[For the entire collection see Zbl 0568.00012.]
Let p be an odd prime. Let \(U_ p\) denote the category of unstable graded modules over the mod p Steenrod algebra. Let V be a finite dimensional \({\mathbb{Z}}_ p\)-vector space and define a functor T from \(U_ p\) to the category of \({\mathbb{Z}}_ p\)-vector spaces by \(T(M)=\operatorname{Hom}_{U_ p}(M, H^*(V; {\mathbb{Z}}_ p)).\) The author announces four theorems. First, the functor T is exact. Second, \(T(\Sigma M)=0\) where \(\Sigma\) is the suspension functor on \(U_ p\). Third, \(T(M){\hat \otimes}T(N)\cong T(M\otimes N).\) Fourth, \({\mathbb{Z}}_ p[\operatorname{Hom}_{{\mathbb{Z}}_ p}(V,W)]\cong T(H^*(W; {\mathbb{Z}}_ p)).\)
The first of these theorems can be used to prove the Segal conjecture for elementary abelian p-groups. A proof that the second theorem implies the following result is given: if \(M\in U_ p\) and \(f\in \operatorname{Hom}_{U_ p}(M, H^*(V; {\mathbb{Z}}_ p))\) such that \(f| M^+=0\) then \(f=0\).