( Background: These is one part of a critera for commutative rings $ f:A \rightarrow B$ to be etale. )
It is claimed that the following two conditions 1 & 2 are equivalent.
- The multiplication map $$ p: B \otimes _A B \rightarrow B$$ is the projection on to a summand. There exists another commutative ring $q: B \otimes _A B \rightarrow R$ such that $p$ and $q$ induces isomorphism $$ B \otimes_A B \rightarrow B \times R $$
and
- There exists an idempotent element $e \in B \otimes_A B$ such that $p$ induces an isomorphism $$(B \otimes_A B)[1/e] \simeq B$$ under localization at $e$.
I can prove 1=>2. But I can't prove 2=>1. Help would be appreciated : )
Let $R$ be a ring and $e$ an idempotent.
So $e^2=e$ or, perhaps written in a more suitable form for this question : $e(e-1) = 0$.
In particular if you invert $e$, then $e-1 = e^{-1} 0 = 0$ so $e=1$. Inverting an idempotent turns it into $1$.
In particular, you have a factorization of the canonical morphism as $R\to R/(e-1)\to R[1/e]$.
Conversely, if you kill $e-1$, you get $e=1$, so $e$ is invertible, so you also get a factorization of the canonical morphism as $R\to R[1/e]\to R/(e-1)$. It's then easy to check that these give you an isomorphism $R[1/e]\cong R/(e-1)$