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 \(0 \rightarrow M \rightarrow UM \rightarrow UM \rightarrow M \otimes \mathbf{Q} \rightarrow 0\), where U denotes the co-free functor, right adjoint to the forgetful functor to \(\mathbf{Z}_{(p)}\)-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 \(\mathbf{Z}_{(p)}[x]\), 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.

Bousfield 引入了一个代数类别的模块，它反映了p局部复杂拓扑K理论检测到的结构。他为该类别中的任何模块M构造了一个自然的 4 项精确序列\(0 \rightarrow M \rightarrow UM \rightarrow UM \rightarrow M \otimes \mathbf{Q} \rightarrow 0\)，其中U表示co-free 函子，紧邻\(\mathbf{Z}_{(p)}\) -modules的健忘函子。克拉克等人。将 Bousfield 类别的对象识别为某个拓扑环A的“离散”模块，作为多项式环的完成而获得\(\mathbf{Z}_{(p)}[x]\)，并在这种情况下简化了布斯菲尔德序列的构造。我们引入了“共基离散” R模块的概念，作为对 Clarke 等人的澄清。模块，注意这些对应于R 对偶的余代数上的协模块。我们研究了其他多项式完成环的 Bousfield 序列的类似物，注意到序列最后一项的各种行为。