Abstract
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.
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Notes
We note with sadness the passing of A.K. Bousfield during the preparation of this paper.
The ring depends on the choice of sequence, but for clarity we omit the sequence from the notation.
This map depends on the choice of module M and, implicitly, on the sequence \((c_1, c_2, \dots )\) defining R. Again, for clarity, we omit both from the notation.
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Acknowledgements
The work described here is substantially based on the second author’s PhD thesis, for which she was supported by an Iraqi Government scholarship. Both authors would like to record their gratitude to the referee for very detailed and helpful reports which led to considerable improvements in the paper, and to Sarah Whitehouse for a number of useful discussions.
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Communicated by Jugal K Verma.
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Crossley, M.D., Khafaja, N.T. Cobasically discrete modules and generalizations of Bousfield’s exact sequence. Indian J Pure Appl Math 53, 261–272 (2022). https://doi.org/10.1007/s13226-021-00019-6
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DOI: https://doi.org/10.1007/s13226-021-00019-6