Let $A$ be an Artinian ring. It's very well known that the spectrum of $A$ is finite and discrete, i.e., $\mathsf{Spec} \: A=\lbrace\mathfrak{m}_1,\ldots,\mathfrak{m}_n\rbrace$, where the $\mathfrak{m}_i$'s are the maximal ideals of $A$.

I want to prove that $$A\simeq\bigoplus_{i=1}^nA_{\mathfrak{m}_i}.$$ Are there any suggestions or references?


This is immediate from basic scheme theory, specifically the theorem that the ring of global sections of the structure sheaf on $\operatorname{Spec} A$ is naturally isomorphic to $A$. Let $X=\operatorname{Spec} A$ and $\mathcal{O}_X$ be its structure sheaf. Since $X$ is discrete, $X=\{\mathfrak{m}_1\}\cup\dots\cup\{\mathfrak{m}_n\}$ is an open cover by disjoint open sets and so the gluing axiom says that $\mathcal{O}_X(X)\cong\prod_i \mathcal{O}_X(\{\mathfrak{m}_i\})$ via the restriction maps. But $\mathcal{O}_X(\{\mathfrak{m}_i\})$ is exactly the localization $A_\mathfrak{m_i}$, so $A\cong \mathcal{O}_X(X)\cong\prod_i A_{\mathfrak{m}_i}$.

[There are lower-tech ways to prove it, but this is my favorite and it makes the result "obvious" once you've developed a little intuition for schemes.]

  • $\begingroup$ Very nice answer! I studied just a bit of scheme theory, however my knowledge about the matter is enought to understand your proof. Thank you very much! $\endgroup$ – Vincenzo Zaccaro Aug 7 '17 at 2:51

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