Künneth formulas for Cotor

Abstract

We investigate the question of how to compute the cotensor product, and more generally the derived cotensor (i.e. Cotor) groups, of a tensor product of comodules. In particular, we determine the conditions under which there is a Künneth formula for Cotor. We show that there is a simple Künneth theorem for Cotor groups if and only if an appropriate coefficient comodule has trivial coaction. This result is an application of a spectral sequence we construct for computing Cotor of a tensor product of comodules. Finally, for certain families of nontrivial comodules which are especially topologically natural, we work out necessary and sufficient conditions for the existence of a Künneth formula for the $0$th Cotor group, i.e. the cotensor product. We give topological applications in the form of consequences for the $E_2$-term of the Adams spectral sequence of a smash product of spectra, and the Hurewicz image of a smash product of spectra.