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Consider the disjoint union Spec$(\mathbb{Q})\coprod_{p}$Spec$(\mathbb{F}_p)$ with its canonical map to Spec$(\mathbb{Z})$. This is bijective on points, but the preimage of any open in Spec$(\mathbb{Z …

For question 5, I think the motivation for thinking of elements of $\hat{F_2}$ as "profinite words" is that $\hat{F_2}$ enjoys the same universal mapping properties as $F_2$, but in the profinite sett …

Here's a counterexample. Let $P=\mathbb{P}^1$, $X=\mathbb{A}^1$, and attach $X$ to $P$ along a single point $\{x\}$. Then there is a global section $f$ which is nonzero on $X$ except at $x$, and is i …

- Separated, excellent, regular: Spec$(k)$. - Separated, excellent, not regular: Spec$(k[\epsilon]/\epsilon^2)$. - Separated, not excellent, regular: See http://en.wikipedia.org/wiki/Excellent_ring …

Primes in a (commutative) Jacobson ring This question was phrased purely algebraically, but I only arrived at the solution by geometric arguments.

Here's a more analytic description of exactly what knowing $V(f)$ tells you. Let us say $f$ ~ $g$ if their vanishing sets are the same, and moreover there exist positive constants $c,C$ such that $c\ …

This will never be the case. By the criterion of Neron-Ogg-Shafarevich, an elliptic curve has good reduction if and only if its $\ell$-adic Tate module is unramified for $\ell\neq p$, where $p$ is t …

Expanding on abx’s comment, let’s suppose $\mathcal{M}_{G,C}$ was representable as a scheme. Then for any $\mathbb{C}$-scheme $S$ there is a canonical isomorphism $Hom_\mathbb{C}(S, \mathcal{M}_{G,C}) …

$\DeclareMathOperator\Spec{Spec}$Actually, your suggested categorical characterization of spectra of fields does work. Edit: (I had written something incorrect here) By Martin's comment below, we just …

In fact there are one-dimensional counterexamples: if $\chi$ is the $l$-adic cyclotomic character, and $k\in \mathbb{Z}_l \backslash \mathbb{Z}$, then $\chi^{(l-1)k}$ is unramified outside $l$, but do …

1) No. The normalization of a ring $R$ is never flat over $R$, unless $R$ was normal in the first place.

A current trend in number theory is understanding how modular forms (or more generally automorphic representations) can be put into $p$-adic families, as well as their associated galois representation …