Is a submodule of the sheaf of sections of a smooth vector bundle necessarily finitely generated? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:15:17Z http://mathoverflow.net/feeds/question/41867 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41867/is-a-submodule-of-the-sheaf-of-sections-of-a-smooth-vector-bundle-necessarily-fin Is a submodule of the sheaf of sections of a smooth vector bundle necessarily finitely generated? Theo Johnson-Freyd 2010-10-12T04:53:18Z 2010-10-12T08:50:04Z <p>Let $X$ be a finite-dimensional smooth manifold, $\mathcal C^\infty(X)$ its algebra of smooth functions, $V\to X$ a finite-dimensional smooth vector bundle, and $\Gamma(V)$ the space of smooth sections of $V$. In particular, $\Gamma(V)$ is a $\mathcal C^\infty(X)$-module. I am interested in $\mathcal C^\infty(X)$-submodules $D \subseteq \Gamma(V)$.</p> <blockquote> <p>Is $D$ necessarily finitely-generated as a $\mathcal C^\infty(X)$-module?</p> </blockquote> <p>If $X$ is not compact (or maybe even if it is?), then $\mathcal C^\infty(X)$ is not Noetherian. So it is not true that submodules of arbitrary finitely-generated modules are finitely generated. So I expect that the answer to my question is "no", but I'm having trouble coming up with a counterexample.</p> <p>Actually, what I really want is for $D$ to receive a ($\mathcal C^\infty$-linear) surjection from $\Gamma(W)$ for some finite-dimensional vector bundle $W$. If $X$ is not compact, then I think it is still the case (using partitions of unity) that $\Gamma(W)$ is globally finitely-generated (the idea is to find a cover for which each open intersects only finitely many others in the cover, and then to double up the generators). But if it isn't, the actual question I want to ask is the one with the word "locally" sprinkled in all the necessary places.</p> http://mathoverflow.net/questions/41867/is-a-submodule-of-the-sheaf-of-sections-of-a-smooth-vector-bundle-necessarily-fin/41869#41869 Answer by Tom Goodwillie for Is a submodule of the sheaf of sections of a smooth vector bundle necessarily finitely generated? Tom Goodwillie 2010-10-12T05:04:17Z 2010-10-12T05:04:17Z <p>The module of all sections of $V$ that vanish to infinite order at a given point of the manifold will not be finitely generated (unless the bundle has rank zero or the manifold has dimension zero). </p> http://mathoverflow.net/questions/41867/is-a-submodule-of-the-sheaf-of-sections-of-a-smooth-vector-bundle-necessarily-fin/41884#41884 Answer by AH for Is a submodule of the sheaf of sections of a smooth vector bundle necessarily finitely generated? AH 2010-10-12T08:50:04Z 2010-10-12T08:50:04Z <p>One simple counterexample is to let X be the real line, let V be the trivial line bundle, and consider the submodule of \Gamma(V) of smooth sections with compact support. The same story works for any vector bundle on any non-compact manifold.</p>