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Let $j: \mathbf{C} - \mathbf{R} \rightarrow \mathbf{C}$ denote the classical $j$-function from the theory of elliptic functions. That is, $j(\tau)$ is the $j$-invariant of the elliptic curve $\mathbf{ …

It is a standard and important fact that any smooth affine group scheme $G$ over a field $k$ is a closed $k$-subgroup of ${\rm{GL}}_n$ for some $n > 0$. (Smoothness can be relaxed to finite type, but …

Every formally smooth morphism between locally noetherian schemes is flat; this is a deep result of Grothendieck. Indeed, the formal smoothness is preserved by localization on target and then likewis …

It is false. I'm not sure what the comment about algebraic spaces has to do with the question, since algebraic spaces do admit an fpqc (even \'etale) cover by a scheme. This is analogous to the fact …

Some time ago I mentioned a certain open question in an MO answer, and Pete Clark suggesting posting the question on its own. OK, so here it is: First, the setup. Let $X$ be a projective scheme over …

Let $k$ be a field complete with respect to a non-archimedean absolute value, and $X$ a separated algebraic space of finite type over $k$. If $X$ is a scheme then $X(k)$ inherits a natural (Hausdor …

The question gives the "wrong" definition of $\operatorname{Fix}(T)$, hence the resulting confusion. A more natural definition of the subfunctor $X^G$ of "$G$-fixed points in $X$" is $$ X^G(T) = \{x \ …

A point worth noting: the proof of fpqc descent for projectivity in Raynaud-Gruson is apparently incorrect (as I learned today from Gabber in connection with something else), but the result is noneth …

Dear Workitout: The list of comments above is getting unwieldy, so let me post an answer here, now that you have finally identified 1.10.1 in Katz-Mazur as (at least one) source of the question. As I …

This is really a comment on Pete's comment for Mikhail's answer, but I am making it an answer because it raises a question which I think should be more widely known. The construction of Aut-scheme us …

There's a more down to earth way to deal with this, which is already explained in Mumford's GIT: make an fppf (or even etale) surjective base change to acquire a section, use that to define the princ …

I've always meant to sit down and figure out some examples. OK, got it. I think the following works over any field (including finite fields and numbers fields) and so must be standard (unless I've ove …

Here is a really cool illustration of the principle which Emerton was outlining. We know that the Picard group of projective $(n-1)$-space over a field $k$ is $\mathbf{Z}$ ($n \ge 2$), generated by $ …

There are smooth counterexamples. Let $S_0$ be a smooth separated irreducible scheme over a field $k$ with dimension $d > 1$, and $s_0 \in S_0(k)$. Blow up $s_0$ to get another such scheme $S_1$ wit …

In the setup in the question, it should really say "we could have invertible meromorphic functions on Spec($A$) that don't come Frac($A)^{\times}$", since those are what give rise to "extra principal …