# Tangent space of a scheme and subschemes of length two

I found in Huybrecht's book Fourier Mukai transforms in Algebraic Geometry the following statement

A tangent vector $$v$$ at $$x \in X$$ is the data of a length two subscheme $$Z$$ concentrated at x

Here $$X$$ is a $$k$$-scheme and $$x$$ is a closed point of $$X$$. I tried to prove this equivalence, but I am not sure whether I did it correctly or not. I know that tangent vectors are in bijection with the set of $$k$$-scheme homomorphisms $$Hom_{k} \left( \text{Spec} \; k[\epsilon], X \right)$$, where I set $$k[\epsilon] = k[\epsilon] \left/ (\epsilon^2) \right.$$. Now my idea was to do the following:

To every $$\phi \in Hom_{k} \left( \text{Spec} \; k[\epsilon], X \right)$$ I attach its schematic image $$Z_{\phi}$$. This is a subscheme of $$X$$ supported at $$x$$ and the stalk at $$x$$ is given by $$\mathcal{O}_{X,x} \left/ \text{ker} \phi_{x} \simeq k[\epsilon] \right.$$ for every morphism which is not the trivial one (i.e. the one factorizing for the inclusion of the closed point, which gives a subscheme of length 1). Therefore, $$Z_{\phi}$$ is the required subscheme.

For every subscheme $$i : Z \rightarrow X$$ of length two let us consider $$\mathcal{O}_{Z,x} \simeq \mathcal{O}_{X,x} \left/ \mathcal{I}_{Z,x} \right.$$. As this is a length 2 module over $$\mathcal{O}_{X,x}$$, and $$\mathcal{O}_{X,x}$$ is a local ring, we have a short exact sequence $$0 \rightarrow k(x) \rightarrow \mathcal{O}_{X,x} \left/ \mathcal{I}_{Z,x} \right. \rightarrow k(x) \rightarrow 0$$ of $$k(x)$$-vector spaces. Therefore, $$\mathcal{O}_{X,x} \left/ \mathcal{I}_{Z,x} \simeq k(x) \oplus k(x) \right.$$ as a vector space, and the multiplication on the left can easily be transferred on the right because $$\left( m_{x} \left/ \mathcal{I}_{Z,x} \right. \right)^2 = 0$$, namely we have $$(a,b) \cdot (c,d) = (ac, ad+bc)$$. We now define $$\mathcal{O}_{X,x} \rightarrow k[\epsilon]$$ as the composition of the projection $$\mathcal{O}_{X,x} \rightarrow \mathcal{O}_{Z,x}$$, the isomorphism $$\mathcal{O}_{X,x} \left/ \mathcal{I}_{Z,x} \simeq k(x) \oplus k(x) \right.$$ and the map $$(a,b) \mapsto a + \epsilon b$$.

Is this correct? I fear it is not, because it seems like all the subschemes have the same structure sheaf, namely $$k[\epsilon]$$, with the same multiplication structure.

• You can have distinct but isomorphic closed subschemes! – loch Jan 22 at 10:47
• So is my answer correct? Therefore, all the subschemes are isomorphic? But then where is the information of the tangent vector? – Federico Jan 22 at 10:55
• @loch Sorry, I forgot to tag you. – Federico Jan 22 at 17:47