Here is a small summary of smooth, etale and unramified maps.

An $R$-algebra $S$ is always assumed to be fintiely presented over $R$.

Theorem

• $S$ is etale over $R$ iff near every prime $Q$, $S_b$ is a standard etale algebra over $R_a$

• $S$ is unramified over $R$ iff near every prime $Q$, $S_b$ is a holomorphic image by a finitely generated ideal of a standard etale algebra over $R_a$

• $S$ is smooth over $R$ iff $S_b$ is etale over a polynomial ring in finitely many variables over $R_a$

Theorem

• $S$ is etale over $R$ iff $S$ is $R$-flat and $\Omega_{S/R}=0$

• $S$ is unramified over $R$ iff $\Omega_{S/R}=0$

• (If $R$ contains rational)$S$ is smooth over $R$ iff $S$ is $R$-flat and $\Omega_{S/R}$ is projective.

# Fiberwise Characteriszation

Theorem

• $S$ is etale over $R$ iff $S$ is $R$-flat and $\kappa_P\otimes S$ is a product of finite seperable algebraic field extension of $\kappa_P$ for any prime $P$

• $S$ is smooth over $R$ iff $S$ is $R$-flat and $\kappa_P\otimes S$ is geometrically regular over $\kappa_P$