bibkey: clavieretal2022localitytensor
authors: Pierre J. Clavier; Loic Foissy; Diego A. López; Sylvie Paycha
year: 2022
title: "Tensor products and the Milnor-Moore theorem in the locality setup"
doi: 10.1007/s40879-025-00822-z
url: https://arxiv.org/abs/2205.14616v1
claim: "The paper develops locality vector spaces and locality tensor products V⊗_⊤W = K(V×_⊤W)/(I_bil ∩ K(V×_⊤W)), and states as Conjectural statement 6.16 that I_bil(V1,V2) ∩ K(V1×_⊤V2) is locality compatible with the relation ⊤_{V1×_⊤V2}, a condition under which the locality tensor product is a locality vector space."
strata_touched:
  - D5/S3/HomologicalAlgebra/LocalityTensorProductRefutation
license: citation-only
triage: anchor

Tensor products and the Milnor-Moore theorem in the locality setup

P. J. Clavier, L. Foissy, D. A. López and S. Paycha, arXiv:2205.14616v1 [math.RA, math-ph, math.QA]; European Journal of Mathematics 11 (2025), article 41. Quotations are from the v1 TeX source.

Locality vector spaces (Definition 1.1):

A {\bf locality $\text{K}$-vector space} or simply {\bf locality vector space} is a pre-locality $\text{K}$-vector space $(V,\top )$ such that the polar set $U^{\top }=\{x\in V:(\forall u\in U),u\top x\}$ of any subset $U\subset V$ is a linear subspace of $V$.

Locality compatibility (Definition 6.7):

Let $(V,\top )$ be a locality vector space and $W\subset V$ a linear subspace. We say that {\bf $W$ is locality compatible with $\top$} if $\forall (x,y,z)\in V^3,\forall w\in W,$ $(x,y)\in \top \wedge (x+w,z)\in \top ⟹(\exists w^{\prime }\in W):(x+w^{\prime },y)\in \top \wedge (x+w^{\prime },z)\in \top .$

Conjectural statement 6.16:

Given a locality vector space $(E,\top )$ and $V_1$, $V_2$ two of its subspaces, the subspace $I_{\mathrm{bil}}(V_1,V_2)\cap \text{K}(V_1{\times }_{\top }{V}_2)\subset \text{K}(V_1{\times }_{\top }{V}_2)$ is locality compatible with ${\top }_{V_1{\times }_{\top }{V}_2}$.

Verified locator

Source: https://arxiv.org/abs/2205.14616v1

• URL: https://arxiv.org/abs/2205.14616v1 (source Clavier_Foissy_Lopez_Paycha_arxiv.tex, md5 1f6041a82cd80114573aa776694ccf20, 2896 lines):
  • locality sets and locality vector spaces, Definition defn:basic_loc_defn, l. 340–359;
  • the locality Cartesian product, l. 387–394;
  • the bilinear relations $I_{\mathrm{bil}}$, l. 422–430, and the relation ${\top }_{V{\times }_{\top }W}$, Eq. eq:locreltoptimes, l. 447–449;
  • the locality tensor product, Definition defn:loctensprod1, l. 556–562;
  • the relation on a free span, Eq. eq:locrelonthefreespan, l. 755–769;
  • locality compatibility, Definition defn:loccompatible, l. 1880–1886;
  • the Remark identifying ${\top }_{V{\times }_{\top }W}$ with $V^{{\times }_{\top }4}$ when $V=W$, l. 1972–1975;
  • compatibility implies that the quotient is a locality vector space, Theorem thm:quotientlocality-vs, l. 1935;
  • Conjectural statement conj:main1, l. 1984–1987.
• DOI: 10.1007/s40879-025-00822-z (https://doi.org/10.1007/s40879-025-00822-z; Crossref lists European Journal of Mathematics, volume 11, article 41, published 16 June 2025). The journal text was not read (subscription); its abstract lists the question whether the locality tensor product of two locality vector spaces is a locality vector space as open.