Definition 111.35.1. For any ring $R$ we denote $R[\epsilon ]$ the ring of dual numbers. As an $R$-module it is free with basis $1$, $\epsilon $. The ring structure comes from setting $\epsilon ^2 = 0$.
111.35 Tangent Spaces
Exercise 111.35.2. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$ be a point, let $s = f(x)$. Consider the solid commutative diagram with the curved arrow being the canonical morphism of $\mathop{\mathrm{Spec}}(\kappa (x))$ into $X$. If $\kappa (x) = \kappa (s)$ show that the set of dotted arrows which make the diagram commute are in one to one correspondence with the set of linear maps In other words: describe such a bijection. (This works more generally if $\kappa (x) \supset \kappa (s)$ is a separable algebraic extension.)
\[ \xymatrix{ \mathop{\mathrm{Spec}}(\kappa (x)) \ar[r] \ar[dr] \ar@/^1pc/[rr] & \mathop{\mathrm{Spec}}(\kappa (x)[\epsilon ]) \ar@{.>}[r] \ar[d]& X \ar[d] \\ & \mathop{\mathrm{Spec}}(\kappa (s)) \ar[r] & S } \]
\[ \mathop{\mathrm{Hom}}\nolimits _{\kappa (x)}( \frac{\mathfrak m_ x}{\mathfrak m_ x^2 + \mathfrak m_ s\mathcal{O}_{X, x}}, \kappa (x)) \]
Definition 111.35.3. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. We dub the set of dotted arrows of Exercise 111.35.2 the tangent space of $X$ over $S$ and we denote it $T_{X/S, x}$. An element of this space is called a tangent vector of $X/S$ at $x$.
Exercise 111.35.4. For any field $K$ prove that the diagram is a pushout diagram in the category of schemes. (Here $\epsilon _ i^2 = 0$ as before.)
\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(K[\epsilon _1]) \ar[d] \\ \mathop{\mathrm{Spec}}(K[\epsilon _2]) \ar[r] & \mathop{\mathrm{Spec}}(K[\epsilon _1, \epsilon _2]/(\epsilon _1\epsilon _2)) } \]
Exercise 111.35.5. Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. Define addition of tangent vectors, using Exercise 111.35.4 and a suitable morphism Similarly, define scalar multiplication of tangent vectors (this is easier). Show that $T_{X/S, x}$ becomes a $\kappa (x)$-vector space with your constructions.
\[ \mathop{\mathrm{Spec}}(K[\epsilon ]) \longrightarrow \mathop{\mathrm{Spec}}(K[\epsilon _1, \epsilon _2]/(\epsilon _1\epsilon _2)). \]
Exercise 111.35.6. Let $k$ be a field. Consider the structure morphism $f : X = \mathbf{A}^1_ k \to \mathop{\mathrm{Spec}}(k) = S$.
Let $x \in X$ be a closed point. What is the dimension of $T_{X/S, x}$?
Let $\eta \in X$ be the generic point. What is the dimension of $T_{X/S, \eta }$?
Consider now $X$ as a scheme over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. What are the dimensions of $T_{X/\mathbf{Z}, x}$ and $T_{X/\mathbf{Z}, \eta }$?
Remark 111.35.7. Exercise 111.35.6 explains why it is necessary to consider the tangent space of $X$ over $S$ to get a good notion.
Exercise 111.35.8. Consider the morphism of schemes Compute the tangent space of $X/S$ at the unique point of $X$. Isn't that weird? What do you think happens if you take the morphism of schemes corresponding to $\mathbf{F}_ p[t^ p] \to \mathbf{F}_ p[t]$?
\[ f : X = \mathop{\mathrm{Spec}}(\mathbf{F}_ p(t)) \longrightarrow \mathop{\mathrm{Spec}}(\mathbf{F}_ p(t^ p)) = S \]
Exercise 111.35.9. Let $k$ be a field. Compute the tangent space of $X/k$ at the point $x = (0, 0)$ where $X = \mathop{\mathrm{Spec}}(k[x, y]/(x^2 - y^3))$.
Exercise 111.35.10. Let $f : X \to Y$ be a morphism of schemes over $S$. Let $x \in X$ be a point. Set $y = f(x)$. Assume that the natural map $\kappa (y) \to \kappa (x)$ is bijective. Show, using the definition, that $f$ induces a natural linear map Match it with what happens on local rings via Exercise 111.35.2 in case $\kappa (x) = \kappa (s)$.
\[ \text{d}f : T_{X/S, x} \longrightarrow T_{Y/S, y}. \]
Exercise 111.35.11. Let $k$ be an algebraically closed field. Let be a morphism of schemes over $k$. This is given by $m$ polynomials $f_1, \ldots , f_ m$ in $n$ variables. Consider the matrix Let $x \in \mathbf{A}^ n_ k$ be a closed point. Set $y = f(x)$. Show that the map on tangent spaces $T_{\mathbf{A}^ n_ k/k, x} \to T_{\mathbf{A}^ m_ k/k, y}$ is given by the value of the matrix $A$ at the point $x$.
\begin{eqnarray*} f : \mathbf{A}_ k^ n & \longrightarrow & \mathbf{A}^ m_ k \\ (x_1, \ldots , x_ n) & \longmapsto & (f_1(x_ i), \ldots , f_ m(x_ i)) \end{eqnarray*}
\[ A = \left( \frac{\partial f_ j}{\partial x_ i} \right) \]
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