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Sep
7
awarded  Nice Answer
Sep
6
comment Can regular functions be specified simply as the sections of a bundle?
Obviously, $p : E \to X$ is not always a local homeomorphism. You should check that you understand the definitions correctly.
Sep
6
comment Can regular functions be specified simply as the sections of a bundle?
Here's an odd fact: if $E$ has the initial topology with respect to $p : E \to X$, then $p : E \to X$ is a local homeomorphism if and only if it is a homeomorphism.
Sep
6
comment is tangent bundle of $S^n$ an algebraic variety?
I have no idea. I originally encountered it as an exercise when I took undergraduate algebraic geometry from Ian Grojnowski.
Sep
6
comment hom-set definition of limit?
First, you need to understand the definition of morphisms in $[\mathcal{D}^\mathrm{op}, \mathbf{Set}]$. If you don't like $\mathcal{D}^\mathrm{op}$ you can replace it with $\mathcal{I}$ if you like – there is no difference. Once you do that, it will be quite easy to see the connection with cones.
Sep
5
comment Commuting with kernels implies left exactness in Abelian category
The statement is imprecise. An additive functor that preserves kernels is left exact.
Sep
5
comment Is every $\mathcal{O}_X$-module homomorphism $\mathcal{O}_X^{\oplus n} \to \mathcal{O}_X^{\oplus m}$ given by a matrix?
@SergioDaSilva $\mathscr{O}_X$-module means (abelian) sheaf with an $\mathcal{O}_X$-action. This is standard terminology.
Sep
5
comment Non-bijective isomorphism in a category of of sets.
Indeed, any isomorphism in any subcategory of $\mathbf{Set}$ must be an isomorphism in $\mathbf{Set}$ itself, i.e. bijective. Are you sure you have not misread the question?
Sep
4
comment Polynomial vanishing on $\mathbb{A}^2$
I presume the interesting part of the question is not the application of the Nullstellensatz but rather recognising that $f(x, xy)$ is identically zero iff $f(x, y)$ is identically zero.
Sep
4
answered Polynomial vanishing on $\mathbb{A}^2$
Sep
4
comment Is there a name for taking the pushforward (ie pushout) over the pullback?
Pushforward is not a synonym of pushout, don't use it like that. In an adhesive category, you might just call this the union.
Sep
3
comment Cartesian product distributes over second factor in tensor product?
Yes. But more directly you could show that $\otimes$ distributes over $\oplus$.
Sep
3
comment Why do we quotient by chain homotopy in the derived category.
You don't really need the Ore condition (as Sunny explained). It is merely convenient for proving some things...
Sep
3
comment Abstract nonsense proof that stalks of $\mathcal{O}_X$ modules are modules over $\mathcal{O}_X$-stalks
The category of sheaves of sets, of course.
Sep
3
comment Abstract nonsense proof that stalks of $\mathcal{O}_X$ modules are modules over $\mathcal{O}_X$-stalks
If you think in terms of action maps, then the relevant fact is that taking stalks preserves finite limits (in particular finite products).
Sep
3
comment Do schemes have a characterisation as an etale space?
No. Read carefully: "Formally, a ringed space $(X, O_X)$ is a topological space $X$ together with a sheaf of rings $O_X$ on $X$."
Sep
3
comment Do schemes have a characterisation as an etale space?
A ring is in particular a set, so you can construct its espace étalé the same way. The rest of your question still indicates some confusion about what a scheme is. A sheaf of rings can't be locally affine – that's a property of ringed spaces; a scheme is not merely a space; etc.
Sep
3
comment Tensor product of injective ring homomorphisms
Yes, indeed. I used Gröbner bases in Mathematica for my calculations.
Sep
3
comment Recommendation on Category theory textbook
You can work purely within ZFC if you like, you just have to pay a lot of attention to size issues. It only confuses beginners.
Sep
2
comment Tensor product of injective ring homomorphisms
@MartinBrandenburg As it turns out, there is an example!