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Since you ask about other situations where this sort of thing occurs, let me describe a general principle (applied to the context of the original question) which is widely applied in EGA and elsewhere …

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{ …

I agree with Adam Smith that the question seems a bit misguided, but let me show anyway that the answer is negative away from certain silly cases. Well, first to make a more well-posed question, one …

Define $F(X) = {\rm{Hom}}_{\mathbf{C}}(X,{\rm{Spec}}(R/tR))$ where $R$ is the valuation ring of an algebraic closure of $\mathbf{C}((t))$. Note that every element of the maximal ideal of $R/tR$ is nil …

Read Katz-Mazur, "Arithmetic moduli of elliptic curves" (and for your purposes you can ignore the last chapter, even though it was their motivation for writing the book).

The non-classical aspect of this setup is that you're using a quasi-coherent sheaf that is not coherent, and beyond the coherent case one cannot expect information about a fiber (e.g., vanishing, 6 ge …

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 …

Initial question has a negative answer even for affine schemes. Let $B$ = Spec($R$) equipped with an action by a finite group $G$, and define $R' = \prod_{g \in G} R$ and $A$ = Spec($R'$). Let $A \ri …

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 …

Here is a partial solution: modulo a problem of constructing "sufficiently generic" elements in the maximal ideal of a reduced noetherian local ring of dimension > 1 (in a sense made precise at the en …

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 …

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 …

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 ( …

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 …

Set $A = B = k[x]$ and figure out for yourself what that is a counterexample. (Hint: rigorously prove that there's no "universal polynomial" over $k$-algebras.)