Nous montrons que la propriété d’un espace spectral d’être un sous-espace spectral du spectre réel d’un anneau commutatif n’est pas exprimable dans le langage infinitaire du premier ordre de son treillis de définition. Ceci généralise un résultat de Delzell et Madden qui dit qu’en général, un espace spectral complètement normal n’est pas un spectre réel.
We show that the property of a spectral space, to be a spectral subspace of the real spectrum of a commutative ring, is not expressible in the infinitary first order language of its defining lattice. This generalises a result of Delzell and Madden which says that not every completely normal spectral space is a real spectrum.
@article{AFST_2012_6_21_2_343_0,
author = {Timothy Mellor and Marcus Tressl},
title = {Non-axiomatizability of real spectra in $\mathcal{L}_\infty \lambda $},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {343--358},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {Ser. 6, 21},
number = {2},
year = {2012},
doi = {10.5802/afst.1337},
mrnumber = {2978098},
zbl = {1254.03075},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1337/}
}
TY - JOUR
AU - Timothy Mellor
AU - Marcus Tressl
TI - Non-axiomatizability of real spectra in $\mathcal{L}_\infty \lambda $
JO - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY - 2012
SP - 343
EP - 358
VL - 21
IS - 2
PB - Université Paul Sabatier, Institut de Mathématiques
PP - Toulouse
UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1337/
DO - 10.5802/afst.1337
LA - en
ID - AFST_2012_6_21_2_343_0
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%0 Journal Article
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%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2012
%P 343-358
%V 21
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%I Université Paul Sabatier, Institut de Mathématiques
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%U https://afst.centre-mersenne.org/articles/10.5802/afst.1337/
%R 10.5802/afst.1337
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%F AFST_2012_6_21_2_343_0
Timothy Mellor; Marcus Tressl. Non-axiomatizability of real spectra in $\mathcal{L}_\infty \lambda $. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. 2, pp. 343-358. doi : 10.5802/afst.1337. https://afst.centre-mersenne.org/articles/10.5802/afst.1337/
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