0
votes
0answers
46 views

Grothendieck topology

Hello : Here is a small parapgraph that i try to understand : If $ M $ is a smooth manifold, then we can recover the underlying set of $ M $ by considering the set $ \mathrm{Hom} ( \{ \star \} , ...
3
votes
2answers
42 views

The two projection maps are different?

I'm reading Vistoli's notes, and I came across something on the bottom of page 31, starting with "The reader might find our definition of sheaf pedantic..." Essentially my problem is the following ...
2
votes
1answer
29 views

About closed map beween schemes

Probabily it's trivial but I've no idea for a proof. Let $f: X \rightarrow Y $ a continuous map between Topological Spaces, with $Im(f)$ closed in $Y$. I know there exist a covering $\{Y_i\}$ of $Y$ ...
2
votes
1answer
86 views

Why do the Zariski distinguished open subsets form a base?

Let $R$ be commutative ring with unit. I have to prove that the distinguished open sets form a base for the Zariski topology i.e. any non-empty open set is a union of distinguished ones. We have that ...
5
votes
1answer
47 views

Concentrated schemes are closed under finite gluings

Let $X$ be a scheme. Assume that $X = \cup_i X_i$ is finite open covering, such that the $X_i$ as well as their intersections $X_i \cap X_j$ are concentrated, i.e. quasi-compact and quasi-separated ...
4
votes
1answer
95 views

Irreducibility preserved under étale maps?

I remember hearing about this statement once, but cannot remember where or when. If it is true i could make good use of it. Let $\pi: X \rightarrow Y$ be an étale map of (irreducible) algebraic ...
2
votes
1answer
131 views

The precise definition of a “sheaf of rings”

Let $X$ be a topological space. Let $Op(X)$ be the category of open sets. A sheaf of abelian groups is a (contravariant) functor $F:\mathcal{C} \to Ab$ satisfying the sheaf condition: the following ...