In the local, unramified case the determinantal functions associated to the group-ring of a finite group satisfy Galois descent. This note examines the obstructions to Galois determinantal descent in the ramified case.
Let $p$ be a prime. We calculate the connective unitary K-theory of the smash product of two copies of the classifying space for the cyclic group of order $p$, using a K{u}nneth formula short exact sequence. As a corollary, using the Bott exact seque
nce and the mod $2$ Hurewicz homomorphism we calculate the connective orthogonal K-theory of the smash product of two copies of the classifying space for the cyclic group of order two.
As an application of the upper triangular technology method of (V.P. Snaith: {em Stable homotopy -- around the Arf-Kervaire invariant}; Birkh{a}user Progress on Math. Series vol. 273 (April 2009)) it is shown that there do not exist stable homotopy c
lasses of $ {mathbb RP}^{infty} wedge {mathbb RP}^{infty}$ in dimension $2^{s+1}-2$ with $s geq 2$ whose composition with the Hopf map to $ {mathbb RP}^{infty}$ followed by the Kahn-Priddy map gives an element in the stable homotopy of spheres of Arf-Kervaire invariant one.
