For questions on the Zariski-open immersion, étale, flat, or other topologies defined via classes of morphisms in a category of schemes. Not to be confused with the usual Zariski topology in commutative algebra and algebraic geometry.
Questions tagged [grothendieck-topologies]
144 questions
16
votes
2 answers
How to obtain Grothendieck’s “Long March Through Galois Theory”
Several works cite "La longue marche a travers la theorie de galois". The work by Leila Schneps "Grothendieck’s "Long March through Galois theory" ( http://webusers.imj-prg.fr/~leila.schneps/SchnepsLM.pdf ) tells us that there is as TeX format of…
Valera Rozuvan
- 449
12
votes
1 answer
Dense topology <=> double negation operator in a constructive metatheory?
The dense topology for a category $\mathbb{C}$ can be defined as follows, writing $\mathscr{H}:\mathbb{C}\to\widehat{\mathbb{C}}$ for the Yoneda embedding (considering sieves on $c$ as subfunctors of $\mathscr{H}(c)$):
$$J_{\mathsf{dense}}(c)…
11
votes
0 answers
Algebraic Geometric Analogue of Brown's Representability
Brown's representability theorem is very usefull to show that the functor
$$X \rightarrow H^i(X,A)$$
is representable. I would be interested to see if there exists an analogue of this statement in the context of algebraic geometry, in particular to…
curious math guy
- 2,008
11
votes
1 answer
Does $\mathsf{Top}$ have interesting Grothendieck topologies, and do they have applications?
In algebraic geometry, the importance of non-trivial Grothendieck topologies is very well-known. One starts out with the Zariski topology on $\mathsf{Sch}$, but concludes that it is 'too coarse' for cohomological techniques to work, and so one…
user542740
11
votes
3 answers
Kummer sequence etale topology
Consider the category $C=Sch/S$ of schemes over $S$ and let $n \in \Gamma(S,\mathcal{O}_S)^{*}$. It is possible to show that $$0 \rightarrow \mu_{n,S} \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 0$$
is exact in the etale topology,…
Tommaso Scognamiglio
- 1,468
10
votes
2 answers
Are category of groups and abelian groups topoi?
I am currently learning Grothendieck topoi and am wondering if Grps and Abel are topoi. I guess they are not topoi but I do not know which one of the Giraud axiom fails. The literature I am using is Categorical logic written by Jacob Lurie.…
Ziqiang Cui
- 491
10
votes
0 answers
Set theoretic issues in the definition of a site in Stacks Project
I've been learning about sites from the Stacks Project, which is generally very precise in its terminology, but I've found some of their conventions very confusing in this part. Their definition of a site is given here. Below the definition, they…
Luke
- 3,795
10
votes
2 answers
Lack of "good covers" in the étale topology
Disclaimer: This question might be terribly naive and almost certainly reflects my own ignorance.
If $X$ is a topological space admitting a finite triangulation, then it admits a "good covering," i.e. an open covering by contractible sets $U_i$ such…
Keenan Kidwell
- 26,770
8
votes
1 answer
Is representability of Zariski sheaves local on the base?
Let $F: \mathsf{Sch_{/S}}^{op} \to \mathsf{Set}$ be a Zariski sheaf on the category of $S$-schemes. $F$ being a sheaf means it satisfies the following property:
Sheaf condition: For every $S$-scheme $X$ and every open cover $\{U_j\} \subset…
Saal Hardali
- 5,059
7
votes
2 answers
points of a grothendieck topos $\mathscr{X}/X$
There is an exercise related to Grothendieck topos that I cannot solve:
Let $e:\text{Set}\rightarrow \mathscr{X}$ be a point, i.e. a geometric morphism $e^*\dashv e_*:\text{Set}\rightarrow \mathscr{X}$. Then for each $X\in \mathscr{X}$ and $x\in…
王夏辉
- 195
7
votes
1 answer
Proving $\operatorname{Spec} k[x]$ is internally a ring of fractions in the internal language of the Zariski topos
Let $\mathcal E$ be a Grothendieck topos and $R$ be a ring object in it. Say $R$ is internally a ring of fractions if $\neg (a=0)\implies a$ is invertible.
I'm trying to prove that in the opposite category of commutative $k$-algebras for $k$ a…
Arrow
- 14,390
7
votes
1 answer
Under-site like subspace topology
Let $X$ be a topological space and $Op(X)$ the category of its open sets. It is well known that $Op(X)$ has a canonical Grothendieck topology which makes of it a site. Let $U\in Op(X)$ be an object (an open set) of this category. It is naturally a…
C. Dubussy
- 9,373
7
votes
2 answers
Why is the fundamental group a sheaf in the etale topology?
In this paper by Minhyong Kim on p5, there is a variety $X$ defined over $\mathbb{Q}$, $G = \pi_1(X(\mathbb{C}),b)$ the topological fundamental group of the associated complex algebraic variety, and $G$^ the profinite completion of $G$.
Kim states…
Somatic Custard
- 1,054
7
votes
1 answer
Grothendieck topology and relation with usual topologies
Recently I stumbled upon the definition of $\textbf{Grothendieck}$ $\textbf{topologies}$ of a category $\mathcal{C}$. I do know that is one of the most interesting parts of the contemporary algebraic approach for topology and geometry as well.…
user321268
7
votes
1 answer
An example of a Grothendieck topology
A Grothendieck topology on a category $\mathcal{C}$ with finite limits consists of, for each object $U$ in $\mathcal{C}$ a collection $\text{Cov}(U)$ of sets $\{ U_i \to U \}$ such that
Isomorphisms are covers, e.g if $V \to U$ is an isomorphism…
Juan S
- 10,526