Maximal ideals of regulous functions are not finitely generated

Czarnecki, Aleksander

doi:10.1016/j.jpaa.2018.05.019

Cited by 2 publications

(1 citation statement)

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“…The concise name ‘regulous’ was coined by Fichou, Huisman, Mangolte and Monnier [FHMM16]. Since the publication of [Kuc09] in 2009 several mathematicians have devoted their attention to regulous maps (see [BKVV13, Cza19, FFQU18, FHMM16, FMQ17, FMQ20, FMQ21b, FMQ21a, KKK18, KN15, Kuc09, Kuc13, Kuc14a, Kuc14b, Kuc15, Kuc16a, Kuc16b, Kuc20, KK16, KK17, KZ18, KK18a, KK18b, Mon18, Zie16, Zie18] and the references therein).…”

Section: Introductionmentioning

confidence: 99%

Approximation and homotopy in regulous geometry

Kucharz

2023

Compositio Math.

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Let $X$, $Y$ be nonsingular real algebraic sets. A map $\varphi \colon X \to Y$ is said to be $k$-regulous, where $k$ is a nonnegative integer, if it is of class $\mathcal {C}^k$ and the restriction of $\varphi$ to some Zariski open dense subset of $X$ is a regular map. Assuming that $Y$ is uniformly rational, and $k \geq 1$, we prove that a $\mathcal {C}^{\infty }$ map $f \colon X \to Y$ can be approximated by $k$-regulous maps in the $\mathcal {C}^k$ topology if and only if $f$ is homotopic to a $k$-regulous map. The class of uniformly rational real algebraic varieties includes spheres, Grassmannians and rational nonsingular surfaces, and is stable under blowing up nonsingular centers. Furthermore, taking $Y=\mathbb {S}^p$ (the unit $p$-dimensional sphere), we obtain several new results on approximation of $\mathcal {C}^{\infty }$ maps from $X$ into $\mathbb {S}^p$ by $k$-regulous maps in the $\mathcal {C}^k$ topology, for $k \geq 0$.

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