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Exercise 111.35.4. For any field $K$ prove that the diagram

\[ \xymatrix{ \mathop{\mathrm{Spec}}(K) \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(K[\epsilon _1]) \ar[d] \\ \mathop{\mathrm{Spec}}(K[\epsilon _2]) \ar[r] & \mathop{\mathrm{Spec}}(K[\epsilon _1, \epsilon _2]/(\epsilon _1\epsilon _2)) } \]

is a pushout diagram in the category of schemes. (Here $\epsilon _ i^2 = 0$ as before.)

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View Exercise 111.35.4 as pdf