Questions tagged [grothendieck-topology]
The grothendieck-topology tag has no usage guidance.
125 questions
3
votes
1
answer
315
views
What to call a morphism of sites inducing an equivalence on categories of sheaves?
Is there a standard term for a morphism of sites $(C,J)\to(C',J')$ which induces an equivalence on sheaf categories $\operatorname{Sh}(C',J')\xrightarrow\sim\operatorname{Sh}(C,J)$? The nLab entry ...
- 63.9k
27
votes
1
answer
966
views
Can we just use effective descent morphisms (pure morphisms) as covers?
There are a number of notions of "cover" for a scheme: etale, faithfully flat, fpqc, fppf, Zariski, Nisnevich, etc. Most of these have a nice property, which is that a cover of that type satisfies ...
- 10.4k
3
votes
1
answer
384
views
fppf-extension of algebraic groups is an algebraic group
The problem is the following:
Let $k$ be a field and let $N,G,Q: (\operatorname{Sch}/k)_{\operatorname{fppf}} \longrightarrow \operatorname{Grps}$
be fppf-sheaves from the big fppf-site of $\...
- 4,746
2
votes
1
answer
678
views
Why care about Grothendieck topology? [closed]
Noah Schweber said here the following:
Why would you want a notion of sheaf theory for objects more general
than topological spaces? Well, the original motivation (to my
understanding) was to ...
- 12.3k
6
votes
1
answer
670
views
Do coherent sheaves on rigid analytic spaces form an abelian category?
It is said that the category of sheaves of abelian groups on a Grothendieck site(topology) is an abelian category. On the other hand, it is known that in usual algebraic geometry, given an variety (...
CommunityBot
- 1
7
votes
1
answer
977
views
Applications of $h$-topology and $h$-descent
This is a technical problem about applications of Grothendieck topologies. In some recent works, the technique of $h$-topology and $h$-descent is very useful, for an introduction see https://stacks....
- 17.4k
4
votes
1
answer
658
views
How to construct cup-product in a general site?
Let $\mathcal{C}$ be a site on the category of schemes for some Grothendieck topology, $X$ a scheme, and let $F,G$ be two sheaves of abelian groups on $\mathcal{C}$. Do we have cup-product as follows?
...
- 37.8k
2
votes
2
answers
731
views
In the definition of big/small étale/fppf/... site, is their covering set really a set?
My definition of a site is a pair $(\mathcal{C},\DeclareMathOperator{\Cov}{Cov}\Cov (\mathcal{C}))$ where $\mathcal{C}$ is a category and $\Cov (\mathcal{C})$ is a set consisting of elements of the ...
- 12.9k
0
votes
0
answers
325
views
Grothendieck topology on a scheme equivalent to the circle
Suppose a class of morphisms of schemes is reasonable enough so that we can associate a small site to any scheme (just like small étale site). Is there a simple class of morphisms such that there is a ...
CommunityBot
- 1
6
votes
0
answers
152
views
Does an fppf cohomological class annihilated by an etale cover come from etale cohomological group?
Let $X$ be a scheme, $F$ a sheaf on the fppf site of $X$, and $\alpha\in H^i_{\mathrm{fppf}}(X,F)$ such that it is trivialized by an etale cover of $X$. Does $\alpha$ lie in the image of the canonical ...
- 997
4
votes
1
answer
180
views
When can a scheme be recovered from its descent groupoid?
Suppose that $ Y $ is a scheme and $ f\colon X\to Y $ a covering of $ Y $ in some Grothendieck topology on the category of schemes (i.e. if $\{ U_i\to Y\}$ is a covering in the topological sense, then ...
- 42.4k
37
votes
4
answers
5k
views
In what sense is the étale topology equivalent to the Euclidean topology?
I have heard it said more than once—on Wikipedia, for example—that the étale topology on the category of, say, smooth varieties over $\mathbb{C}$, is equivalent to the Euclidean topology. I have not ...
- 47.5k
3
votes
1
answer
360
views
Why are sheaves of a coverage the same as those on its generated Grothendieck coverage?
Preliminaries: There are lots of variations on the settings in which we define sheaves. I am concerned with the details linking the general definition below to the Grothendieck topology it generates; ...
- 352
3
votes
0
answers
307
views
Locality in Grothendieck Topologies
Let $\mathcal{C}$ be a category and $\mathcal{J}$ be a Grothendieck topology on it (i.e., $(\mathcal{C},\mathcal{J})$ is a site). Then what is a good notion of locality in it?
I came up with the ...
- 438
2
votes
3
answers
1k
views
Classical point-set topology using Grothendieck topologies
Its well known that the category of opens $O(X) $of a topological space $X$ can be endowed with a Grothendieck topology making it into a site. I am looking for references which take the reader through ...
- 66.8k
7
votes
0
answers
219
views
Pushout of Nisnevich sheaves
Let us consider the projective line $\mathbb{P}^1$ over a field $k$ and take the following open embeddings
$$j_{\epsilon}\colon \mathbb{P}^1\setminus\{0,\infty\} \to \mathbb{P}^1\setminus\{\epsilon\}$$...
- 303
21
votes
1
answer
846
views
Is there a category of topological spaces such that open surjections admit local sections?
The class of open surjections $Q \to X$ is a Grothendieck pretopology on the category $Top$ of spaces, and includes the class of maps $\amalg U_\alpha \to X$ where $\{U_\alpha\}$ is an open cover of $...
- 41.8k
10
votes
1
answer
506
views
What is the total space of a stack after all?
From my general experience I think for myself of what follows as some kind of taboo question for some reason: in my imagination, everybody wants an answer to this but somehow thinks it shall not be ...
- 66.8k
30
votes
1
answer
3k
views
Cohomology of sheaves in different Grothendieck topologies
Suppose I have a sheaf $\mathcal{F}$ on the (small) étale site over $X$. By restriction, $\mathcal{F}$ is also a sheaf on $X$ (with the Zariski topology). When is it that the sheaf cohomologies (i.e. ...
- 23
9
votes
0
answers
369
views
Topologies (and sheaves) on Cat and CAT
I've been wondering lately what sort of Grothendieck (pre)topologies there are on $Cat$ (the category of small categories) and $CAT$ (the v. large category of large categories - to forestall criticism ...
- 15
21
votes
1
answer
2k
views
Points in sites (etale, fppf, ... )
I asked a part of this in an earlier question, but that part of my question didn't receive precedence.
Etale site is useful - examples of using the small fppf site?
Let $X$ be a scheme (assume it ...
- 73
12
votes
0
answers
990
views
Stacks in the fpqc topology
This is related to Matt Satriano's earlier question about an analog of Artin's theorem for stacks with an fpqc cover by a scheme.
Suppose one developed the theory of stacks in the fpqc topology and ...
CommunityBot
- 1
9
votes
1
answer
2k
views
Torsors and the fpqc topology
Fix a scheme $S$, a group scheme $G/S$ (let us say smooth, maybe even affine with some finiteness conditions if you like), and suppose I have some other $S$-scheme $P$ with a right $G$-action. We want ...
- 43
11
votes
1
answer
892
views
Are all Grothendieck topologies on Set equivalent?
The category $\textbf{Set}$ can be given a Grothendieck topology where the covering families are jointly surjective families of set inclusions $\{X_i\stackrel{\phi_i}{\hookrightarrow} X\}\in\mathrm{...
- 55.9k
0
votes
1
answer
158
views
If $J$-coverings can be glued $I$-locally is $J$-locality an $I$-local property? (Reducing descent problems to simpler ones)
Let $(C,J)$ be a category with a grothendieck topology. For every object $X \in C$ there's (I hope) a little site which is the full subcategory of the slice category $C_{/X}$ whose objects are the ...
- 8,972
20
votes
3
answers
2k
views
Voevodsky's Triangulated Categories of Motives and their Relationships
As we know, Voevodsky constructed several candidates for the triangulated category of motives using different constructions and topologies (h, qfh, etale, and Nisnevich).
I would like to know what ...
- 251
2
votes
1
answer
125
views
Exercise on "locality" in topos theory
Let $\mathcal E=\mathsf{Sh}(\mathsf C,J)$. Let $A\rightarrowtail \Omega$ be a fixed subobject. For each $X$ in $\mathcal E$, define $T_A(X)$ to be a set of subobjects of $X$ as follows. $U\...
- 42.4k
4
votes
2
answers
626
views
Grothendieck topologies on $\mathbb{C}$
I would like to consider three sheaves $\mathcal{C}^0$, $\mathcal{H}$ and $\mathcal{S}$ on $\mathbb{C}$ (endowed with the euclidean topology): the first is the sheaf of continuous $\mathbb{C}$-valued ...
CommunityBot
- 1
7
votes
1
answer
464
views
Needless axiom for Grothendieck topologies?
Hi,
The first axiom for a Grothendieck (pre)topology on a category $C$ says that for every object $X\in C$, the family consisting of just the identity $1_X : X\to X$ should be a covering family.
Why ...
- 42.4k
7
votes
2
answers
886
views
Nisnevich topology on non-(locally) Noetherian schemes
Background
Lurie has in DAG XI a definition (given below) of a Nisnevich cover for arbitrary commutative rings, which reduces to the usual one for Noetherian rings. It boils down to being a etale ...
- 35.5k
13
votes
0
answers
481
views
Making the conceptual leap from locales to Grothendieck topologies?
I find the definition for locales and sheaves on locales to be straightforward, but I'm stumbling over the idea of a Grothendieck topology. Is there a nice way to see roughly how the latter ...
- 373
7
votes
1
answer
347
views
Which dense inclusions of sites are ∞-dense?
An inclusion of sites f: D→C is dense if it induces an equivalence between the categories of sheaves on C and D.
Likewise, f is ∞-dense is it induces an equivalence between the ∞-categories of ∞-...
CommunityBot
- 1
3
votes
1
answer
275
views
Sections of morphisms up to fppf covering
Let $f:X\to S$ be a finite type affine morphism of schemes where $S$ is an integral noetherian affine regular scheme whose function field is of characteristic zero.
Assume that all geometric fibers ...
- 5,019
51
votes
3
answers
7k
views
What is the purpose of the flat/fppf/fpqc topologies?
There have been other similar questions before (e.g. What is your picture of the flat topology?), but none of them seem to have been answered fully.
As someone who originally started in topology/...
CommunityBot
- 1
1
vote
0
answers
273
views
Isomorphism in a derived category of chain complexes with rational coefficients
Let $C$ be the category of quasi-projective schemes over a base field $k$ (maybe I will need some assumptions on $k$). Let $Ab(C_{\tau})$ be the category of Abelian sheaves on a site $C_{\tau}$, where ...
- 11
2
votes
0
answers
516
views
Reference for cdh topology
Let $f:X\rightarrow Y$ be a proper surjective morphism over some base scheme $S$ of finite type, suppose $f$ restricts to an isomorphism over some open $U$ of $X$, we also suppose both $X$ and $Y$ are ...
- 997
3
votes
2
answers
548
views
Morphism on schemes induced by continuous morphism of sites
I am beginner in the theory of Grothendieck topologies and I have the following question.
Let $X, Y$ be schemes over an algebraically closed field $k$. Denote by $X_{et}$ and $Y_{et}$ the Etale sites ...
- 15.5k
9
votes
3
answers
2k
views
Gerbes and Stacks
The definition of a gerbe on a smooth manifold that I know is that - after fixing an open cover $U_i$, a gerbe consists of the data of line bundles $L_{ij}$ on two-fold-intersections $U_{ij}$, ...
- 15.5k
12
votes
1
answer
2k
views
Reference request: Book of topology from "Topos" point of view
Question: Is there any book of topology in the modern language of topos theory?
Motivation:
In "Sheaves in Geometry and Logic" Mac Lane and Moerdijk say: "For Grothendieck, topology became the ...
- 545
10
votes
1
answer
743
views
Stable motivic cohomology with finite coefficients?
In this question, which attracted no responces so far, I've asked whether it is possible to extend the Beilinson-Lichtenbaum etale descent rule for motivic cohomology to singular varieties, in ...
CommunityBot
- 1
3
votes
0
answers
715
views
Two functorial definitions of schemes
I have been reading a bit about the "functor of points" theory for schemes. There seem to be two ways of going about defining schemes this way:
Equip the category $\textbf {Psh}=\operatorname{Fun}(\...
- 2,011
8
votes
1
answer
791
views
Hypercovers of sheaves in classical and quasi-categories
I am interested in relating the definition of hypercovers in the $\infty$-topos of sheaves on an $\infty$-Grothendieck site to the classical definition of hypercovers of presheaves on a Grothendieck ...
- 15.5k
3
votes
0
answers
189
views
Are there any useful Grothendieck topologies for which the H1 of $GL_n$ is not the set of rank $n$ vector bundles
Let n be a positive integer and X a scheme. Then for all the Grothendieck topologies I know (Zariski, etale, fppf) the set $H^1(X,GL_n)$ is the set of (isomorphism classes of) rank $n$ vector bundles.
...
- 31
2
votes
0
answers
115
views
Cohomology and quotients for the canonical topology
Recall that for any category $\mathcal C$, there is a unique finest topology, the canonical topology on $\mathcal C$ for which all representable functors are sheaves. I am interested in the example $\...
- 5,759
0
votes
1
answer
374
views
Sheaffication using a $\lambda$-transfinite colimit
I asked this question on mathstack (long time ago), however I received no answers, so I'm trying it here. I don't know whether it's suitable for this site, anyway.
I was reading this article http://...
- 12.3k
2
votes
0
answers
486
views
Fine and acyclic sheaves on locales
Hey all. Here's the thing, so I have a measure space. According to Johnstone's 'Topos theory' (page 213), let $(X,\Sigma,\mu)$ be a measure space, we can define a Grothendieck pretopology on it (and ...
CommunityBot
- 1
10
votes
1
answer
838
views
Is there a direct proof that affine schemes are fppf quasi-compact?
Let $A$ be a (commutative) ring. A family $(B_i)_{i\in I}$ of $A$-algebras is said to be an fppf cover if it satisfies three properties: (1) each $B_i$ is flat as an $A$-module, (2) each $B_i$ is ...
- 53.3k
31
votes
1
answer
3k
views
What is your picture of the flat topology?
I recently tried to explain the fppf site to a differential geometer. I started with the etale site, where I had two motivating claims:
If X is a smooth projective variety over the complexes, the ...
- 1,254
1
vote
1
answer
622
views
What was the original/historical motivation for introducing Grothendieck (pre-)topologies
The title essentially explains it, but I'll give some background:
I'm giving a talk to some fellow grad students about the relative Picard functor which requires introducing Grothendieck (pre-)...
- 1,323
4
votes
0
answers
261
views
Can one construct Freyd-Mitchell's embeddings that respect sheafifications?
For a certain presheaf $P$ with values in an abelian category $A$ satisfying AB5 and its sheafification $S$ (with respect to a small Grothendieck site) I would like to prove: $S(f):S(X)\to S(Y)$ is ...
CommunityBot
- 1