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user14449
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In Milne's EC, he defines an etale neighbourhood of a point $x \in X$ by a pair $(Y,y)$ with an etale morphism $f: Y \to X$ where $f^{-1}(x) = y$$f(y) = x$, such that $k(x) = k(y)$.

What I don't understand is this last condition. Is there an intuitive reason why this condition is necessary? Explanations are welcome, although examples would be more appreciated.

In Milne's EC, he defines an etale neighbourhood of a point $x \in X$ by a pair $(Y,y)$ with an etale morphism $f: Y \to X$ where $f^{-1}(x) = y$, such that $k(x) = k(y)$.

What I don't understand is this last condition. Is there an intuitive reason why this condition is necessary? Explanations are welcome, although examples would be more appreciated.

In Milne's EC, he defines an etale neighbourhood of a point $x \in X$ by a pair $(Y,y)$ with an etale morphism $f: Y \to X$ where $f(y) = x$, such that $k(x) = k(y)$.

What I don't understand is this last condition. Is there an intuitive reason why this condition is necessary? Explanations are welcome, although examples would be more appreciated.

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user14449
user14449
  • 371
  • 1
  • 9

Definition of etale neighbourhood

In Milne's EC, he defines an etale neighbourhood of a point $x \in X$ by a pair $(Y,y)$ with an etale morphism $f: Y \to X$ where $f^{-1}(x) = y$, such that $k(x) = k(y)$.

What I don't understand is this last condition. Is there an intuitive reason why this condition is necessary? Explanations are welcome, although examples would be more appreciated.