Introduction

I remember Frank Adams describing certain recent developments in topology as follows: It is the business of homotopy theorists to compute [X, Y], and, while traditionally X has been a finite complex, now we can let X = BG. This was, of course, initiated by H. Miller in [M], and is ongoing.

Underlying the topological theorems were some wonderful new results about H8* (V), the mod p cohomology of an elementary abelian p-group V, viewed as an object in the category of unstable modules over the Steenrod algebra Ap. These results were ultimately given an elegant treatment by J. Lannes, who considers the functor left adjoint to H* (V) ⊗. Many of us have learned the mantra “Tv is exact and takes tensor products to tensor products”. (Most people have been content to believe the survey [L1], but the nontrivial details have finally appeared in [L2].)

At the Manchester conference, I lectured on a representation theoretic framework for understanding Steenrod algebra “technology”, as in [HLS] and [K1]. Here I wish to give an exposition of Tv and its properties from this point of view.

The key observation is as follows. Let (q) be the category with objects functors

F: finite Fq -vector spaces → Fq -vector spaces,

where Fq is the field with q elements. Morphisms in (q) are the natural transformations.