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Jun
11
comment '$R$-rational points,' where $R$ is an arbitrary ring
However, when $R$ is a field, this is not to be confused with the notion of $R$-rational points...
Jun
11
answered Hartshorne Lemma(II 6.1)
Jun
11
comment What is the difference between $\ell$-adic cohomology and cohomology with coefficient in $Z_\ell$?
Étale cohomology with coefficients in an infinite constant sheaf is sometimes not what you expect. For example, in $H^1$, under good conditions, for a constant abelian group $A$, $H^1_{\text{ét}} (X, A) \cong \mathrm{Hom}(\pi_1^{\text{ét}} (X), A)$, where the RHS is the group of continuous homomorphisms. Since $\pi_1^{\text{ét}} (X)$ is profinite, this will be trivial if every finite subgroup of $A$ is trivial.
Jun
11
revised What is the difference between $\ell$-adic cohomology and cohomology with coefficient in $Z_\ell$?
added 162 characters in body; edited tags
Jun
11
comment When a divisor on an algebraic curve is canonical
Riemann and Roch are two people, so it's just the "Riemann–Roch theorem".
Jun
10
comment Characterization of injective objects in abelian categories
@xyzzyz You mean left exact.
Jun
9
revised Galois group of $K(X)/K$
edited tags
Jun
9
answered Galois group of $K(X)/K$
Jun
9
comment algebraic-geometric interpretation of the principal ideal theorem
The plane $z = 0$ and the point $(x, y, z) = (0, 0, 1)$.
Jun
9
answered algebraic-geometric interpretation of the principal ideal theorem
Jun
9
comment Is $\mathbf{Rel}$ pre-additive?
$\Delta : X \to X \times X$ is the diagonal morphism, and $\nabla : X \amalg X \to X$ is the codiagonal morphism.
Jun
9
comment morphisms on topological spaces
Epimorphisms in $\mathbf{Top}$ are surjective. (Easy proof: the forgetful functor $\mathbf{Top} \to \mathbf{Set}$ has a right adjoint.) You're thinking of $\mathbf{Haus}$.
Jun
9
comment morphisms on topological spaces
For (4), it suffices to show that $\mathbf{Top}$ has equalisers.
Jun
9
comment Multiplication of rings is an abelian group homomorphism
@DonAntonio $R$ is an abelian group under addition!
Jun
9
comment Axiom UB on Grothendieck Universes
It depends on your definition of category. If you require the hom-sets to be disjoint, then yes. But then you will see the same phenomenon whether you use Bourbaki set theory or ZFC set theory.
Jun
9
comment Axiom UB on Grothendieck Universes
You should go look at the appendix in [SGA 4, Exposé I]. In ZFC, one can prove that any category whose hom-sets are isomorphic to sets in $\mathscr{U}$ is in fact isomorphic as a category to one whose hom-sets are in $\mathscr{U}$, so there is no loss of generality in restricting attention to those. The lack of canonicity is just something one has to live with when there is no global choice operator.
Jun
9
comment Axiom UB on Grothendieck Universes
There isn't one. ZFC does not have anything like $\tau$.
Jun
9
comment Functors that have a natural Isomorphism
What do you mean by $Rng$? For instance, the identity functor on the category of unital commutative rings has no interesting automorphisms.
Jun
9
answered Axiom UB on Grothendieck Universes
Jun
9
revised What is the $\tau$ symbol in the Bourbaki text?
added 3 characters in body