I lied (This has nothing to do with the title.)

This is actually a question that's gone unanswered since Monday. The previous title was "Formally étale at all primes does not imply formally étale." I hope my ploy actually works.

Real Question

All rings are assumed to be commutative and unital, with all homomorphisms unital as well.

On last week's homework, there was a mistake in one of the questions:

(2.5) Let $R\to S$ be a morphism of commutative rings giving $S$ an $R$-algebra structure. Suppose that the induced maps $R\to S_{\mathfrak{p}}$ are formally étale for all prime ideals $\mathfrak{p}\subset S$. Then $R\to S$ is formally étale.

According to our professor, the exercise should have stated additionally that $S$ was finitely presented over $R$ (which allows us to prove that $S$ is in fact étale over $R$ rather than just formally étale). This is not too hard to do and is left as an exercise. (You can also find it in EGA).

However, I'm interested in seeing either a counterexample or a proof for the stronger claim in the grey box.

I lied (This has nothing to do with the title.)

This is actually a question that's gone unanswered since Monday. The previous title was "Formally étale at all primes does not imply formally étale." I hope my ploy actually works.

Real Question

All rings are assumed to be commutative and unital, with all homomorphisms unital as well.

On last week's homework, there was a mistake in one of the questions:

(2.5) Let $R\to S$ be a morphism of commutative rings giving $S$ an $R$-algebra structure. Suppose that the induced maps $R\to S_{\mathfrak{p}}$ are formally étale for all prime ideals $\mathfrak{p}\subset S$. Then $R\to S$ is formally étale.

According to our professor, the exercise should have stated additionally that $S$ was finitely presented over $R$ (which allows us to prove that $S$ is in fact étale over $R$ rather than just formally étale). Does anyone know of a counterexample to the original (non-finitely-presented) claim?

Edit: Any sort of geometric reformulation of the question This is obviously just as interesting, but if you choose not too hard to answer this as a geometry question, remember that étale here means finitely presented do and not locally finitely presented, so I'd appreciate it if you clarified which definition you're usingis left as an exercise.

Edit 2: (To clarifyYou can also find it in EGA).

However, we're looking for I'm interested in seeing either a counterexample to or a proof for the statement stronger claim in the gray box.)

Edit 3: Is there a reason why coming up with a counterexample is particularly difficult? I'd be interested to hear what everyone's thinkinggrey box.

All rings are assumed to be commutative and unital, with all homomorphisms unital as well.

On last week's homework, there was a mistake in one of the questions:

(2.5) Let $R\to S$ be a morphism of commutative rings giving $S$ an $R$-algebra structure. Suppose that the induced maps $R\to S_{\mathfrak{p}}$ are formally étale for all prime ideals $\mathfrak{p}\subset S$. Then $R\to S$ is formally étale.

According to our professor, the exercise should have stated additionally that $S$ was finitely presented over $R$ (which allows us to prove that $S$ is in fact étale over $R$ rather than just formally étale). Does anyone know of a counterexample to the original (non-finitely-presented) claim?

Edit: Any sort of geometric reformulation of the question is obviously just as interesting, but if you choose to answer this as a geometry question, remember that étale here means finitely presented and not locally finitely presented, so I'd appreciate it if you clarified which definition you're using.

Edit 2: (To clarify, we're looking for a counterexample to the statement in the gray box.)

Edit 3: Our professor has checked Is there a reason why coming up on this via e-mail but hasn't gotten with a response yet. Any ideascounterexample is particularly difficult? I'd be interested to hear what everyone's thinking.

All rings are assumed to be commutative and unital, with all homomorphisms unital as well.

On last week's homework, there was a mistake in one of the questions:

(2.5) Let $R\to S$ be a morphism of commutative rings giving $S$ an $R$-algebra structure. Suppose that the induced maps $R\to S_{\mathfrak{p}}$ are formally étale for all prime ideals $\mathfrak{p}\subset S$. Then $R\to S$ is formally étale.

According to our professor, the exercise should have stated additionally that $S$ was finitely presented over $R$ (which allows us to prove that $S$ is in fact étale over $R$ rather than just formally étale). Does anyone know of a counterexample to the original (non-finitely-presented) claim?

Edit: Any sort of geometric reformulation of the question is obviously just as interesting, but if you choose to answer this as a geometry question, remember that étale here means finitely presented and not locally finitely presented, so I'd appreciate it if you clarified which definition you're using.

Edit 2: (To clarify, we're looking for a counterexample to the statement in the gray box.)

Edit 3: Our professor has checked up on this via e-mail but hasn't gotten a response yet. Any ideas?

All rings are assumed to be commutative and unital, with all homomorphisms unital as well.

On last week's homework, there was a mistake in one of the questions:

*

(2.5)* Let $R\to S$ be a morphism of commutative rings giving $S$ an $R$-algebra structure. Suppose that the induced maps $R\to S_{\mathfrak{p}}$ are formally étale for all prime ideals $\mathfrak{p}\subset S$. Then $R\to S$ is formally étale.

According to our professor, the exercise should have stated additionally that $S$ was finitely presented over $R$ (which allows us to prove that $S$ is in fact étale over $R$ rather than just formally étale). Does anyone know of a counterexample to the original (non-finitely-presented) claim?

Edit: Any sort of geometric reformulation of the question is obviously just as interesting, but if you choose to answer this as a geometry question, remember that étale here means finitely presented and not locally finitely presented, so I'd appreciate it if you clarified which definition you're using.

Edit 2: (To clarify, we're looking for a counterexample to the statement in the gray box.)

All rings are assumed to be commutative and unital, with all homomorphisms unital as well.

On last week's homework, there was a mistake in one of the questions:

*(2.5)*Let $R\to S$ be a morphism of commutative rings giving $S$ an $R$-algebra structure. Suppose that the induced maps $R\to S_{\mathfrak{p}}$ are formally étale for all prime ideals $\mathfrak{p}\subset S$. Then $R\to S$ is formally étale.

According to our professor, the exercise should have stated additionally that $S$ was finitely presented over $R$ (which allows us to prove that $S$ is in fact étale over $R$ rather than just formally étale). Does anyone know of a counterexample to the original (non-finitely-presented) claim?

Edit: Any sort of geometric reformulation of the question is obviously just as interesting, but if you choose to answer this as a geometry question, remember that étale here means finitely presented and not locally finitely presented, so I'd appreciate it if you clarified which definition you're using.

Edit 2: (To clarify, we're looking for a counterexample to the statement in the gray box.box.)

All rings are assumed to be commutative and unital, with all homomorphisms unital as well.

On last week's homework, there was a mistake in one of the questions:

*(2.5)*Let $R\to S$ be a morphism of commutative rings giving $S$ an $R$-algebra structure. Suppose that the induced maps $R\to S_{\mathfrak{p}}$ are formally étale for all prime ideals $\mathfrak{p}\subset S$. Then $R\to S$ is formally étale.

According to our professor, the exercise should have stated additionally that $S$ was finitely presented over $R$ (which allows us to prove that $S$ is in fact étale over $R$ rather than just formally étale). Does anyone know of a counterexample to the original (non-finitely-presented) claim?

Edit: Any sort of geometric reformulation of the question is obviously just as interesting, but if you choose to answer this as a geometry question, remember that étale here means finitely presented and not locally finitely presented, so I'd appreciate it if you clarified which definition you're using.

Edit 2: To clarify, we're looking for a counterexample to the statement in the gray box.

All rings are assumed to be commutative and unital, with all homomorphisms unital as well.

On last week's homework, there was a mistake in one of the questions:

*(2.5)*Let $R\to S$ be a morphism of commutative rings giving $S$ an $R$-algebra structure. Suppose that the induced maps $R\to S_{\mathfrak{p}}$ are formally étale for all prime ideals $\mathfrak{p}\subset S$. Then $R\to S$ is formally étale.

According to our professor, the exercise should have stated additionally that $S$ was finitely presented over $R$ (which allows us to prove that $S$ is in fact étale over $R$ rather than just formally étale). Does anyone know of a counterexample to the original (non-finitely-presented) claim?

Edit: Any sort of geometric reformulation of the question is obviously just as interesting, but if you choose to answer this as a geometry question, remember that étale here means finitely presented and not locally finitely presented, so I'd appreciate it if you clarified which definition you're using.

All rings are assumed to be commutative and unital, with all homomorphisms unital as well.

On last week's homework, there was a mistake in one of the question, which asked us to show that if questions:

*(2.5)*Let $R\to S$ is be a morphism of commutative rings giving $S$ an $R$-algebra, and R$-algebra structure. Suppose that the induced maps $R\to S_{\mathfrak{p}}$ is are formally étale for all prime ideals $\mathfrak{p}\subset S$, then . Then $R\to S$ is formally étale. The

According to our professor, the exercise should have stated additionally that $S$ was finitely presented over $R$ (which proves allows us to prove that $S$ is in fact étale over $R$ rather than just formally étale), according to our professor. Does anyone know of a counterexample to the original (non-finitely-presented) claim?

Edit: Of course, $S$, $R$ commutative rings with unity and all homomorphisms are unital.

On last week's homework, there was a mistake in the question, which asked us to show that if $S$ is an $R$-algebra, and $R\to S_{\mathfrak{p}}$ is formally étale for all prime ideals $\mathfrak{p}\subset S$, then $R\to S$ is formally étale. The exercise should have stated additionally that $S$ was finitely presented over $R$ (which proves that $S$ is in fact étale over $R$ rather than just formally étale), according to our professor. Does anyone know of a counterexample to the original (non-finitely-presented) claim?

Edit: Of course, $S$, $R$ commutative rings with unity and all homomorphisms are unital.

On last week's homework, there was a mistake in the question, which asked us to show that if $S$ is an $R$-algebra, and $R\to S_{\mathfrak{p}}$ is formally étale for all prime ideals $\mathfrak{p}\subset S$, then $R\to S$ is formally étale. The exercise should have stated additionally that $S$ was finitely presented over $R$ (which proves that $S$ is in fact étale over $R$ rather than just formally étale), according to our professor. Does anyone know of a counterexample to the original claim?

Edit: Of course, $S$, $R$ commutative rings with unity and all homomorphisms are unital.