Example 109.7.1 (0DSU): Non-reduced automorphism group

Example 109.7.1 (Non-reduced automorphism group). Let $k$ be an algebraically closed field of characteristic $2$. Set $Y = Z = \mathbf{P}^1_ k$. Choose three pairwise distinct $k$-valued points $a, b, c$ in $\mathbf{A}^1_ k$. Thinking of $\mathbf{A}^1_ k \subset \mathbf{P}^1_ k = Y = Z$ as an open subschemes, we get a closed immersion

\[ T = \mathop{\mathrm{Spec}}(k[t]/(t - a)^2) \amalg \mathop{\mathrm{Spec}}(k[t]/(t - b)^2) \amalg \mathop{\mathrm{Spec}}(k[t]/(t - c)^2) \longrightarrow \mathbf{P}^1_ k \]

Let $X$ be the pushout in the diagram

\[ \xymatrix{ T \ar[r] \ar[d] & Y \ar[d] \\ Z \ar[r] & X } \]

Let $U \subset X$ be the affine open part which is the image of $\mathbf{A}^1_ k \amalg \mathbf{A}^1_ k$. Then we have an equalizer diagram

\[ \xymatrix{ \mathcal{O}_ X(U) \ar[r] & k[t] \times k[t] \ar@<1ex>[r] \ar@<-1ex>[r] & k[t]/(t - a)^2 \times k[t]/(t - b)^2 \times k[t]/(t - c)^2 } \]

Over the dual numbers $A = k[\epsilon ]$ we have a nontrivial automorphism of this equalizer diagram sending $t$ to $t + \epsilon $. We leave it to the reader to see that this automorphism extends to an automorphism of $X$ over $A$. On the other hand, the reader easily shows that the automorphism group of $X$ over $k$ is finite. Thus $\mathit{Aut}(X)$ must be non-reduced.

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