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5 votes
Accepted

Making clear the definition of 'affine variety' in Mumford's book.

To the pair. If it referred to only $đť‘‹$, it would be only a topological isomorphism, i.e., a homeomorphism. That's too weak. You want the composition of any regular function with the map to be a ...
Deane's user avatar
  • 7,767
5 votes
Accepted

Functoriality for presheaves

In general, if $J \subseteq I$, and you have an inverse system of algebraic objects $X_i$, then you have a canonical map $$\varprojlim_{i\in I} X_i \to \varprojlim_{i\in J} X_i.$$ One way to see this ...
Daniel Schepler's user avatar
4 votes
Accepted

$C(X, A) \otimes C(X, B) \cong C(X, A \otimes B)$?

You define the tensor product of condensed abelian groups like this for the same reason you do it this way in any category of sheaves of abelian groups: it gives the tensor product the expected ...
Daniël Apol's user avatar
  • 5,337
3 votes

Functoriality for presheaves

Hint: The limit of a diagram $D$ has a canonical map to the limit of a subdiagram $D'$, induced from projection maps out of the limit of $D$, using the universal property of the limit of $D'$.
Alex Kruckman's user avatar
3 votes
Accepted

Examples of non-localic topos

There are many ways to construct toposes, besides from locales or (directly) as a classifying topos of some geometric theory. One of the easiest constructions is probably that of a presheaf topos, so ...
Mark Kamsma's user avatar
2 votes

Universal property of global spec of a quasicoherent sheaf of algebra

I can only guess what Vakil intended the insight of the problem to be, but if I were to write an exposition of the relative spectrum I would include this exercise to get the reader to work out a "...
John Dougherty's user avatar
2 votes

So many different 'varieties', which one is this? Serre's algebraic variety

The answer to the question: How are “varieties in the sense of FAC” related to “varieties in the sense of scheme theory”? is They are the same up to equivalence of categories (if the base field is ...
Elías Guisado Villalgordo's user avatar
2 votes
Accepted

Existence of the restriction of a global section on a sheaf.

$T$ is by definition a sheaf on $X$, so the inclusion of the open set $U\subseteq X$ induces by definition a restriction map $\mathrm{res}^X_U\colon T(X)\rightarrow T(U)$. This is part of the data ...
Thorgott's user avatar
  • 12.3k
2 votes
Accepted

Does $\mathscr F$ have an injective resolution such each injective sheaf is an quasi-coherent $\mathcal O_X$-module?

In this answer one can find the following lemma: Lemma Let $X$ be a locally Noetherian scheme, then a sheaf $\mathcal F$ is quasicoherent and injective if and only it's an injective object in the ...
Lukas Heger's user avatar
  • 21.6k
2 votes
Accepted

Contradiction: Exact sequence of constant sheaves, but not exact when using long exact sequence?

Your contradiction is resolved by the fact that the connecting map $M \to H^1(U;N_X)$ is zero.
Sasha's user avatar
  • 17.4k
2 votes
Accepted

Which representation of $\mathrm{SL} (2; \mathbb{C})$ does the tautological bundle on $ \mathbb{CP}^1 $ correspond to?

The last part of the first statement is false; for any $n$, we get $\mathcal{O}(n)$ as the line bundle $\mathcal{L}_{-n}$ associated to weight $-n$ (along with a $G-$equivariant structure). The ...
I'm Representable's user avatar
1 vote
Accepted

The symbol of de Rham cohomology on Stacks Project

If $A = \Gamma(X, \mathscr{O}_X)$, then $R\Gamma(X,\cdot) : D(X) \to D(A)$ is the right derived functor of the global sections functor $\Gamma(X,\cdot) : (\mathscr{O}_X-\text{Mod}) \to (A-\text{Mod})$;...
kobe's user avatar
  • 42.1k
1 vote
Accepted

Definition of degree of a vector bundle?

Any vector bundle $E$ on a curve $C$ has a rational section, i.e., a section defined on the complement of a finite set of points. Such a section extends to a fiberwise embedding $L \to E$ from a line ...
Sasha's user avatar
  • 17.4k
1 vote
Accepted

Definition of isomorphism of locally ringed spaces

This follows from the fact that $f$ is a homeomorphism. It namely holds that the canonical map $\mathrm{colim}_{f(x)\subset V\,\mathrm{open}}\,F(f^{-1}(V))\to\mathrm{colim}_{x\in U\,\mathrm{open}}\,F(...
Daniël Apol's user avatar
  • 5,337
1 vote

Transition function of the tautological bundle on $\mathbb{P}^n_k$

$X$ is a closed subscheme of $\mathbb A^{n+1}\times \mathbb P^n$, so the coordinate rings (denoting $x_{ij}=x_i/x_j$ on $x_j\neq0$) $k[x_0,\ldots,x_n]\otimes k[x_{0j},\ldots,x_{nj}]$ are also subject ...
Ben's user avatar
  • 6,931
1 vote

Abstract nonsense proof that stalks of $\mathcal{O}_X$ modules are modules over $\mathcal{O}_X$-stalks

A sheaf of $\mathcal{O}_X$ modules can be thought of as a sheaf $\mathcal{F}$ with two sheaf homomorphisms i) $\mathcal{F}\times \mathcal{F}\rightarrow \mathcal {F}$ ii) $\mathcal{O}_X\times \mathcal{...
Babai's user avatar
  • 5,075

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