Cobasically discrete modules and generalizations of Bousfield’s exact sequence

Bousfield introduced an algebraic category of modules that reflects the structure detected by p-localized complex topological K-theory. He constructed, for any module M in this category, a natural 4-term exact sequence $$0 \rightarrow M \rightarrow UM \rightarrow UM \rightarrow M \otimes \mathbf{Q} \rightarrow 0$$ 0 → M → U M → U M → M ⊗ Q → 0 , where U denotes the co-free functor, right adjoint to the forgetful functor to $$\mathbf{Z}_{(p)}$$ Z ( p ) -modules. Clarke et al. identified the objects of Bousfield’s category as the ‘discrete’ modules for a certain topological ring A, obtained as a completion of the polynomial ring $$\mathbf{Z}_{(p)}[x]$$ Z ( p ) [ x ] , and simplified the construction of the Bousfield sequence in this context. We introduce the notion of ‘cobasically discrete’ R-modules as a clarification of the Clarke et al. modules, noting that these correspond to comodules over the coalgebra that R is dual to. We study analogues of the Bousfield sequence for other polynomial completion rings, noting a variety of behaviour in the last term of the sequence.