# Questions tagged [sheaf-theory]

For questions about sheaves on a topological space.

947
questions

8
votes

1
answer

368
views

### 6-functor formalism for topological stacks

I am trying to understand the 6-functor formalism of sheaves on topological stacks. As explained in this answer, there is a 6-functor formalism of sheaves for locally compact Hausdorff spaces, which ...

3
votes

0
answers

154
views

### Dévissage of stratified structures in Grothendieck's "Esquisse d’un programme"

I have a question about the intuition behind Grothendieck's proposed notion of so called "Tame topology" in his Esquisse d’un programme. Grothendieck insisted that theory should admit “...

4
votes

1
answer

270
views

### Bott & Tu differential forms Example 10.1

In Bott & Tu's "Differential forms", Example 10.1 states:
$\textbf{Example 10.1}$ Let $\pi: E \to M$ be a fiber bundle with fiber $F$. Define a presheaf on $M$ by $\mathcal F(U) = H^q(\...

0
votes

0
answers

102
views

### Local freeness of dualizing sheaf

I am reading the dualizing sheaf and duality theorems from Hartshorne’s algebraic geometry book. I am wondering about the following.
When does the dualizing sheaf of a projective scheme is an locally ...

2
votes

0
answers

136
views

### Equivalence of cohomology with compact support

Let $𝑋$ be a connected CW complex, $𝜌:\pi_1(𝑋)→\mathrm{Aut}(G)$ a representation, and $G_\rho$ the associated sheaf. It follows from here, that the following two cohomologies are isomorphic.
(1)The ...

3
votes

0
answers

166
views

### Category of sheaves of vector spaces on BG

Let $G$ be an affine group scheme over $\mathbb{C}$. I am interested in understanding the differences between different notions of sheaves on the stack $pt/G = BG$. For any algebraic stack $X$ one can ...

6
votes

1
answer

238
views

### Spectral sequence generalizing Čech cohomology

Let $X$ be a 'nice' topological space. Let $\left(U_i\right)_{i\in I}$ be a finite open covering of $X$. Let $\mathcal{F}$ be a sheaf of abelian groups.
For a subset $A\subset I$ denote $$U_A:=\cap_{...

3
votes

0
answers

84
views

### Cohomology of differentiable stacks: should the sheaf be fine?

I'm reading these Behrend's notes on cohomology of stacks, and I can't get over a detail in the fifth page.
Let $X_\bullet=(X_1\rightrightarrows X_0)$ be a Lie groupoid and let $\mathcal{N}$ be its ...

4
votes

0
answers

61
views

### Topos as a totally cocomplete object in a 2-category CART

In the preface to Sketches of an elephant, Peter Johnstone gives a list of characterizations of topos, some applicable to elementary (ii- finite limits and power objects) other to Grothendieck (i- ...

0
votes

0
answers

93
views

### Higher direct images of locally constant etale sheaf under smooth proper map locally constant

Let $f:X \to Y$ a surjective smooth proper map between Noetherian schemes and $F$ a locally constant sheaf on small etale site of $X$.
Question: Refering to Donu Arapura's answer here, how to see that ...

4
votes

1
answer

235
views

### Origin of the name Trace resp Integral symbol for the trace map of Dualizing Sheaf

Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) ...

0
votes

2
answers

244
views

### Vakil exercise on sheaf associated to the divisor of rational section

This is exercise 15.4.G. of Vakil's notes.
Let $\mathscr{L}$ be an invertible sheaf on an irreducible normal scheme $X$ with $s$ a rational section of $\mathscr{L}$. We want that $\mathscr{O}_X(\text{...

1
vote

1
answer

113
views

### System of local isos gives system of local epis

Suppose that $W$ is a system of local isomorphisms on a presheaf topos $\mathbf{Pre}(\mathcal{C})$. We say a map in $W$ is a $W$-local isomorphism, and we say that a map of presheaves $f: X \to Y$ is ...

19
votes

0
answers

301
views

### Are these comparison morphisms between Čech and Grothendieck cohomology the same?

For better or for worse, there is more than one approach to comparing Čech cohomology $\smash{\check{H}}^\bullet(\mathfrak{U},X;\mathscr{F})$ of a sheaf $\mathscr{F}$ on a space $X$ w.r.t the cover $\...

2
votes

0
answers

157
views

### Hodge bundles associated to a family of complex manifolds

I'm reading Voisin's books on Hodge theory. In the first volume she claimed but didn't prove this theorem:
Theorem 10.10 (Voisin) Let $\varphi:\chi\rightarrow B$ be a family of compact complex ...

6
votes

0
answers

83
views

### Is the derived category of sheaves localised at pointwise homotopy equivalences locally small?

In order to define the cup and cross products in sheaf cohomology, Iversen makes computations in an intermediate derived category. If $K(X;k)$ is the triangulated category of cochain complexes of ...

3
votes

1
answer

137
views

### (Derived category of) sheaves over an infinite union

The short version of my question is:
Suppose $X$ is a (reasonably nice) topological space such that $X = \bigcup_{n \ge 1} X_n$ for an increasing sequence of (closed) subspaces $X_1 \subset X_2 \...

3
votes

1
answer

218
views

### Etale cohomology of relative elliptic curve

Let $E_a: y^2 = x(x-1)(x-a)$ be a smooth proper relative elliptic curve over $\text{Spec}(A)$, with $a\in A$, and assume $\text{Spec}(A)$ is a $\text{Spec}(\mathbb{Q}_p)$-scheme.
Let $R^1f_*\mathbb{Q}...

7
votes

1
answer

186
views

### Variation on definition of logical functors avoiding power objects

Without power sets in meta-theory not every Grothendieck topos is an elementary topos, Set is still Grothendieck, but it lacks power objects.
Now I am looking for a definition of a logical functor ...

0
votes

0
answers

53
views

### Lifting of quadrics containing hyperplane section for projectively normal curves

Let $C \subset \mathbb{P}^r$ be a projective curve (over $k=\mathbb{C}$), smooth, irreducible and nondegenerate of degree $d$, ie the embedding line bundle $\mathcal{O}_C(1)=(\mathcal{O}_{\mathbb{P}^r}...

4
votes

1
answer

315
views

### Gluing objects of derived category of sheaves

Let $X$ be a locally compact topological space (may be assumed to be a stratified space with finite stratification).
Let $\{U_i\}$ be an open finite covering. Assume that over each $U_i$ we are given ...

1
vote

0
answers

216
views

### Vakil's Generalization of qcqs Lemma

(This was also simultaneously asked on math stack exchange: https://math.stackexchange.com/questions/4857715/vakils-generalization-of-qcqs-lemma)
In the most recent notes of Vakil, this is problem 15....

3
votes

0
answers

186
views

### When inverse image presheaf is already a sheaf

Following proof from Milne's Étale Cohomology (page 94) contains an equality I not understand.
Setting: assume $X$ is a variety (=absolutely reduced, irreducible scheme of finite type over base field ...

5
votes

1
answer

354
views

### Day convolution and sheafification

$\DeclareMathOperator\Psh{Psh}\DeclareMathOperator\Sh{Sh}\newcommand\copower{\mathrm{copower}}$I was looking through Bodil Biering's thesis On the Logic of Bunched Implications - and its relation to ...

2
votes

1
answer

166
views

### Reference for original Leray definition of a sheaf

Leray originally defined sheaves over closed sets. Is there any easily readable (i.e. obtainable through the Internet and written in English) reference that explicitly states the definition using ...

0
votes

0
answers

130
views

### A stalk criterion for unit map to be an isomorphism on étale site

Let $f: X \to Y$ be a morphism of schemes and $\mathcal{F}$ sheaf of sets/Abelian groups on the small étale site $Y_{ét}$. Assume we manage somehow to show thatat every geometric point $\overline{y} \...

3
votes

1
answer

194
views

### Čech cohomology refinement mapping

Let us consider the map $t_{AB}^*:H^1(A,F)\to H^1(B,F)$ between the cohomology groups, induced by the refinement map $t_{AB}:J\to I$, where $F$ is a sheaf of abelian groups on $X$, $A$ and $B$ are ...

3
votes

1
answer

439
views

### Characterization of étale locally constant sheaves over a normal scheme

I have a question about the verification of remark 1.2 in James Milne's book Étale Cohomology stated on page 156:
Assume $X$ be a normal & connected scheme with generic
point $g: \eta \to X$.
Then ...

0
votes

1
answer

112
views

### Do covector fields correspond to homomorphisms of $ \mathscr C^\infty $-modules from the sheaf of vector fields to the sheaf of smooth functions?

This question has been crossposted from MSE since there it received no attention. Please notify me if questions like these are not appropriate for this platform.
The question
Let $ M $ be a smooth ...

1
vote

0
answers

100
views

### Site structure on smooth fibered manifolds

Let $\mathsf{FB}_{s}$ be the category whose objects are smooth fibered manifolds, and whose morphisms are smooth strong projectable maps. Recall that given fibered manifolds $(\pi:Y\rightarrow X)$ and ...

5
votes

0
answers

159
views

### Why equaliser of product and terminal object is coproduct?

I’m reading “Sheaves in geometry and logic”, in page 80:
Please refer to [1]: https://i.sstatic.net/INrU0.jpg
It says “…,therefore $FU=\coprod_{x\in U} fx$. The space…”.
So could anyone please explain ...

1
vote

0
answers

160
views

### Line bundles on curve with nodal singularity

Let $C$ be be an irreducible reduced curve over alg closed field $k$ with only one single nodal singularity $x$ and $f:N \to C$ it's normalization with $f^{-1}(x)=\{x_1,x_2\}$ (as set), and an iso ...

2
votes

1
answer

85
views

### Are the injections of a coproduct a cover in the canonical pretopology?

Assume we're in a category $C$ with all pullbacks and finite coproducts.
Recall that the canonical coverage of $C$ is the finest Grothendieck (pre) topology for which all representables are sheaves. A ...

2
votes

1
answer

227
views

### Derived category of local systems of finite type on a $K(\pi,1)$ space: an explicit counterexample

Let $X$ be a nice enough topological space. I am mostly interested in smooth complex algebraic varieties. One may ask whether the bounded derived category of the category $\mathrm{Loc}(X)$ of local ...

2
votes

1
answer

369
views

### Hypersheaves vs derived category of sheaves

This question arose from Peter Scholze's notes on six functor formalisms, specifically lecture VII in the proof of proposition 7.1.
We fix a LCH space $X$ and consider the functor $D(\mathrm{Ab}(X)) \...

0
votes

0
answers

75
views

### Alternative definition of $\chi_{k}(x)$

Assume $q:B\rightarrow I$ is a local homeomorphism, and $A\subseteq B$ is open. Consider arbitrary $x\in B$, and $S$ is an open nbhd of $x$ such that $q\upharpoonright S$ is homeomorphism (locally).
...

2
votes

1
answer

682
views

### why is counit called the trace map

Let $f: X \to Y$ be a morphism of schemes, then
$f_*$ and $f^*$ form an adjoint pair inducing natural
correspondence
$\text{Hom}_{\mathcal{O}_X}(f^*\mathcal{G},\mathcal{F})=
\text{Hom}_{\mathcal{O}_Y}(...

6
votes

1
answer

308
views

### Relationship between canonical topology on a topos and its site of definition

The canonical (Grothendieck) topology for a category $C$ is the largest (finest) topology such that every representable presheaf over $C$ is a sheaf.
According to First Order Categorical Logic Lemma 1....

2
votes

1
answer

137
views

### Is the slice of a subcanonical site also subcanonical?

A subcanonical site is one for which every representable functor is a sheaf.
For a subcanonical site $C$, the fundamental theorem of topos theory says that there is an equivalence $Sh(C/c)\cong Sh(C)/...

2
votes

0
answers

59
views

### Coequalizers and pullbacks in $\infty$-topoi

In an $\infty$-topos, suppose we have two cartesian diagrams of the form
$$
\require{AMScd}
\begin{CD}
\overline{A} @>>> \overline{B} \\
@VVV @VVV \\
A @>>> B .
\end{CD}
$$
Let
$$
\...

8
votes

0
answers

403
views

### Sheaf of compact Hausdorff spaces but not a condensed anima

Consider the site $\mathbf{CHaus}$ of compact Hausdorff spaces together with the finitely jointly surjective families of maps as coverings. Restriction induces an equivalence of categories $$ \mathbf{...

2
votes

1
answer

228
views

### Compatibility of Beck Chevalley condition: sheaves

Given a (not necessarily Cartesian) square of spaces
$$\require{AMScd}\begin{CD}
X @>g>> \overline{X} \\
@VVfV @VV\overline{f}V \\
Y @>\overline{g}>> \overline{Y}
\end{CD}$$
does the ...

1
vote

0
answers

69
views

### Idempotent completeness

We say a category $\mathcal{N}$ is exact if it is additive and is endowed with an exact structure. In brief, it is an additive category with a predetermined class of short exact sequences in its ...

9
votes

1
answer

328
views

### G-topological spaces and locales

Consider the following generalization of topological spaces:
Definition: Let $X$ be a set. A G-topology on $X$ is given by certain distinguished subsets $U \subset X$, called admissible open subsets, ...

3
votes

1
answer

387
views

### Finding global sections of a sheaf of sets using (some kind of) sheaf cohomology?

Let $X$ be a compact manifold, say, and $G$ a Lie group, and $H$ a closed Lie subgroup such that $M \cong G/H$ is a homogeneous space. (For my purposes, $X$ and $M$ would be a smooth projective ...

1
vote

1
answer

160
views

### Field of fractions of etale stalk of Dedekind domain (Example from Milne's LEC)

Let $X=\operatorname{Spec}(A)$ be an affine Dedekind domain with field of fractions $K$. Let $\widetilde{A}$ be the integral closure of $A$ in separable closure $ K^{\text{sep}}$. A closed point $x$ ...

2
votes

1
answer

242
views

### Calculate stalk of etale derived pushforward sheaf (Milne's LEC)

Assume $X=\operatorname{Spec}(A)$ is connected and normal (especially integral), and let $g:\eta \hookrightarrow X$ be the inclusion of the generic point of $X$. In Milne's LEC script on Etale ...

2
votes

0
answers

164
views

### The conormal sheaf is the sheaf of sections of the conormal bundle for smooth manifolds

$\def\sO{\mathcal{O}}
\def\d{\mathrm{d}}$In ringed spaces theory, there is a notion of “conormal sheaf of an immersion” (mainly used in scheme theory), whereas in smooth manifold theory, there is the ...

1
vote

2
answers

315
views

### If $\mathcal{F}$ globally generated, then counit map $f^*f_* \mathcal{F} \to \mathcal{F} $ surjective

Let $f: X \to S$ be a morphism, and $\mathcal{F}$
be quasi-coherent $\mathcal{O}_X$-module generated by global sections (eg if
$X$ projective, then this holds for the twisted sheaf $\mathcal{F}(n)$ ...

1
vote

1
answer

172
views

### Second fundamental exact sequence of sheaves of differentials. Sufficient condition to get a splitting s.e.s

$\def\spec{\operatorname{Spec}}$I am trying to understand the proof of Lemma 0474 of the Stacks Project. I'll give some context to its statement before discussing its proof: In commutative algebra, if ...