Let $A$ be a ring. Is the sequence \begin{align} \textstyle A \to \prod_{\mathfrak{p}} A_{\mathfrak{p}} \rightrightarrows \prod_{\mathfrak{p}_{1},\mathfrak{p}_{2}} A_{\mathfrak{p}_{1}} \otimes_{A} A_{\mathfrak{p}_{2}} \end{align} exact? Here the products are over all prime ideals of $A$.

*Thoughts:*

- Since $A \to \prod_{\mathfrak{p}} A_{\mathfrak{p}}$ is faithfully flat, the sequence \begin{align} \textstyle A \to \prod_{\mathfrak{p}} A_{\mathfrak{p}} \rightrightarrows (\prod_{\mathfrak{p}} A_{\mathfrak{p}}) \otimes_{A} (\prod_{\mathfrak{p}} A_{\mathfrak{p}}) \end{align} is exact, so it is enough to show that \begin{align} \textstyle (\prod_{\mathfrak{p}} A_{\mathfrak{p}}) \otimes_{A} (\prod_{\mathfrak{p}} A_{\mathfrak{p}}) \to \prod_{\mathfrak{p}_{1},\mathfrak{p}_{2}} A_{\mathfrak{p}_{1}} \otimes_{A} A_{\mathfrak{p}_{2}} \end{align} is injective.
- If $A$ is an integral domain, the question is equivalent to asking that $A = \bigcap_{\mathfrak{p}} A_{\mathfrak{p}}$ inside its fraction field. See e.g. this and a related question here.
- Here I'm asking more-or-less whether sections of $\mathcal{O}_{X}$ satisfies descent with respect to the morphism $\coprod_{x \in X} \operatorname{Spec} \mathcal{O}_{X,x} \to X$ (which is not an fpqc cover). It's at least true that morphisms of quasi-coherent sheaves do not satisfy descent for such maps, see this.