This note investigate some finiteness properties of the category U of unstable modules. One shows finiteness properties for the injective resolution of finitely generated unstable modules. One also shows a stabilization result under Frobenius twist for Ext-groups.

The controlled finiteness obstruction and torsion are defined using controlled algebra, giving a more algebraic proof of the topological invariance of torsion and the homotopy finiteness of compact ANRs.

In this paper we use "ring changed'' Gorenstein homologicaldimensions to define Cohen-Macaulay injective, projective and flatdimensions. For doing this we use the amalgamated duplication of thebase ring with semi-dualizing ideals. Among other results, we prove that finiteness of these new dimensions characterizes Cohen-Macaulay rings with dualizing ideals.

Understanding input effects is crucial for delineating the roles of Universal Grammar and learning in grammatical development. Unlike the input effects seen in vocabulary development (Huttenlocher, Haight, Bryk, Seltzer, & Lyons, 1991; Hart & Risley, 1995), effects on grammatical development have been more difficult to identify (Newport, Gleitman, & Gleitman, 1977; Hoff-Ginsberg, 1998; Huttenlo...