Algebraic stacks thickening of an algebraic space
Thanks to Matthew Emerton, Toby Gee, and Brandon Levin
Matthew writes:
"Please find attached a PDF file, and the underlying LaTeX file, giving
a proof of the following result, which I couldn't find in the stacks
project, but which might belong there:
An algebraic stack thickening of an algebraic space is again an
algebraic space
and some related results (replace ``algebraic space'' by
``Deligne--Mumford stack'', ``quasi-DM stack'', etc.).
First, apologies if this is already there, and I missed it.
Second, this write-up arose out of discussions with Brandon Levin and
Toby Gee, and reflects contributions of all three of us. Brandon also
told me that he heard of this result from Bhargav Bhatt, but didn't know
of a reference.
<--------snip--------->
It would be great if this could be incorporated into the Stack project.
I think that Toby and I might need things like this in our work at some
point, and I think maybe Brandon does as well, and having the stacks
project as a reference for it would be very convenient!
Cheers,
Matt"
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Improve lemmas on thickenings and add one
Thanks to Matthew Emerton, Brandon Levin, and Toby Gee who sent these
improvements as well as the new lemma together with other material and
the following comments:
"The write-up begins with some preliminary material related to
http://stacks.math.columbia.edu/tag/05ZJ
In particular, Lemma 1 of the file recalls
http://stacks.math.columbia.edu/tag/05ZJ
and adds the condition of being representable (by schemes) to the
conditions already there.
Lemma 2 is
http://stacks.math.columbia.edu/tag/09ZZ
but adds the conditions of being a monomorphism, or being an
immersion.
Lemma 3 is a variant on Lemma 2, where we remove the hypothesis of
being a finite order thickening, but impose the condition that
both f and f' are locally of finite type."
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