Question:
Consider $X\equiv\mathbb{P}^2$ (the complex projective space), and a point $p\in\mathbb{P}^2$. Let $\mathfrak{m}_p$ be the maximal ideal of $p$.
Is it the case that $\mathcal{O}_X/\mathfrak{m}_p$ and $\mathfrak{m}_p/\mathfrak{m}_p^2$ are both sheaves supported only at $p$ and isomorphic to $\mathbb{C}$ and $\mathbb{C}^2$ respectively? How should one think about this in terms of functions on the patches of $\mathbb{P}^2$?
What I would like to understand is how to think about these as sheaves, and how to see from the definition of sheaves and quotient sheaves that this is their behaviour, in particular in terms of functions on the usual coordinate patches on $\mathbb{P}^2$. I am used to thinking about line bundles, and their transition functions, and I usually use $\mathbb{P}^1$ as a simple example in which to illustrate ideas, but I am not used to thinking about sheaves like this. It would be nice to be able to think clearly about these quotients on $\mathbb{P}^1$ first.
Context and thoughts:
In Lemma 1 of Chapter 3 of Friedman's Algebraic Surfaces and Holomorphic Vector Bundles, Friedman considers a blow-up $\rho:\tilde{X}\to X$ of a surface $X$ at the point $p\in X$, with the exceptional divisor $E=\rho^{-1}(p)\cong\mathbb{P}^1$. The following theorem is then stated.
Lemma 1: Let $\mathfrak{m}_p$ be the maximal ideal of $p$ in $X$. Then $$ \rho_*\mathcal{O}_{\tilde{X}}(aE)= \begin{cases} \mathfrak{m}_p^n,~~\textit{if }a=-n<0, \\ \mathcal{O}_X,~~\textit{if }a\geq0. \end{cases} $$
Now consider the short exact sequence, $$ 0 \to \mathcal{O}_{\tilde{X}}(-E) \to \mathcal{O}_{\tilde{X}} \to \mathcal{O}_E \to 0 \,. \tag{1} \label{1} $$ After tensoring with $\mathcal{O}_{\tilde{X}}(-E)$ we get, $$ 0 \to \mathcal{O}_{\tilde{X}}(-2E) \to \mathcal{O}_{\tilde{X}}(-E) \to \mathcal{O}_E(1) \to 0 \,, \tag{2} \label{2} $$ where we have noted that $E^2=-1$, so that $\mathcal{O}_E\otimes\mathcal{O}_{\tilde{X}}(-E)=\mathcal{O}_E(1)\cong\mathcal{O}_{\mathbb{P}^1}(1)$.
Now, it can also be shown that $R^i\rho_*\mathcal{O}_{\tilde{X}}(aE)=0$ for $i>0$ when $a\leq0$ - this is stated for example here at the bottom of page 592. So now consider the long exact sequences of pushforwards and higher direct images associated with the short exact sequences \eqref{1} and \eqref{2} above. Since $R^1\rho_*\mathcal{O}_{\tilde{X}}=R^1\rho_*\mathcal{O}_{\tilde{X}}(-E)=0$, in both cases we get a short exact sequence involving just the pushforwards. Making use of the above lemma, these can be written respectively as follows, $$ 0 \to \mathfrak{m}_p \to \mathcal{O}_X \to \rho_*\mathcal{O}_E \to 0 \,, \tag{3} \label{3} $$ $$ 0 \to \mathfrak{m}_p^2 \to \mathfrak{m}_p \to \rho_*\mathcal{O}_E(1) \to 0 \,. \tag{4} \label{4} $$ Hence we can identify, $$ \rho_*\mathcal{O}_E\cong\mathcal{O}_X/\mathfrak{m}_p ~~\text{and}~~ \rho_*\mathcal{O}_E(1)\cong\mathfrak{m}_p/\mathfrak{m}_p^2\,. $$ Now, the blow-down $\rho$ acts on the $\mathbb{P}^1$ as a map from the whole space to a point, and the sheaf $\mathcal{O}_E(n)$ effectively lives only on the $\mathbb{P}^1$. By the definition of the pushforward $f_*\mathcal{F}$ as the sheaf associated to the presheaf $U\mapsto H^0(f^{-1}(U),\mathcal{F}_{f^{-1}(U)})$ (for a map $f$ of topological spaces), since in the map of $\mathbb{P}^1$ to a point the inverse image of the point is just the whole $\mathbb{P}^1$, we expect that $\rho_*\mathcal{O}_E(n)$ will be a sheaf supported only at $p$ and isomorphic to $H^0(\mathcal{O}_E(n))\cong\mathbb{C}^{n+1}$ (using $\binom{n+1}{n}=n+1$) for $n\geq0$ (and isomorphic to $0$ otherwise).
Hence in our particular case, we expect $\rho_*\mathcal{O}_E$ and $\rho_*\mathcal{O}_E(1)$ to be supported only at $p$ and to be isomorphic to $\mathbb{C}$ and $\mathbb{C}^2$ respectively. So recalling \eqref{3} and \eqref{4}, we see that this must be true of $\mathcal{O}_X/\mathfrak{m}_p$ and $\mathfrak{m}_p/\mathfrak{m}_p^2$, respectively. Since I know that $\mathcal{O}_X/\mathfrak{m}_p\cong i_*\mathcal{O}_P$ by a general result (with $i:p\to X$ the inclusion map), and since clearly $i_*\mathcal{O}_P\cong\mathbb{C}$, the first of these results is at least clearly correct, though I do not have a good 'feel' for this by thinking about the quotient sheaf.
This is a local question. Take a point $p \in \mathbb{P}^2$, projective plane has a standard covering by 3 affine open sets, point $p$ is in one of these sets $p \in \mathbb{A}^2$. There rest is commutative algebra.
Let $m$ be the maximal ideal of $\mathbb{A}^2 = \text{Spec} k[x_1,x_2]$ corresponding to $p$. Then $\mathcal{O}_{X,p} \cong k[x_1,x_2]_m$, and $m_p$ is the maximal ideal in $\mathcal{O}_{X,p}$. The residue field $\mathcal{O}_{X,p}/m_p \cong k(p)$ is isomorphic to $k$ if $k$ is algebraically closed. The quotient $m_p/m_p^2$ is by definition a conormal bundle of ${p}$, which is in the case of a closed point is just the fiber of cotangent bundle at the point, indeed $\dim m_p/m_p^2 =\dim \mathbb{P}^2=2$.
These are local question, it is enough to work in one affine chart and no need to work with transition functions etc