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See EGA III$_1$ 2.5.3 and EGA IV$_2$ 5.3. The elegant generalization there which incorporates an auxiliary coherent sheaf opens the door to using Grothendieck's unscrewing lemma (EGA III$_1$ 3.1.2) to …

A sufficient condition is that $f:X \rightarrow B$ is flat and locally finite type map between locally noetherian schemes such that its fibers are local complete intersection morphism (see EGA IV$\_4$ …

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 …

The point must be to avoid separatedness hypotheses on $f$. (D. Knutson proved algebraic spaces locally quasi-finite and separated over schemes are schemes; he may have had noetherian hypotheses, in w …

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 $ …

The entirety of sections 8--12 (apart from 10) of EGA IV is devoted to fleshing out in remarkably exhaustive and elegant generality the entire yoga of "spreading out and specialization" of which this …

The following seems to give a reasonable affirmative answer which avoids computing the coordinate ring directly, and replaces condition (2) with the more natural condition that the subset $\Sigma := X …

Since nobody gave a reference yet, in my paper "Artithmetic moduli of generalized elliptic curves" I included a proof that an Artin stack whose geometric points have trivial automorphism schemes is ne …

EGA IV$_4$, 19.3.8 (and 19.3.6); this addresses openness upstairs without properness, and (as an immediate consequence) the openness downstairs if $f$ is proper (which I assume you meant to require). …

An example is Deligne's theorem on the existence of good notion of quotient $X/G$ of a separated algebraic space $X$ under the action of a finite group $G$, or relativizations or generalizations (with …

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 …

In fact something better is true (properly formulated!) over any field $k$, using $k$-rational conjugacy and maximal $k$-split $k$-tori. I will give a precise statement and proof below, with $G$ any …

Does your critic dislike that the argument seems not applicable over general rings? But it is: if there's an isomorphism over some ring $R$ then we can descend to a finitely generated subring and pas …

Rational representations are directed unions of finite-dimensional ones, on which all linear representations of $G$ are completely reducible (either by an ad hoc definition of "reductive group" or a t …

Since the case of interest is $W_2(k)$ with perfect $k$ of characteristic $p > 2$, the answer is given by Ioan Berbec's 2009 paper "Group schemes over artinian rings and applications. In that paper ( …