José L. Bueso

Homological Computations in PBW Modules

In this paper the Poincaré–Birkhoff–Witt (PBW) rings are characterized. Gröbner bases techniques are also developed for these rings. An explicit presentation of Exti(M,N) is provided when N is a centralizing bimodule.

RE-FILTERING AND EXACTNESS OF THE GELFAND-KIRILLOV DIMENSION

Abstract We prove that any multi-filtered algebra with semi-commutative associated graded algebra can be endowed with a locally finite filtration keeping up the semi-commutativity of the associated graded algebra. As consequences, we obtain that Gelfand–Kirillov dimension is exact for finitely generated modules and that the algebra is finitely partitive. Our methods apply to algebras of current interest like the quantized enveloping algebras, iterated differential operators algebras, quantum matrices or quantum Weyl algebras.

EFFECTIVE COMPUTATION OF THE GELFAND-KIRILLOV DIMENSION*

In this note we propose an effective method based on the computation of a Grobner basis of a left ideal to calculate the Gelfand-Kirillov dimension of modules.

PRIMALITY TEST IN ITERATED ORE EXTENSIONS

We give a primality test for two-sided ideals over rings belonging to a class of iterated Ore extensions of a field, which includes differential operators rings and coordinate rings of quantum affine spaces. When applied to ideals of commutative polynomial rings, the test boils down to the given in (Gianni et al. J. Symb. Comput. 1988, 6, 149–167).

Decomposition of injective modules relative to a torsion theory

IfR is a right noetherian ring, the decomposition of an injective module, as a direct sum of uniform submodules, is well known. Also, this property characterises this kind of ring. M. L. Teply obtains this result for torsion-free injective modules. The decomposition of injective modules relative to a torsion theory has been studied by S. Mohamed, S. Singh, K. Masaike and T. Horigone. In this paper our aim is to determine those rings satisfying that every torsion-freeτ-injective module is a direct sum ofτ-uniformτ-injective submodules and also to determine those rings with the same property for everyτ-injective module.

A Generalization of Semisimple Modules

This communication generalizes the concepts of the socle of a module and of the semisimple module, by replacing simples with τ-critical modules for a hereditary torsion theory τ, showing that t is strongly semiprime if and only if τ = X(M)for a τ-semi-critical module. As a further application, properties of the endomorphism ring of the τ-injective hulls of these modules are related to its structure.

Duality, localization and completion

Abstract In this paper we study relative duality theory, with respect to an idempotent kernel functor σ over some commutative ring R and prove that σ -dualizing R -modules are not only locally injective, but (somewhat surprisingly) globally injective. Using a relative version of completion, we show that the endomorphism ring of a σ -dualizing module coincides with the completion of R with respect to σ . In the final part of the paper we consider relative Gorenstein rings, giving an explicit calculation of their generalized local cohomology groups.

On the cotorsion hull in torsion theory

The concept of cotorsion was first introduced in the category of Abelian groups (Fuchs [l] ). Matlis [5], studied the cotorsion modules over integral domains. Henderson and Orzech [4], Fuchs [2], and Mines [6], replaced the classical notion of torsion by a torsion theory (T,F) on R-mod, where R is not necessarily commutative ring. In this paper we find conditions on the torsion theory in order to get a T-cotorsion hull for every module. This generalizes the result of Fuchs [2].

Semiartinian modules relative to a torsion theory

In the first section we generalize the concept of the socle of a module by replacing simples with τ-simple modules for a hereditary torsion theory τ. The second section is concerned with the τ-Loewy series, and finally these general results are applied in the section 3 to the notions of τ-semiartinian rings and modules.

Generalized Local Cohomology and Quasicoherent Sheaves

Although idempotent kernel functors 161, or equivalently, abstract localization theory with respect to hereditary torsion theories [S] or Gabriel topologies [3] were originally introduced to generalize traditional localization theory to noncommutative rings and modules, they have also been applied to sheaf theory. Consider an arbitrary ringed space (X, Q,). In [ 121 K. Suominen introduces cohomology with support in the category of sheaves of &-modules, as derived functors of certain idempotent kernel functors f,, associated to not-necessarily closed subsets Z of X, thus generalizing some work of A. Grothendieck’s [7]. On the other hand, in [2] J. P. Cahen considers localization theory in the category of quasicoherent sheaves of Q,-modules over a locally noetherian scheme X. Finally, in [14, 151 F. Van Oystaeyen and one of the authors introduce abstract localization theory for presheaves and study the behaviour of sheaves in this theory. In this note we study the relationship between these points of view and settle some problems that remained open in [ 161. In fact, it appears that the three theories are essentially equivalent, when restricted to the category of quasicoherent sheaves on a locally noetherian scheme.