Library of Mathexandria

This is an article on why using Max-Spec and Spec do not matter much if the ring $A$ is a Jacobson ring.  Definition. A ring $A$ is a Jacobson ring if every prime ideal of $A$ is an intersection of maximal ideals. In other words, \[ \mathrm{Nil}(A) = \bigcap_{\mathfrak{p}} \mathfrak{p} = \bigcap_{\mathfrak{m}} \mathfrak{m} = \mathrm{Jac}(A) \]  Proposition.  Let $X =\mathrm{Spec}(A)$ and $X_0 = \mathrm{MaxSpec}(A)$, i.e. the set of closed points of $X$. The $X_0$ is a very dense subset of $X$. We show that some topological properties are preserved. Let's first describe what it means for $\mathrm{Spec}(A)$ to be disconnected. By this means that there exist disjoint closed subsets $V(I)$ and $V(J)$ that cover $\mathrm{Spec}(A)$. In other words, 1) $V((0))=\mathrm{Spec}(A) = V(I) \cup V(J) = V(I \cap J)$.  2) $V((1)) = \varnothing = V(I) \cap V(J) = V(I + J)$. This is same as saying that 1') $\sqrt{I \cap J} = \sqrt{(0)} = \mathrm{Nil}(A)$, 2'

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 $\mathcal{O}/\pi^r \mathcal{O}$.