TITLE: How to think of the Zariski tangent space
QUESTION [21 upvotes]: The Zariski tangent space at a point $\mathfrak m$ is defined as the dual of $\mathfrak m/\mathfrak m ^2$. While I do appreciate this definition, I find it hard to work with, because we are not given an isomorphism from $\mathfrak m/\mathfrak m^2$ to $(\mathfrak m/\mathfrak m ^2)^\vee$ (which I'd wish for at least in the finite dimensional case so that I could put my hands on something concrete).
So my question is: how does one go from this abstract definition to actually writing down what is the $T_{X,p}$ as a scheme? To take a simple case, we might consider $$X=k[x,y,z]/(x+y+z^2,x+y+z^3); \qquad p=(x-0,y-0,z-0)$$
Then, the cotangent space is easy to calculate. It is the plane cut out by $x+y$, i.e. it is the scheme $k[x,y,z]/(x+y)$. But what is the tangent space as a scheme?

REPLY [11 votes]: I'd like to briefly explain how the Zariski tangent space can be worked with and where the abstract definition comes from. A good introduction to this is Shafarevich's book on affine and projective varieties. He first shows that if an affine variety $X$ is defined by an ideal $I$, then the tangent space (say at $0$, assuming $0$ is in $X$) is defined by the zero set of the polynomials $\{dG:G\in I\}$, where $dG$ is the differential of $G$ at $0$; that is, the zero set of all homogeneous elements of degree 1 that appear in the polynomials in the ideal $I$. For example, the tangent space to $x-y+x^2+y^4=0$ at $0$ is defined by the equation $x-y=0$. Shafarevich shows that the variety defined by these equations is naturally what one would expect the tangent space to be. 
He then goes on to show that abstractly, as a vector space, this space is isomorphic to $(\frak{m}_0/\frak{m}_0^2)^\vee$ in a natural way. Now, why pick $(\frak{m}_0/\frak{m}_0^2)^\vee$ instead of $\frak{m}_0/\frak{m}_0^2$? For functorial reasons. In topology, a map $f:X\to Y$ between two manifolds induces a map $T_pX\to T_{f(p)}Y$ for a point $p\in X$, where $T_pX$ denotes the tangent space of $X$ at $p$. If we take $\frak{m}_p/\frak{m}_p^2$ as the definition of tangent space and we have a map $f:X\to Y$ between two varieties, then we only get the pullback map $f^*:{\frak{m}}_{f(p)}/{\frak{m}}_{f(p)}^2\to \frak{m}_0/\frak{m}_0^2$ and not the other way around, as we would expect. That's why taking the dual space works perfectly.
In practice, if you want explicit equations for the tangent space of a variety, then the first method is very concrete. If you want more theoretical calculations, then the abstract method is the way to go. In general for a scheme $X$ over a field $k$, you can also think of the tangent space at $p$ as $\mbox{Mor}((\mbox{Spec}(k[\epsilon]/\epsilon^2),(\epsilon)),(X,p))$.