Why is $ \mathfrak{m} / \mathfrak{m}^2 $ a vector space?

Question

I am trying to teach myself what the Zariski tangent space is using the following set of notes: https://math.mit.edu/~mckernan/Teaching/09-10/Spring/18.726/l_8.pdf

Let $X$ be a variety and let $p \in X$ be a point of $X$. The Zariski tangent space of $X$ at $p$, denoted $T_p X$, is equal to the dual of the quotient $$ \mathfrak{m} / \mathfrak{m}^2, $$ where $\mathfrak{m}$ is the maximal ideal of $\mathcal{O}_{X, p}$. Note that $\mathfrak{m} / \mathfrak{m}^2$ is a vector space. Suppose that we are given a morphism $$ f: X \longrightarrow Y, $$ which sends $p$ to $q$. In this case there is a ring homomorphism $$ f^*: \mathcal{O}_{Y, q} \longrightarrow \mathcal{O}_{X, p} $$ which sends the maximal ideal $\mathfrak{n}$ into the maximal ideal $\mathfrak{m}$. Thus we get an induced map $$ d f: \mathfrak{n} / \mathfrak{n}^2 \longrightarrow \mathfrak{m} / \mathfrak{m}^2 $$

At the moment, I am still unable to see why $ \mathfrak{m} / \mathfrak{m}^2, $ a vector space? What is it a vector space over?

For some background, I have not taken commutative algebra, so I am just learning whatever I need as I read these notes. I know what localization is and that $m$ is the unique maximal ideal.

Answer 1

In general, if $M$ is a module over a (commutative) ring $R$, and $I\subseteq R$ is an ideal then we can define the submodule:

$IM=\{\sum\limits_{i=1}^n a_im_i: a_i\in I, m_i\in M, n\geq 0\}$

Then it is easy to see that $M/IM$ is a module over the quotient ring $R/I$ by the action $(r+I)(m+IM)=rm+IM$. It's not difficult to check that this is well defined.

So in your case take $R=\mathcal{O}_{X, p}$ and $M=I=\mathfrak{m}$. The module $\mathfrak{m}/\mathfrak{m}^2$ is a module over the field $R/\mathfrak{m}$, i.e a vector space over that field.

Answer 2

$\mathfrak{m}/\mathfrak{m}^2\cong\mathfrak{m} \otimes_{\mathcal{O}_{X,p}} \mathcal{O}_{X,p}/\mathfrak{m}$, so $\mathfrak{m}/\mathfrak{m}^2$ is naturally a vector space over $\mathcal{O}_{X,p}/\mathfrak{m}\cong \kappa(p)$, the residue field of $p\in X$.