Sheaf of modules that is not locally free

I am a beginner to sheaf theory; so please bear with me.

Consider the sheaf of differentiable functions on $\mathbb R^2$, and the subsheaf consisting of functions vanishing at $0$.

Question 1:

How would you show that $\mathcal F$ is not a locally free sheaf over $\mathcal A$?

Question 2:

If instead of $\mathbb R^2$ we were considering $\mathbb R$, then is the ideal sheaf at $0$ locally free?

Let $\mathcal{F}$ denote the subsheaf in question. If it were locally free then there would be some open set containing $0$ on which $\mathcal{F}\vert_U \cong \mathcal{O}^{J}_{\mathbb{R}^2}\vert_U$, where $J$ is some set. But on every open set containing zero, the latter sheaf has either all nonzero stalks or all zero stalks (if $J = \varnothing$) whereas the former has stalks of both types. Isomorphisms of sheaves induce isomorphisms of stalks, so we have a contradiction.
• The stalk at $0$ of the ideal sheaf is not zero, if that is what you are suggesting. The stalks consists of equivalence classes of smooth functions vanishing at $0$, with the relation that they agree on some neighborhood of $0$. It is easy to find two such functions that are not equivalent under this relation. For example, $f(x, y)=x$ and $g(x,y) =x^2$ will do. – Shinigami Feb 10 '13 at 6:03
• Ah, I think your example can be fixed in the following way. Consider instead $\mathcal F_x/\mathcal M_x \mathcal F_x$ where $\mathcal M_x$ is the maximal ideal in the stalk of the structure sheaf at $x$, and $\mathcal F_x$ is the stalk of $\mathcal F$ at $x$. Then, this has dimension $2$ at $0$ and $1$ everywhere else. This also suggests that if we were to consider $\mathbb R$ instead of $\mathbb R^2$, the ideal sheaf might indeed be locally free. – Shinigami Feb 10 '13 at 6:19