Mixing
Unramified implies local-etale
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I'm trying to understand the proof of the following lemma in Freitag/Kiehl, "Etale Cohomology...":
1.5 Lemma. Let $A rightarrow B$ be a finitely generated local homomorphism. We assume that it is injective and that $A$ is a normal ring. If $B$ is unramified over $A$, then $B$ is a local-etale $A$-algebra.
It had previously been shown that $B=tildeB/mathfrak a$, where $tilde B$ is local-etale over $A$. Taking this for granted, the proof continues:
As $A$ is normal, so is $tilde B$ ... ,and thus without zero divisors. From the injectivity of $A rightarrow B$ we conclude $mathfrak a = 0$.
I don't understand how we conclude that $mathfrak a =0$ based on the preceding information. (I accept that $tildeB$ is normal.) Would someone be kind enough to help me understand?
Note: $f: A rightarrow B$ local-etale means:
- $f$ is a localization of a finitely generated morphism
- $f$ is flat and unramified
commutative-algebra
asked Jul 22 at 16:22
rj7k8
89110
add a comment |Â
up vote
0
down vote
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I'm trying to understand the proof of the following lemma in Freitag/Kiehl, "Etale Cohomology...":
1.5 Lemma. Let $A rightarrow B$ be a finitely generated local homomorphism. We assume that it is injective and that $A$ is a normal ring. If $B$ is unramified over $A$, then $B$ is a local-etale $A$-algebra.
It had previously been shown that $B=tildeB/mathfrak a$, where $tilde B$ is local-etale over $A$. Taking this for granted, the proof continues:
As $A$ is normal, so is $tilde B$ ... ,and thus without zero divisors. From the injectivity of $A rightarrow B$ we conclude $mathfrak a = 0$.
I don't understand how we conclude that $mathfrak a =0$ based on the preceding information. (I accept that $tildeB$ is normal.) Would someone be kind enough to help me understand?
Note: $f: A rightarrow B$ local-etale means:
- $f$ is a localization of a finitely generated morphism
- $f$ is flat and unramified
commutative-algebra
asked Jul 22 at 16:22
rj7k8
89110
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I'm trying to understand the proof of the following lemma in Freitag/Kiehl, "Etale Cohomology...":
1.5 Lemma. Let $A rightarrow B$ be a finitely generated local homomorphism. We assume that it is injective and that $A$ is a normal ring. If $B$ is unramified over $A$, then $B$ is a local-etale $A$-algebra.
It had previously been shown that $B=tildeB/mathfrak a$, where $tilde B$ is local-etale over $A$. Taking this for granted, the proof continues:
As $A$ is normal, so is $tilde B$ ... ,and thus without zero divisors. From the injectivity of $A rightarrow B$ we conclude $mathfrak a = 0$.
I don't understand how we conclude that $mathfrak a =0$ based on the preceding information. (I accept that $tildeB$ is normal.) Would someone be kind enough to help me understand?
Note: $f: A rightarrow B$ local-etale means:
- $f$ is a localization of a finitely generated morphism
- $f$ is flat and unramified
commutative-algebra
asked Jul 22 at 16:22
rj7k8
89110
I'm trying to understand the proof of the following lemma in Freitag/Kiehl, "Etale Cohomology...":
1.5 Lemma. Let $A rightarrow B$ be a finitely generated local homomorphism. We assume that it is injective and that $A$ is a normal ring. If $B$ is unramified over $A$, then $B$ is a local-etale $A$-algebra.
It had previously been shown that $B=tildeB/mathfrak a$, where $tilde B$ is local-etale over $A$. Taking this for granted, the proof continues:
As $A$ is normal, so is $tilde B$ ... ,and thus without zero divisors. From the injectivity of $A rightarrow B$ we conclude $mathfrak a = 0$.
I don't understand how we conclude that $mathfrak a =0$ based on the preceding information. (I accept that $tildeB$ is normal.) Would someone be kind enough to help me understand?
Note: $f: A rightarrow B$ local-etale means:
- $f$ is a localization of a finitely generated morphism
- $f$ is flat and unramified
commutative-algebra
asked Jul 22 at 16:22
rj7k8
89110
edited Jul 22 at 16:33
asked Jul 22 at 16:22
rj7k8
89110
asked Jul 22 at 16:22
rj7k8
89110
asked Jul 22 at 16:22
rj7k8
89110
89110
add a comment |Â
add a comment |Â
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