I am reading Freitag’s etale cohomology and Weil conjecture. He says that A finitely generated flat algebra $A \rightarrow B$ is etale if and only if the following condition is satisfied- For every prime ideal $p \subset A$ , the algebra $B \underset{A} {\otimes}k(p)$ is finite separable over $k(p)$. Here $k(p)$ is quotient field of $A/p$ and finite separable means that it can be written as finite cartesian product of finite seperable extensions of $k(p)$.
Can someone help me on this?
Edit:
A ring homomorphism $A \rightarrow$ B is called etale if the following conditions are satisfied:
a) $ B $ is finitely generated $A $ algebra.
b) $B$ is a flat $A$ module.
c) $A \rightarrow B$ is unramified, that is , all localisations $A_q \rightarrow B_p $ are unramified ($p$ is prime ideal of $B$ , $q = p \cap A$ ).
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A local homomorphism is called unramified if the following conditions are satisfied:
a) $m(A).B = m(A)$, and
b)the residue field extension is finite and separable.