The ring $A = \prod_{n=1}^{\infty} \mathbb{F}_2$ has some interesting/disturbing properties. For example, the affine scheme $X := {\rm{Spec}}(A)$ has non-open connected components (since it has ...

The necessary and sufficient condition on a schemes $\{U_i\}$ and gluing isomorphisms $\varphi_{ij}:U_{ij} \simeq U_{ji}$ between opens $U_{ij} \subset U_i$ ($i, j \in I$) that the gluing $X$ be ...

Note that it is not necessary to say to avoid GAGA, as GAGA has no relevance in the absence of compactness assumptions. Anyway, something much more general (and satisfying) is true: all topological ...

The reference is "Auslander-Buchsbaum formula". What matters is that $X$ and $Y$ are smooth of some common pure dimension $n$ (which forces flatness due to the quasi-finiteness, by the way), not that ...

Here are some remarks on the answers of Charles Matthews, Kevin Buzzard, and Victor Protsak. For justification of Kevin Buzzard's claim, see G. Prasad, "An elementary proof of a theorem of Bruhat-Tits-...

To paraphrase Igor Pak: OK, this I know. It is remarkable how difficult it is to track down a reference which gives an actual proof for this fact (moreover applicable to all global fields). The ...

It is a birational invariant (for smooth proper connected schemes over a field, ultimately due to Zariski-Nagata purity of the branch locus), and its formation is compatible with products (for proper ...

There is no example of an algebraic number of degree $> 2$ for which the boundedness or not of the entries of the continued fraction has been determined. In a 1976 Annals of Math paper, Baum & ...

Serre introduced a notion of "thin set" in the $k$-rational points of a $k$-variety (such as $k^n$ viewed as the $k$-rational points of affine $n$-space, or likewise for a Zariski-dense open locus in ...

Etale cohomology, whose definition and study would be inconceivable without the use of categorical methods, underlies the construction of many interesting Galois representations as well as the entire ...

To be honest, I have never in my life paid any attention to this stuff (except for its sieving application when reading postdoc job applications). I read papers because of the title/abstract, or ...

Indeed since the map on the base is not an isomorphism (apart from silly cases such as when $X$ is empty: the Frobenius endomporphism of a locally finite type $\mathbb{F}_p$-scheme is an isomorphism ...

I give an argument using the correct definition of isotriviality (relating it to the weak definition), and the reader who is familiar with Deligne-Mumford stacks can see that the method is actually ...

It sounds like the question you mean to ask is the following: if $X$ is an integral noetherian scheme with generic point $\eta$ and $C^{\bullet}$ is a finite complex of coherent sheaves on $X$ such ...

The group $D$ is the preimage of $E$ in $N(F)$, so it is as you expect. The finiteness hypothesis can be weakened, which is important for many applications. Things become clearer if one thinks ...

EGA IV$_4$, 3.1.1 for any quasi-coherent sheaf on any (pre)scheme: the commutative algebra definition on stalks. So one needs 3.1.2 and 3.1.3 there to get useful alternative formulations of this ...

It all works out as well as you could want in every possible sense because of the freeness of the action. As you know, you can make a cover by $G$-stable affine opens, so the real work is in that case....

Yes. Inject $f_ {!}$ into $f_ {\ast}$ to convert it into a claim concerning equality among "base change morphisms" (not generally isomorphisms) relating topological pushforward and pullback. Then it ...