A converse to Cartan's Theorem B: The extension property for real analytic and Nash sets

Abstract

In 1957 Cartan proved his celebrated Theorem B and deduced that if $\Omega\subset{\mathbb R}^n$ is an open set and $X$ is a coherent real analytic subset of $\Omega$, then $X$ has the analytic extension property: Each real analytic function on $X$ extends to a real analytic function on $\Omega$. The converse implication remains unproven. In the literature only special cases of non-coherent real analytic sets $X\subset\Omega$ without the extension property appear: mainly real analytic sets $X\subset\Omega$ that have a visible `tail'. We prove the converse implication: If $X\subset\Omega$ has the analytic extension property, it is a coherent real analytic subset of $\Omega$. Taking advantage of cohomology of sheaves, we provide for each non-coherent real analytic set $X$, `many' failing analytic functions on $X$ (that have no analytic extension to $\Omega$), yielding an almost complete description of the possible non-extendability sets. We extend the previous characterization to the Nash case, which is somehow more demanding, because of its finiteness properties and its disappointing behavior with respect to cohomology of sheaves of Nash function germs, and we provide again an almost complete description of the possible non-extendability sets. Let $\Omega\subset{\mathbb R}^n$ be an open semialgebraic set and let $X\subset\Omega$: Each Nash function on $X$ extends to a Nash function on $\Omega$ if and only if $X\subset\Omega$ is a coherent Nash set. The `if' implication goes back to some celebrated results of Coste, Ruiz and Shiota. If $M\subset{\mathbb R}^n$ is a Nash manifold, ${\mathcal C}^\infty$ semialgebraic functions on $M$ coincide with Nash functions on $M$. As an application of our results, we provide a full characterization of the semialgebraic sets $S\subset\Omega$ for which ${\mathcal C}^\infty$ semialgebraic functions on $S$ coincide with Nash functions on $S$.