Lemma 10.21.1. Let $R$ be a ring. Let $e \in R$ be an idempotent. In this case

**Proof.**
Note that an idempotent $e$ of a domain is either $1$ or $0$. Hence we see that

Similarly we have

Since the image of $e$ in any residue field is either $1$ or $0$ we deduce that $D(e)$ and $D(1-e)$ cover all of $\mathop{\mathrm{Spec}}(R)$. $\square$

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