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$\DeclareMathOperator{\Id}{Id} \require{cancel}$ Jet Nestruev's "Smooth Manifolds and Observables" contains following exercise:

Exercise. Show that $P$ is geometric if and only if the two modules $P$ and $\Gamma(P)$ are isomorphic.

It is clear that there is an injective map from $P$ to $\Gamma(P)$ iff $P$ is geometric. However, I do not see why this map needs to be surjective. I doubt that it holds, cause $\Gamma(P)$ is "sort of bidual" of $P.$


Notions and the question.

Let $P$ be a $C^\infty(M)$-module. We define following objects. For any $x\in M$ $$\mu_x:=\lbrace f\in C^\infty(M): f(x)=0\rbrace \hspace{5pt}\text{and}\hspace{5pt}P_x:=P/\mu_xP \hspace{5pt}\text{with}\hspace{5pt}\pi_x:P\to P_x.$$ Next we define $$|P|:=\bigcup_{x\in M}P_x\hspace{5pt}\text{with}\hspace{5pt}\pi_P:|P|\to M\hspace{5pt}\text{and}\hspace{5pt}\Gamma(P):=\overbrace{\cancel{\lbrace s:M\to|P|: \pi_P\circ s=\Id_M\rbrace}}^{\color{red}{\text{MY MISREAD! IT IS DEFINED AS }\phi(P)}}.$$

We will say that $P$ is geometric if $$\bigcap_{x\in M}\mu_xP=0.$$

For convenience consider natural $C^\infty(M)$-module homomorphism $\phi:P\to\Gamma(P)$ $$\phi(p)(x):=\pi_x(p)$$

Question. Why is $\phi$ surjective? Is geometricity needed for $\phi$ to be surjective?


Some note after Wille Liou's comment.

It is suspicious that they made no constraints on sections (elements of $\Gamma(P))$. If we consider $P=C^\infty(M)$, then $\Gamma(P)$ contains all functions from $M$ to $\mathbb{R}.$

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  • $\begingroup$ What topology is put on $|P|$? $\endgroup$ – Wille Liou Aug 17 '17 at 11:08
  • $\begingroup$ @WilleLiou Section 11.12 is devoted to topology in $|P|.$ Shortly, it is the Zariski topology of the symmetric algebra $S(P^\vee)$ viewed as functions on $|P|.$ $\endgroup$ – Fallen Apart Aug 17 '17 at 11:25
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The exercise is fine. I misread the definition of $\Gamma(P).$ Jet Nestruev defines $\Gamma(P)$ as an image $\phi(P).$ Hence surjectivity follows obviously from the definition of $\Gamma(P)$.

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    $\begingroup$ You can accept this answer to take your question off the unanswered list. $\endgroup$ – David Roberts Sep 17 '17 at 7:28

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