Example 29.9.5. Bijectivity is not stable under base change, and so neither is injectivity. For example consider the bijection $\mathop{\mathrm{Spec}}(\mathbf{C}) \to \mathop{\mathrm{Spec}}(\mathbf{R})$. The base change $\mathop{\mathrm{Spec}}(\mathbf{C} \otimes _{\mathbf{R}} \mathbf{C}) \to \mathop{\mathrm{Spec}}(\mathbf{C})$ is not injective, since there is an isomorphism $\mathbf{C} \otimes _{\mathbf{R}} \mathbf{C} \cong \mathbf{C} \times \mathbf{C}$ (the decomposition comes from the idempotent $\frac{1 \otimes 1 + i \otimes i}{2}$) and hence $\mathop{\mathrm{Spec}}(\mathbf{C} \otimes _{\mathbf{R}} \mathbf{C})$ has two points.

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