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Peer-reviewedCC BY 4.0Bulletin of the London Mathematical Society

Tate modules as condensed modules

  • Valerio Melani
  • Hugo Pourcelot
  • Gabriele Vezzosi
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Abstract

We prove that the category of countable Tate modules over an arbitrary discrete ring embeds fully faithfully into that of condensed modules. If the base ring is of finite type, we characterize the essential image as generated by the free module of infinite countable rank under direct sums, duals and retracts. In the ∞-categorical context, under the same assumption on the base ring, we establish a fully faithful embedding of the ∞-category of countable Tate objects in perfect complexes, with uniformly bounded tor-amplitude, into the derived ∞-category of condensed modules. The boundedness assumption is necessary to ensure fullness, as we prove via an explicit counterexample in the unbounded case.

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Cite

BibTeXRIS
BibTeX
@article{melani2026tate,
  title = {{Tate modules as condensed modules}},
  author = {Valerio Melani and Hugo Pourcelot and Gabriele Vezzosi},
  journal = {Bulletin of the London Mathematical Society},
  year = {2026},
  month = jun,
  doi = {10.1112/blms.70418},
  eprint = {2501.07481},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-04910970v2},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Tate modules as condensed modules
AU  - Valerio Melani
AU  - Hugo Pourcelot
AU  - Gabriele Vezzosi
PY  - 2026
DA  - 2026/06/19
JO  - Bulletin of the London Mathematical Society
DO  - 10.1112/blms.70418
UR  - https://hal.science/hal-04910970v2
AB  - We prove that the category of countable Tate modules over an arbitrary discrete ring embeds fully faithfully into that of condensed modules. If the base ring is of finite type, we characterize the essential image as generated by the free module of infinite countable rank under direct sums, duals and retracts. In the ∞-categorical context, under the same assumption on the base ring, we establish a fully faithful embedding of the ∞-category of countable Tate objects in perfect complexes, with uniformly bounded tor-amplitude, into the derived ∞-category of condensed modules. The boundedness assumption is necessary to ensure fullness, as we prove via an explicit counterexample in the unbounded case.
SN  - 0024-6093
SN  - 1469-2120
ER  -