We show that the infinite symmetric product of a connected graded-commutative algebra over is naturally isomorphic to the free graded-commutative algebra on the positive degree subspace of the original algebra. In particular, the infinite symmetric product of a connected commutative (in the usual sense) graded algebra over is a polynomial algebra. Applied to topology, we obtain a quick proof of the Dold–Thom theorem in rational homotopy theory for connected spaces of finite type. We also show that finite symmetric products of certain simple free graded-commutative algebras are free; this allows us to determine minimal Sullivan models for finite symmetric products of complex projective spaces.
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Mots clés : Symmetric products, Dold–Thom theorem
@article{CRMATH_2022__360_G3_275_0,
author = {Hu, Jiahao and Milivojevi\'c, Aleksandar},
title = {Infinite symmetric products of rational algebras and spaces},
journal = {Comptes Rendus. Math\'ematique},
pages = {275--284},
publisher = {Acad\'emie des sciences, Paris},
volume = {360},
number = {G3},
year = {2022},
doi = {10.5802/crmath.298},
language = {en},
url = {http://www.numdam.org/articles/10.5802/crmath.298/}
}
TY - JOUR AU - Hu, Jiahao AU - Milivojević, Aleksandar TI - Infinite symmetric products of rational algebras and spaces JO - Comptes Rendus. Mathématique PY - 2022 SP - 275 EP - 284 VL - 360 IS - G3 PB - Académie des sciences, Paris UR - http://www.numdam.org/articles/10.5802/crmath.298/ DO - 10.5802/crmath.298 LA - en ID - CRMATH_2022__360_G3_275_0 ER -
%0 Journal Article %A Hu, Jiahao %A Milivojević, Aleksandar %T Infinite symmetric products of rational algebras and spaces %J Comptes Rendus. Mathématique %D 2022 %P 275-284 %V 360 %N G3 %I Académie des sciences, Paris %U http://www.numdam.org/articles/10.5802/crmath.298/ %R 10.5802/crmath.298 %G en %F CRMATH_2022__360_G3_275_0
Hu, Jiahao; Milivojević, Aleksandar. Infinite symmetric products of rational algebras and spaces. Comptes Rendus. Mathématique, Tome 360 (2022) no. G3, pp. 275-284. doi : 10.5802/crmath.298. http://www.numdam.org/articles/10.5802/crmath.298/
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