Profinite tensor powers
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910105570639872 |
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| author | Treumann, David Wong, C. -M. Michael |
| author_facet | Treumann, David Wong, C. -M. Michael |
| contents | We discuss the problem of defining a tensor product of profinitely many copies of a vector space $V$, and propose a definition $\bigotimes_X^{\mathrm{mcc}} V$ in the special situation that (1) $V$ is finite-dimensional over $\mathbf{F}_2$, and (2) the profinite $X$ indexing the tensor factors is acted on with finitely many orbits by a pro-$2$-group. The "mcc" on the tensor sign stands for "magnetized and conditionally convergent." A variant construction makes sense when $V$ is a bimodule over a ring of the form $\mathbf{F}_2 \times \cdots \times \mathbf{F}_2$, and the index set $X$ has the profinite version of a cyclic order. The definition organizes some computations in Heegaard Floer homology: it can be pitched as a computation of the Heegaard Floer theory of some pro-$3$-manifolds, though we do not know how to define such a thing. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04367 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Profinite tensor powers Treumann, David Wong, C. -M. Michael Rings and Algebras Geometric Topology 15A69 (Primary) 16D20, 20E18, 57K18 (Secondary) We discuss the problem of defining a tensor product of profinitely many copies of a vector space $V$, and propose a definition $\bigotimes_X^{\mathrm{mcc}} V$ in the special situation that (1) $V$ is finite-dimensional over $\mathbf{F}_2$, and (2) the profinite $X$ indexing the tensor factors is acted on with finitely many orbits by a pro-$2$-group. The "mcc" on the tensor sign stands for "magnetized and conditionally convergent." A variant construction makes sense when $V$ is a bimodule over a ring of the form $\mathbf{F}_2 \times \cdots \times \mathbf{F}_2$, and the index set $X$ has the profinite version of a cyclic order. The definition organizes some computations in Heegaard Floer homology: it can be pitched as a computation of the Heegaard Floer theory of some pro-$3$-manifolds, though we do not know how to define such a thing. |
| title | Profinite tensor powers |
| topic | Rings and Algebras Geometric Topology 15A69 (Primary) 16D20, 20E18, 57K18 (Secondary) |
| url | https://arxiv.org/abs/2604.04367 |