The Stacks project

fix small typos

an error in 0F40

Thanks to 李一笑 who writes

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(in the paragraph starting with defining V_{12}) The morphism p_1 may
not be an isomorphism over an open neighbourhood of Z_{1,2}, and is
only a bijective closed immersion in general, since we only assumed
U_1 \cap U_2 to be set-theoretically dense in U.

To fix the problem we may instead define V_{12} = V_1 \times_U V_2,
and X_{12} the scheme-theoretic image of V_{12} \to X_1 \times_S X_2.
Then the remaining parts of the proof need not be changed.

p.s. If you want to fix the original proof,
the equality p_1^{-1} V_1 = p_2^{-1} V_2 defining V_{12} is true,
but it is not obvious, and may require proof.

---------------------------

All of this is correct (I think). I have fixed the proof by taking
the first suggestion. All mistakes are mine!

Fix error

Thanks to Bogdan Zavyalov

The error was to silently assume that X is dense in any compactification
of X but in the definition of compactifications earlier in the Stacks
project we didn't require this. This was a bit hard to fix because we
then in later lemmas used the flexibility afforded by the more general
definition of a compactification.

I fixed this by leaving the definition of a compactification alone and
by very carefully arguing that inside the category of all
compacfifications of a given scheme X the ones where X is dense is an
initial category.

I also much clarified the proof of Lemma 0B6T to boot

I also checked all other uses of compactifications in the Stacks project
and the lemmas I fixed are the only ones using compactifications in the
wrong manner...

Interestingly in the case of defining etale cohomology with proper
supports we don't make the same mistake because there we have a stronger
lemma about lower shriek functors which makes the thing work more
smoothly.

Tags: Added new tags

Fix word order and one horrible typo

Somehow Nagata's theorem was stated completely wrong. Not sure how this
happened...

Additional lemma in proof Nagata compactification

Clean up and finish proof Nagata compactification

First pass at Nagata's theorem on compactification