# Tangent space of a scheme and subschemes of length two

Mathematics Asked by Federico on October 19, 2020

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.

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

This is a crucial point. Remember, that a closed subscheme $$Z subset X$$ is an equivalence class of closed immersions $$Z hookrightarrow X$$, where two such morphisms $$f, g$$ are equivalent if and only if there is an isomphism $$phi: Z to Z$$, making the following diagram commute: begin{align*}Z & xrightarrow{f} X\ phi downarrow & nearrow g \ Zend{align*}

So both things can happen:

1. You can have two different morphisms $$f, g: Z hookrightarrow X$$, which define the same subscheme $$Z subset X$$.
2. You can have two different subschemes $$Z_1, Z_2 subset X$$, which are abstractly isomorphic as schemes $$Z_1 cong Z_2$$, but not as subschemes of $$X$$, even if they have the same points in $$X$$.

So much for the general theory, now to your question.

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

$$DeclareMathOperator{Spec}{Spec}$$I don't belive the claim of Huybrechts is true as written.

1. You already mentioned the first problem I see: The zero-vector, which is the morphism $$Spec(k[epsilon]) to Spec(k) to X$$, does not define a subscheme of length two.

2. Two different morphisms $$f,g: Spec(k[epsilon]) to X$$(i.e. two different tangent vectors), define the same subscheme in $$X$$, if there is an isomorphism of $$k[epsilon]$$ commuting with $$f$$ and $$g$$. But any such isomorphism is given by a map $$k[epsilon] to k[epsilon], epsilon mapsto c cdot epsilon$$ for any $$c in k setminus {0}$$. So two tangent vectors define the same subscheme if and only if they differ by a scalar.

Those two points show, that the length two subschemes concentrated at $$x$$ are rather the projectification of the tangent vectors, i.e. the space $$mathbb{P}(T_x)$$ (or $$mathbb{P}(T_x^*)$$, depending on your notation).

Answered by red_trumpet on October 19, 2020

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