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Oct
21
comment Subrings generated by a set
Though, it is worth noting that the polynomial algebra $R[x]$ is indeed generated by $x$ as a subring of itself, which is important when describing the $R$-algebra homomorphisms $R[x] \to A$.
Oct
21
revised Subrings generated by a set
added 1 characters in body
Oct
21
comment An elementary question about gluing affine schemes
We must take care to glue it in the right way, though – otherwise we'll get the affine line with the origin doubled!
Oct
21
comment Monomorphisms vs pullbacks
That's false in general.
Oct
21
comment Monomorphisms vs pullbacks
Please clarify your question. Do you mean to ask, "if $X \times_Y X$ exists and the two projections $X \times_Y X \to X$ are equal, then $X \to Y$ is a monomorphism"?
Oct
21
comment A model structure on $\bf Cat$
Acyclic fibrations are indeed strictly surjective on objects.
Oct
21
comment Question about the residue field of a localization
It is not hard to show that $A_\mathfrak{m} / \mathfrak{m}_\mathfrak{m} \cong A / \mathfrak{m}$, but to determine that $A / \mathfrak{m} \cong k$ (as $k$-algebras!) requires a form of the Nullstellensatz.
Oct
20
comment A model structure on $\bf Cat$
No, I don't agree. Why should an acyclic cofibration be strictly surjective on objects?
Oct
20
comment How can function fields have different degrees over the projective line
Yes, it is the unique embedding that sends $t$ to $f$.
Oct
20
answered How can function fields have different degrees over the projective line
Oct
20
comment Why is the Borel Algebra on R not equal the powerset?
And to see why it's difficult to give a completely explicit example of a set which isn't in the Borel $\sigma$-algebra, see this MO answer.
Oct
20
comment Why is the Borel Algebra on R not equal the powerset?
There are lots of subsets of $\mathbb{R}$. What makes you think that every subset of $\mathbb{R}$ can be written as a countable union of countable intersections of open and closed subsets?
Oct
19
comment Relationships between zero morphisms and least morphisms
No, I don't see any reason why bi-monotone composition should imply that the bottom morphism (assuming it even exists!) is preserved.
Oct
19
comment What is algebraic geometry?
@anon: That's one way of doing things, but that is precisely not what algebraic geometry does...
Oct
19
comment Are Indizations cocomplete
This category is more properly called the "ind-completion of $\mathcal{C}$".
Oct
19
comment How to see $\operatorname{Spec} k[x]$ for non necessarily algebraic closed field $k$?
Can you say more about why $G$ acts transitively on $S$, when $\bar{k}$ is possibly non-separable over $k$?
Oct
18
comment A question about the definition of fibre bundle
You need to use the condition $\textrm{proj}_U \circ \phi_U = \pi |_U$.
Oct
18
comment $n$-sheeted branched covering
This is not algebra but geometry – specifically complex geometry. How familiar are you with this subject?
Oct
18
comment Other ways of proving that the set of all countable ordinals is uncountable
Not Russell's paradox so much as Burali-Forti's paradox, but they're closely related.
Oct
18
comment Projective module over a ring
$R^\kappa$ for an infinite cardinal $\kappa$ is not free in general. You mean $R^{\oplus \kappa}$.