THE HOMOTOPY GROUPS OF THE L2-LOCALIZATION OF THE MODULO p MOORE SPECTRUM SMASHING WITH THE FIRST RAVENEL SPECTRUM

Let BP be the Brown-Peterson spectrum at an odd prime p, and L2 denote the Bousfield localization functor with respect to [Formula: see text]. The Ravenel spectrum T(1) is characterized by BP*(T(1)) = BP*[t1] on the primitive generator t1. In this paper, we determine the homotopy groups π*(L2M ∧ T(1)) for the mod p Moore spectrum M.

Higher derivations on rings and modules

Letτbe a hereditary torsion theory onModRand suppose thatQτ:ModR→ModRis the localization functor. It is shown that for allR-modulesM, every higher derivation defined onMcan be extended uniquely to a higher derivation defined onQτ(M)if and only ifτis a higher differential torsion theory. It is also shown that ifτis a TTF theory andCτ:M→Mis the colocalization functor, then a higher derivation defined onMcan be lifted uniquely to a higher derivation defined onCτ(M).

The co-localization of an Artinian module

For a multiplicative set S of a commutative ring R we define the co-localization functor HomR(Rs,⋅). It is a functor on the category of R-modules to the category of Rs-modules. It is shown to be exact on the category of Artinian R-modules. While the co-localization of an Artinian module is almost never an Artinian Rs-module it inherits many good properties of A, e.g. it has a secondary representation. The construction is applied to the dual of a result of Bourbaki, a description of asymptotic prime divisors and the co-support of an Artinian module.