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Questions tagged [grothendieck-topology]

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2
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2answers
181 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 ...
4
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0answers
148 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....
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0answers
221 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 ...
6
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0answers
136 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 ...
4
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1answer
149 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 ...
2
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1answer
168 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; ...
3
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0answers
174 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 ...
2
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3answers
574 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 ...
7
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0answers
119 views

Pushout of Nisnevich sheaves

Let us consider the projective line $\mathbb{P}^1$ over a field $k$ and take the following open embeddings $$j_{\epsilon}\colon \mathbb{P}^1\setminus\{0,\infty\} \to \mathbb{P}^1\setminus\{\epsilon\}$$...
6
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1answer
285 views

Equivalence of the definitions of a sheaf in SGA4 and in “Categories and Sheaves”

I asked this question on Mathematics Stack Exchange, but got no answer. I don't understand why the definition of a sheaf (Definition 17.3.1 (ii)) given in the book [KS] Categories and Sheaves by ...
10
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1answer
392 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 ...
0
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1answer
139 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 ...
7
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1answer
620 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{...
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0answers
189 views

Reference for the Brown-Gersten property for smooth manifolds

A classical result by Brown and Gersten says that to verify the homotopy descent property for the Zariski topology it suffices to verify it for Zariski squares and the empty cover of the empty scheme. ...
7
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1answer
339 views

Is there a way to “puncture” a topos?

Let $E$ be a (Grothendieck) topos, e.g. $E = \text{Sh}(X)$ for a topological space $X$. And let $p = (p^*, p_*):\text{Set}\to E$ be a point of $E$, is there a way to "puncture" $E$ in some sense? By "...
2
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1answer
102 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\...
16
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3answers
1k 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 ...
19
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0answers
456 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 ...
3
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2answers
451 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 ...
22
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1answer
1k views

Flat versus etale cohomology

Although the definition of etale ($\ell$-adic) cohomology is scary, I have at least some intuition for how it should behave: for instance, when it makes sense, I expect that it should be ``similar'' ...
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0answers
304 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 ...
3
votes
1answer
200 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 ...
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0answers
221 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 ...
41
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3answers
4k 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/...
2
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0answers
299 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 ...
3
votes
2answers
374 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 ...
7
votes
3answers
1k 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 $U_{ij}$, ...
12
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1answer
1k 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 ...
2
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0answers
275 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}(\...
5
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1answer
352 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 ...
3
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0answers
176 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. ...
2
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0answers
89 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 $\...
0
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1answer
256 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://...
6
votes
1answer
966 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 ...
8
votes
1answer
234 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 ∞-...
0
votes
1answer
463 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-)...
4
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0answers
214 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 ...
6
votes
2answers
705 views

Can Inequivalent Topologies Have Same Sheaves/Cohomology?

Let $C$ be a fixed category, and let $T_1$ and $T_2$ be two Grothendieck (pre)topologies on $C$. We say $T_1$ is subordinate to $T_2$ if every covering in $T_1$ has a refinement in $T_2$. We say $T_1$ ...
6
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1answer
394 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 ...
9
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1answer
652 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 ...
4
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0answers
238 views

cech cohomology in topos

Hi, The following result seems to be well known, but I can't come up with a proof. Suppose that $C$ is a topos and that $F\to G$ is an effective epimorphism in $C$. If $P$ is any abelian sheaf on $C$...
7
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4answers
884 views

Grothendieck topology for a non-small category

To define a Grothendieck topology of a category, we usually require that the category is small. Question 1: Why do we need to require the category to be small? I thought that the problem was that ...
3
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0answers
214 views

Examples of Sheafification via Hypercovers

For a presheaf $F$ on a category equipped with a pretopology, one has the sheafification $F^{\sharp}$ of $F$. I know well the plus-construction of sheafification, which is presented in Artin's paper "...
16
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1answer
1k 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 ...
4
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1answer
378 views

Numerable covers from the point of view of Grothendieck topologies

Let $G$ be a topological group. Recall that its classifying space $BG$ is a CW-complex which is the base of a locally trivial principal bundle of group $G$, with contractible total space $EG$. It ...
2
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1answer
161 views

Is there a name for this “weak compatibility” between Grothendieck (pre)topologies?

(I find it easier to think in terms of Grothendieck pretopologies, instead of topologies. If this annoys any experts, please forgive me.) Suppose that $C$ is a (full) subcategory of a category $D$. ...
8
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3answers
963 views

Representable Presheaf

I have a very quick question. Is there an easy example of a representable presheaf on a site that is not a sheaf? This certainly can't happen on a small FPPF site so I would expect a counterexample to ...
9
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4answers
2k views

Grothendieck Topologies versus Pretopologies

The wikipedia article(s) as well as the nlab article(s) about Grothendieck topologies and Grothendieck pretopologies are careful to differentiate the two very emphatically and to point out that ...
5
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1answer
345 views

Coverage, itself considered as a presheaf

A coverage $J$ on a category $C$ assigns to an object $U$ of $C$ a set of covering families $J(U)$. The covering families are required to be stable under pullback, which amounts to requiring that for ...
30
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1answer
2k 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 ...