Question. when does a morphism of schemes send a closed point to a closed point? 

 1) Integral Morphisms

 This amounts to saying that $B/A$ integral extension, then $B$ is a field if and only if $A$ is a field.

 2) Projective => Proper => Closed => preserves closed point.

 Reference. mathSE post






Let $X$ be a scheme of finite type over a finite (or algebraically closed) field $k$. Let $K$ be a finite extension of $\mathbb{Q}_p$ with residue field $k$. 

 We would like to define $D_c^b(X, \overline{\mathbb{Q}}_l)$.

 1) $D_c^b(X, \mathcal{O}_r)$ where $\mathcal{O}_r$ is $\mathcal{O}/\pi^r \mathcal{O}$.

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