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That should be read "B is etale over A". This happens when the map from A->B is an etale ring map, which means that its dual map is an etale morphism of affine schems from SpecB->SpecA, which is defined:

http://en.wikipedia.org/wiki/Etale_morphism

As with most things in ring theory, this condition is somewhat more trivial when A is a field. We get flatness since the only stalk of specA is A (spec A has one point), which is a field, so all of its modules are free, and hence flat. Unramifiedness will not always hold, but it's also lot easier because k is a field. If k is of characteristic zero, the extension is automatically separable, so then we only need to restrict to it being finite.

There's a more direct definition which says that the morphism A->B is a smooth ring map with relative dimension zero.

If you'd like to read a section on them in more generality, you can check out Stacks-Git Chapter 7 section 85 (7.85) on page 366 .

http://www.math.columbia.edu/algebraic_geometry/stacks-git/

I'm sure it's also in Hartshorne.

$\DeclareMathOperator{\Spec}{Spec}$ That should be read "$B$ is etale over $A$". This happens when the map from $A\to B$ is an etale ring map, which means that its dual map is an etale morphism of affine schems from $\Spec B \to \Spec A$, which is defined:

http://en.wikipedia.org/wiki/Etale_morphism

As with most things in ring theory, this condition is somewhat more trivial when $A$ is a field $k$. We get flatness since the only stalk of $\Spec A$ is $A$ ($\Spec A$ has one point), which is a field, so all of its modules are free, and hence flat. Unramifiedness will not always hold, but it's also lot easier because $k$ is a field. If $k$ is of characteristic zero, the extension is automatically separable, so then we only need to restrict to it being finite.

There's a more direct definition which says that the morphism $A\to B$ is a smooth ring map with relative dimension zero.

If you'd like to read a section on them in more generality, you can check out Stacks Project Tag 00U0.

I'm sure it's also in Hartshorne.

That should be read "B is etale over A". This happens when the map from A->B is an etale ring map, which means that its dual map is an etale morphism of affine schems from SpecB->SpecA, which is defined:

http://en.wikipedia.org/wiki/Etale_morphism

As with most things in ring theory, this condition is somewhat more trivial when A is a field. We get flatness since the only stalk of specA is A (spec A has one point), which is a field, so all of its modules are free, and hence flat. Unramifiedness will not always hold, but it's also lot easier because k is a field. If k is of characteristic zero, the extension is automatically separable, so then we only need to restrict to it being finite.

If you'd like to read a section on them in more generality, you can check out Stacks-Git Chapter 7 section 83 (7.83) on page 363.

http://www.math.columbia.edu/algebraic_geometry/stacks-git/

I'm sure it's also in Hartshorne.

That should be read "B is etale over A". This happens when the map from A->B is an etale ring map, which means that its dual map is an etale morphism of affine schems from SpecB->SpecA, which is defined:

http://en.wikipedia.org/wiki/Etale_morphism

As with most things in ring theory, this condition is somewhat more trivial when A is a field. We get flatness since the only stalk of specA is A (spec A has one point), which is a field, so all of its modules are free, and hence flat. Unramifiedness will not always hold, but it's also lot easier because k is a field. If k is of characteristic zero, the extension is automatically separable, so then we only need to restrict to it being finite.

There's a more direct definition which says that the morphism A->B is a smooth ring map with relative dimension zero.

If you'd like to read a section on them in more generality, you can check out Stacks-Git Chapter 7 section 85 (7.85) on page 366 .

http://www.math.columbia.edu/algebraic_geometry/stacks-git/

I'm sure it's also in Hartshorne.

That should be read "B is etale over A". This happens when the map from A->B is an etale ring map, which means that its dual map is an etale morphism of affine schems from SpecB->SpecA, which is defined:

http://en.wikipedia.org/wiki/Etale_morphism

As with most things in ring theory, this condition is somewhat more trivial when A is a field. We get flatness since the only stalk of specA is A (spec A has one point), which is a field, so all of its modules are free, and hence flat. Unramifiedness will not always hold, but it's also lot easier because k is a field. If k is of characteristic zero, the extension is automatically separable, so then we only need to restrict to it being finite.

That should be read "B is etale over A". This happens when the map from A->B is an etale ring map, which means that its dual map is an etale morphism of affine schems from SpecB->SpecA, which is defined:

http://en.wikipedia.org/wiki/Etale_morphism

As with most things in ring theory, this condition is somewhat more trivial when A is a field. We get flatness since the only stalk of specA is A (spec A has one point), which is a field, so all of its modules are free, and hence flat. Unramifiedness will not always hold, but it's also lot easier because k is a field. If k is of characteristic zero, the extension is automatically separable, so then we only need to restrict to it being finite.

If you'd like to read a section on them in more generality, you can check out Stacks-Git Chapter 7 section 83 (7.83) on page 363.

http://www.math.columbia.edu/algebraic_geometry/stacks-git/

I'm sure it's also in Hartshorne.