All Questions
Tagged with sheaves or sheaf-theory
979 questions
3
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Which limits are preserved by prolongation of presheaves?
Let $F:C \to D$ be a full and faithful functor between small categories. Then we get a triple of adjoint functors $F_! \dashv F^* \dashv F_*$, with $$F_!:Set^{C^{op}} \to Set^{D^{op}}.$$
Notice that $...
- 15.5k
3
votes
1
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735
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About direct image of ideal sheaves
Let $\mu:X'\rightarrow X$ be a birational morphism of normal complex projective varieties.
Consider the two ideal sheaves $I_1= \mu_*\mathcal{O}_{X'}(-\sum d(E)E)$, $I_2=\mu_*\mathcal{O}_{X'}(-\sum(...
- 1,150
3
votes
1
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844
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A form of cohomology and base change
Let $f \colon X \to Y$ be a proper morphism of (Noetherian) schemes, $\mathcal{F} \in \mathop{Coh}(X)$. Let $i_Z \colon Z \hookrightarrow Y$ be a closed subscheme and take the inverse image $W := X \...
- 14.7k
3
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2
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517
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Equivalence of two definitions of sheaves on a site
I'd like to prove that two definitions of sheaves on a site are equivalent, but I'm having trouble proving one direction.
Let $C$ be a category with pullbacks. Let $(C,T)$ be a site defined through a ...
- 155
3
votes
2
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2k
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Relation between sheaf and group cohomology
Let $E=\mathbb{C}/L$ be an elliptic curve. Then $\mathbb{C}$ is contractible, and $L$ is the fundamental group of $E$. What's interesting is that we can find the cohomology of $E$, which is the same ...
- 15.4k
3
votes
1
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240
views
Cohomology of the complement of a subvariety
Let $X$ be a complex manifold, $Y\subset X$ a subvariety, and $U:=X\setminus Y$ of codimension $d$. It is well known that the restriction map
$$
H^i(X,\mathbb Q)\to H^i(U,\mathbb Q)
$$
is an ...
- 178
3
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1
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149
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(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 \...
- 197
3
votes
1
answer
212
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Algebraic spaces in the étale topology (proof from Stacks project)
I have a question about the proof of Lemma 78.12.1 from Stacks Project. The aim of the last paragraph of the proof is to verify that the map of sheaves in the étale topology $F \to U/R$ is an ...
- 6,038
3
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1
answer
479
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K-injective (also known as hoinjective) complexes of sheaves of modules
Let $(X,\mathcal O_X)$ be a ringed space (if necessary, assume that it is a scheme with suitable hypotheses). Given two complexes of sheaves $\mathcal F$ and $\mathcal G$ of $\mathcal O_X$-modules, ...
- 2,079
3
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1
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342
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On the notion of conelike stratified (cs-) space
The notion of cs-stratification of a topological space is apparently due to Siebenmann, see also the paper by N. Habegger and L. Saper in the paper "Intersection cohomology of cs-spaces and Zeeman's ...
- 21.8k
3
votes
1
answer
159
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Homology in the sections of an infinite exact sequence of injective sheaves of $\mathcal O_X$-modules?
Let $(X, \mathcal O_X)$ be a scheme and the following an infinite, exact sequence of injective sheaves of $\mathcal O_X$-modules:
$$
\cdots \overset{f_5}\longrightarrow I_5\overset{f_4}\longrightarrow ...
- 355
3
votes
1
answer
378
views
Given an exact category, viewed as a site, do there exist non-additive sheaves?
Suppose given an exact category $\mathcal{C}$. The following question arises while proving the Gabriel-Quillen-Laumon embedding theorem following Laumon [1].
Laumon constructs an abelian category $\...
- 33
3
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1
answer
818
views
Serre duality for sheaves of logarithmic differentials
This question is motivated by a comment of Donu Arapura here dimension of compact support cohomology
Let $D$ be a divisor with normal crossings on some smooth projective (complex algebraic) variety $...
- 31
3
votes
1
answer
214
views
For which sites are all constant presheaves separated?
I'm intererested in open surjective geometric morphisms induced by fibrations of sites $S\to T$ a la Moerdijk, but as a warm-up, let's consider the case $S \to \ast$. In the case that $S$ is a poset $...
- 35.5k
3
votes
1
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480
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Sequences of groups, exact not just in étale but also in the Zariski topology
Let $A=k[[x,y]]$, with say $k=\mathbb{C}$ and $X = \operatorname{Spec}(A)$. Let $B$ denote a free $A$-algebra of rank $e^2$; actually, we have $B=A[u,v]$ with $u^e=x$, $v^e=y$ and $uv = \xi_e vu$, ...
- 1,391
3
votes
1
answer
587
views
Functoriality of base change
Let $a:W\rightarrow X$, $c:X\rightarrow Z$, $b:W\rightarrow Y$ and $d:Y\rightarrow Z$ be a pull-back diagram in the category of topological spaces. Then one can construct a natural isomorphism $\kappa$...
- 1,457
3
votes
1
answer
550
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 ...
- 6,038
3
votes
1
answer
258
views
Automorphisms of Frobenius liftings and degeneration of the Hodge-de Rham spectral sequence
I am still studying Deligne and Illusie's paper (https://eudml.org/doc/143480), and I am again stuck, this time on pages 262/263.
Assume $X\longrightarrow S$ is a smooth morphism of $\mathbb{F}_{p}$-...
- 1,805
3
votes
1
answer
129
views
Infinite uniform dimension $\Rightarrow$ infinitely many idempotents in a localization of a quotient
Let $R$ be a commutative ring with $1$ such that its uniform dimension is infinity, equivalently, $$\sup\{k \mid R \text{ contains a direct sum of $k$ nonzero ideals }\}=\infty.$$ How can we ...
- 31
3
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1
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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 ...
- 18.7k
3
votes
1
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89
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The sheaf propagation is open in the zero section
Let $X$ a smooth manifold and $F$ a sheaf (let's say of abelian groups) on $X$. We will say that $F$ propagates at $x\in X$ in the (co)-direction $p \in T_x^*X$ if for all $C^1$-function $\phi$ ...
- 1,017
3
votes
1
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203
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Sheaves with $\mathbb{R}$-constructible proper direct image closed under dualizing?
I learned about $\mathbb{R}$-constructible sheaves very recently, so I hope this isn't a silly question:
Let $M$ be a real analytic manifold, and $U\subset M$ an open subanalytic subspace. Denote the ...
- 163
3
votes
1
answer
245
views
Families of local rings coming from a locally ringed space
Let $X$ be a topological space. For every point $x \in X$ let $R_x$ be a local ring. Under what (necessary / sufficient / necessary and sufficient) conditions is there a sheaf ${\cal O}_X$ such that $(...
- 63.1k
3
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1
answer
463
views
For an l-adic sheaf (F_n), why is the complex F_n of finite Tor dimension?
Let $X$ be a variety and let $\mathcal{F}=(\mathcal{F}_n)_{n\geq 0}$ be a (constructible) $\ell$-adic sheaf on $X$. Let $K_n$ be the object in the derived category $D(X,\mathbf{Z}/\ell^{n+1})$ of ...
- 9,497
3
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1
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226
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
228
views
The sheaf of generalized functions on compact subsets
For $K\subseteq \mathbb{R}^d$ compact, let $C_{\mathrm{c}}^{\infty}(K)$ denote the space of smooth functions on (an open neighborhood of) $K$ with compact support contained in $K$ with the usual ...
- 1,270
3
votes
1
answer
221
views
Sheaf on a filtered topological space?
Is there any nice way of defining a sheaf of abelian groups on a filtered topological space?
Let $X$ equipped with filtration $X_0\subset X_1\subset X_2\subset ... \subset X_k=X$ be an object in the ...
- 913
3
votes
1
answer
1k
views
Chern classes of the direct image of an ideal sheaf resp. skyscraper sheaf
Given a double cover $\pi: X\rightarrow \mathbb{P}^2$ of the projective plane by choosing a square root $S$ of $O_{\mathbb{P}^2}(Q)$, where $Q$ is a quartic in the plane.
Choose a closed point $p\...
- 1,391
3
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0
answers
199
views
When a fully faithful functor from an abelian category to itself will be an equivalence
Let $A$ be an abelian category. Suppose $i:A\to A$ is a fully faithful functor from $A$ to itself. I wonder when the functor will be an equivalence.
If $A$ is a "nice" category, I think $i$ ...
- 253
3
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0
answers
45
views
Sheaf with no singular support in $\mathbb{R}$-direction decomposes as an external tensor product
Consider $X$ to be a smooth manifold. Denote by $p_1, p_2$ respectively the projections $X\times Y$ to $X$ and $Y$ respectively. Recall the definition of microsupport/singular support of a sheaf (...
- 71
3
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0
answers
205
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 ...
- 53
3
votes
0
answers
99
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 ...
- 105
3
votes
0
answers
215
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 ...
- 237
3
votes
0
answers
177
views
Sheaf theory in TDA
I was wondering wether anyone had any examples as to why it more useful to consider a sheaf theory approach to TDA problems.
I am familiar with some of the benefits of using cellular cosheaves to ...
- 343
3
votes
0
answers
280
views
Cellular (co)Sheaves and applications
In the last few years there have been efforts made to generalise the theory of peristence homology and cohomology to deal with sheaves for example Russold - Persistent sheaf cohomology. This as i ...
- 343
3
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0
answers
186
views
The site and the space
There is a (seemingly simple) statement in the literature on sheaf theory, namely,
If $E$ is the site of opens of a topological space $X$, the notion of sheaf over $X$ coincides with that of sheaf of ...
- 894
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0
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154
views
When the sheaf of principal parts is reflexive?
Following EGA IV, we can construct the sheaf of principal parts of a sheaf $\mathcal{E}$ over a scheme $X$ by $\mathcal{P}_X^n(\mathcal{E}) = \mathcal{P}_X^n \otimes \mathcal{E}$, where $\otimes$ is ...
3
votes
0
answers
172
views
Is the Grothendieck topology equivalent to its Singleton Grothendieck topology?
I'm using the definition of a Grothendieck topology in Angelo Vistoli’s 2007 Notes on Grothendieck topologies,
fibered categories
and descent theory and I found on nLab about superextensive site, that ...
- 119
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0
answers
142
views
Johnstone's Elephant - Lemma C2.1.7 confusion
I don't understand the proof of (ii) in the Johnstone's Elephant:
Lemma 2.1.6 is:
Now consider $\bigcup_{f \in R} f \circ f^*(S)$. This is my notation for the sieve Johnstone references in the proof ...
3
votes
1
answer
244
views
Compatibility of pullbacks with an equivalence relation
This question was originally posted last week in Math Stack Exchange (see here).
I'm currently working on the proof of the existence of the sheafification in Angelo Vistoli’s 2007 Notes on ...
- 119
3
votes
0
answers
530
views
Flasque sheaves on a site
This is a cross-post from MathStackexchange.
We define a flasque sheaf on a site as one whose first Čech cohomology vanishes for every covering of every object of the site. I know this definition is ...
- 153
3
votes
0
answers
194
views
Hypercovers with sieves
Consider a covering family $\{Y_i \to X\}$ and the induced sieve $R \subseteq X$, the subpresheaf of all maps to $X$ that factor through some $Y_i$. The family gives me an induced Cech nerve $C_\...
- 1,094
3
votes
0
answers
83
views
Do rationally contractible presheaves have rationally contractible injective resolution
Given a presheaf $\mathcal{F}: Sm/k\rightarrow Ab$ we define a new presheaf $C\mathcal{F}= \varinjlim\limits_{X\times \{0,1\}\subset U \subset X\times \mathbb{A}^1}\mathcal{F}(U)$. The presheaf $\...
- 5,901
3
votes
0
answers
1k
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Saturation of sheaves
Let $(X, \mathcal{O}_X)$ be a complex manifold, which we can take to be projective. A coherent subsheaf $\mathscr{F}$ of some sheaf $\mathscr{G}$ is said to be saturated in $\mathscr{G}$ if the ...
- 651
3
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0
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460
views
Does isomorphism on local rings imply the global isomorphism for the sheaf of spectra?
Let's assume we have a sheaf of spectra on some scheme. As an example I will assume that we are working with the $K$-theory sheaf. There are certain local to global spectral sequences, like descent ...
- 5,901
3
votes
0
answers
83
views
Embedding abelian categories into abelian sheaves
The Yoneda functor from an abelian category into sheaves of abelian groups is shown to be exact in The Stacks Project, Lemma 19.9.2. I like this proof because it is constructive and it doesn't use ...
3
votes
0
answers
70
views
Characterization of degeneracy of spectral sequence of a fiber bundle at the second term
Let $f\colon E\to B$ be a fiber bundle of compact manifolds with fiber $F$. Assume that the push-forward $Rf_*(\underline{\mathbb{F}})$ in the derived category of the constant sheaf with coefficients ...
- 21.8k
3
votes
0
answers
202
views
What's wrong with higher dimensional nearby cycles?
Suppose we have a complex algebraic variety $X$ with a map $f: X \to \mathbb{C}$ with $Y=f^{-1}(0)$. Let $\overset{\sim}{\mathbb{C}}$ be the universal cover of $\mathbb{C}-\{0\}$ and consider the ...
- 3,019
3
votes
0
answers
125
views
Topological Brauer group and sheaf of $C^{\infty}$-functions
Given a smooth manifold $X$; by definition the topological Brauer group $B(X)$ is the group of classes of Azumaya algebras over the sheaf $\mathcal{O}_X$ of $\mathbb{C}$-valued functions on $X$.
If ...
- 181
3
votes
0
answers
81
views
Image of Obstruction Map for Relative Quot-scheme
Let $f: X \to S$ be a projective morphisms between finite-type projective schemes, and $\mathcal{O}_{X}(1)$ an $f$-ample line bundle. Given an $S$-flat coherent $\mathcal{O}_{X}$-module $\mathcal{H}$ ...
- 1,701