Questions tagged [sheaf-theory]

For questions about sheaves on a topological space.

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3 votes
1 answer
110 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
186 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}...
6 votes
1 answer
164 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
48 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
301 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 ...
3 votes
1 answer
178 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 ...
2 votes
1 answer
349 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)) \...
1 vote
0 answers
185 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
175 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
333 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
148 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 ...
3 votes
1 answer
410 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
0 answers
125 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} \...
0 votes
1 answer
108 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
98 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.stack.imgur.com/INrU0.jpg It says “…,therefore $FU=\coprod_{x\in U} fx$. The space…”. So could anyone please ...
1 vote
0 answers
154 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 ...
97 votes
10 answers
13k views

equivalence of Grothendieck-style versus Cech-style sheaf cohomology

Given a topological space $X$, we can define the sheaf cohomology of $X$ in I. the Grothendieck style (as the right derived functor of the global sections functor $\Gamma(X,-)$) or II. the Čech ...
2 votes
1 answer
81 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
226 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 ...
10 votes
0 answers
494 views

Isbell duality between algebras and sheaves

nLab says on Isbell duality, the following: A general abstract adjunction $(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves} \leftrightarrows \mathrm{Presheaves}$ relates (higher) ...
6 votes
1 answer
664 views

Definition of the cotangent complexes of Artin stacks

I am studying the notion of the cotangent complexes of Artin stacks reading LMB's book and Olsson's paper. According to them, the cotangent complexes are defined as projective systems in their derived ...
0 votes
0 answers
74 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
658 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
283 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....
4 votes
2 answers
310 views

Sheafification of presheaf of trivial vector bundles is the stack of vector bundles

This is a deliberately vague question, possibly obvious to experts. Let $F$ be a field. Over the (say, fpqc) site of $F$-schemes, we may define a presheaf $T^{\textrm{pre}}$ that takes a scheme $S$ ...
2 votes
1 answer
129 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)/...
8 votes
1 answer
315 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, ...
2 votes
0 answers
58 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 $$ \...
2 votes
0 answers
139 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 ...
8 votes
0 answers
387 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{...
63 votes
4 answers
11k views

Is there a good way to think of vanishing cycles and nearby cycles?

Once in a while I run into literature that invokes vanishing cycle machinery with a cryptic sentence like, "this follows from a standard vanishing cycle argument." Is there a good way to look at ...
15 votes
2 answers
1k views

Which topological spaces have the property that their sheaves of continuous functions are determined by their global sections?

I hope I'm using the terminology correctly. What I mean is this: fix $K = \mathbb{R}$ or $\mathbb{C}$ (I'm interested in both cases). Which topological spaces $X$ have the property that for every ...
1 vote
1 answer
217 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 ...
4 votes
3 answers
430 views

Locally ringed space with noetherian stalks and a non-coherent structural sheaf

I am looking for a locally ringed space the stalks of which are noetherian and such that the structural sheaf is not coherent over itself. Can you provide me an example of this? Notice that one may ...
3 votes
1 answer
382 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 ...
8 votes
2 answers
2k views

Can we relate Cech cohomology and derived functor cohomology even when the cover we choose isn't nice?

In my algebraic geometry class this semester, we've learned about Leray's Theorem, which states that for a sheaf $\mathcal{F}$ on a topological space $X$, and $\mathcal{U}$ a countable cover of $X$, ...
8 votes
1 answer
1k views

Fpqc sheafification and localisation

I am slightly confused about sheafification at the moment. I first learned sheaves defined as a subcategory of presheaves, then I was told that sheaves are also a localisation of presheaves, then I ...
1 vote
1 answer
154 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
228 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 ...
1 vote
2 answers
282 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
149 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 ...
5 votes
2 answers
821 views

Can a nontrivial presheaf have a trivial sheafification?

I'm studying Vakil's Foundations of Algebraic Geometry, and working through the exercises in 2.4 about compatible germs as a method for constructing a sheafification of a given presheaf. Can this ...
7 votes
1 answer
413 views

Equivariant perverse sheaves and orbit stratification

Let $X$ be a complex algebraic variety with an action of a connected algebraic group $G$. The forgetful functor from the category of $G$-equivariant perverse sheaves on $X$ to the category of perverse ...
3 votes
1 answer
148 views

Sheaves and gratings

A grating is a notion in algebraic topology from the 1940 introduced by Alexander. Cartan extended it as follows. A grating (carapace in french) is defined by a topological space $X$, a module (or a ...
4 votes
0 answers
185 views

Is there a simple counterexample to étale proper base change on the unbounded derived category?

The best non-derived version of proper base change on the étale site of a scheme I know is that for $f : X \to Y$ proper and $g : Y' \to Y$ arbitrary, the base change morphism $g^{-1} R f_\star \...
4 votes
0 answers
135 views

Do presheaf toposes satisfy the full fan theorem?

Presheaf toposes satisfy LPO and (edit: if over categories with binary products) PAx and countable choice internally, so they automatically satisfy the stable fan theorem (every bar which is the ...
3 votes
0 answers
188 views

How to read the definition of Grothendieck Pretopology in SGA4?

In SGA4, the first axiom of a Grothendieck pretopology is given as: PT0: Pour tout objet $X$ de $C$, les morphismes des familles de morphismes de $Cov(𝑋)$ sont quarrables. (Rappelons qu’un morphisme ...
6 votes
2 answers
1k views

Non-zero sheaf cohomology

Let $\mathbb{R}$ denote the real line with its usual topology. Does there exist a sheaf $F$ of abelian groups on $\mathbb{R}$ whose second cohomology group $H^{2}\left(\mathbb{R},F\right)$ is non-zero?...

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