M. Received: 13 November 1997 / Revised version: 6 March 1998     

A new approximation theory which unifies spherical and Cohen-Macaulay approximations

This paper gives a new approximation theory for finitely generated modules over commutative Noetherian rings, which unifies two famous approximation theorems; one is due to Auslander and Bridger and the other is due to Auslander and Buchweitz. Modules admitting such approximations shall be studied.     

A Krull-Schmidt theorem for one-dimensional rings of finite Cohen-Macaulay type

This paper determines when the Krull-Schmidt property holds for all finitely generated modules and for maximal Cohen-Macaulay modules over one-dimensional local rings with finite Cohen-Macaulay type. We classify all maximal Cohen-Macaulay modules over these rings, beginning with the complete rings where the Krull-Schmidt property is known to hold. We are then able to determine when the Krull-Schmidt property holds over the non-complete local rings and when we have the weaker property that any two representations of a maximal Cohen-Macaulay module as a direct sum of indecomposables have the same number of indecomposable summands.     

Castelnuovo-Mumford regularity by approximation

The Castelnuovo-Mumford regularity of a module gives a rough measure of its complexity. We bound the regularity of a module given a system of approximating modules whose regularities are known. Such approximations can arise naturally for modules constructed by inductive combinatorial means. We apply these methods to bound the regularity of ideals constructed as combinations of linear ideals and the module of derivations of a hyperplane arrangement as well as to give degree bounds for invariants of finite groups.     

Euler class groups and a theorem of Roitman

We define the Euler class group of a polynomial algebra in lower codimension and prove an analogue of a result of Roitman (on projective modules) for the Euler class groups.     

On the Gorenstein injective dimension and Bass formula

In this paper, we compare Krull dimension, Gorenstein injective dimension and injective dimension of a module in several cases. In fact, we establish some generalizations of the Bass formula. To this end, we generalize the Grothendieck non-vanishing theorem to a class of modules larger than finitely generated modules. Received: 21 May 2007