Unramified morphism

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In algebraic geometry, an unramified morphism is a morphism f : X → Y of schemes such that (a) it is locally of finite presentation and (b) for each x ∈ X and y = f(x), we have that
1. The residue field k(x) is a separable algebraic extension of k(y). 2. f^(#)(𝔪_(y))𝒪_(x, X) = 𝔪_(x), where f^(#) : 𝒪_(y, Y) → 𝒪_(x, X) and 𝔪_(y), 𝔪_(x) are maximal ideals of the local rings.
A flat unramified morphism is called an étale morphism. Less strongly, if f satisfies the conditions when restricted to sufficiently small neighborhoods of x and y, then f is said to be unramified near x.
Some authors prefer to use weaker conditions, in which case they call a morphism satisfying the above a G-unramified morphism.

Source: https://en.wikipedia.org/wiki/Unramified_morphism
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