Ringed space: interpretation of quotient.

Question

Let $(X,O_X)$ a locally ringed space, I'd like to define the tangent space $T_x$ for $x \in X$. We can consider the local ring (stalk) $R_x$ in $x$ with maximal ideal $m_x$ and from invertibility we have that $k_x:= R_x/m_x$ is a field. How can I interpreter the notation $m_x/m_x^2$? What is the square of a ideal? How can I prove that $m_x/m_x^2$ is a $\mathbb{R} \simeq R_x/m_x$-module?

Answer 1

The square is the product with itsself, $I^2 = I*I$. And you should be familiar with the product of ideals.

Instead of $R_x$ one usually writes $\mathcal{O}_{X,x}$. The residue field is usually denoted by $\kappa(x) = \mathcal{O}_{X,x} / \mathfrak{m}_x$. Of course this doesn't have to be isomorphic to $\mathbb{R}$, it could be any field (namely, every field can be considered as a locally ringed space with one point, and the residue field is the given field). It is isomorphic to $\mathbb{R}$ when we consider a real smooth manifold together with its smooth functions as a locally ringed space.

Observe that $\mathfrak{m}_x/\mathfrak{m}_x^2$ is annihilated by $\mathfrak{m}_x$, hence it is a module over $\mathcal{O}_{X,x}/\mathfrak{m}_x = \kappa(x)$. Here we use the general fact that a module over a quotient ring $A/I$ is the same as a module over $A$ which is annihilated by $I$.

The Zariski tangent space of a locally ringed space $X$ at a point $x$ is defined to be the dual of the $\kappa(x)$-vector space $\mathfrak{m}_x/\mathfrak{m}_x^2$. This definition can be applied for example to smooth manifolds (where it gives the usual tangent space) and to schemes.