The tag has no wiki summary.

learn more… | top users | synonyms

9
votes
1answer
318 views

Homotopical descent information contained in the Dwyer-Kan function complexes of a presheaf category?

Recall that the category of sheaves on some site $C$ equipped with a grothendieck topology $\tau$ is equivalent to the localization of the category of presheaves $W^{-1}Psh(C)$ at $W$ where $W$ is the ...
6
votes
1answer
1k views

How to “globalize” the inverse function theorem?

Let $F: V \times W\rightarrow Z$, where $V,W,Z$ are finite-dimensional smooth (or analytic) manifolds and $F$ is smooth (or analytic). Assume that $\dim W=\dim Z$ and the usual inverse function ...
3
votes
2answers
491 views

finite-dimensionality of cohomology groups on compact riemann surfaces

does the finite dimensionlity of the first cohomology group ($ H^1 $) of the sheaf of meromorphic sections of a holomorphic line bundle on a compact riemann surface follow easily from the finite ...
6
votes
1answer
244 views

Automorphisms of constant sheaves

Let E be a Grothendieck topos, such as the category of sheaves of sets on a topological space. Then there is a unique geometric morphism $(\Delta \dashv \Gamma)\colon E\to \mathrm{Set}$, where ...
3
votes
1answer
507 views

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 ...
7
votes
3answers
3k views

Serre's FAC versus Hartshorne as an introduction to sheaves in algebraic geometry

I just found an English translation of Serre's FAC at Richard Borcherds' Algebraic Geometry course web page. I really want to read it sometime. I am beginner in Algebraic Geometry, just started ...
1
vote
0answers
347 views

Another stupid question on l-adic sheaves: does a generically zero constructible sheaf vanish on an open subvariety?

Suppose that the stalk of a constructible l-adic ($\mathbb{Z}_l$-adic or $\mathbb{Q}_l$-adic) etale sheaf $S$ over a generic (Zariski) point of a variety $V$ is zero. Does this imply that $S$ vanishes ...
14
votes
6answers
3k views

How should one think about sheafification and the difference between a sheaf and a presheaf

The first time I got in touch with the abstract notion of a sheaf on a topological space $X$, I thought of it as something which assigns to an open set $U$ of $X$ something like the ring of continuous ...
3
votes
2answers
314 views

Colimits of covers

Suppose I have category $C$ equipped with a Grothendiek pretopology of covers, and let $y:C \to Sh(C)$ be the Yoneda embedding into sheaves and $y/c:C/c \to Sh(C)/y(c)\cong Sh(C/c)$. How can I show ...
11
votes
2answers
2k views

Wikipedia's definition of 'locally free sheaf'

Let $R$ be a, say, noetherian ring and $M$ an $R$-module. The Wikipedia article on 'locally free sheaf' tells me that the following two statements are equivalent: The module $M$ is locally free ...
4
votes
1answer
580 views

When are non-quasi-coherent sheaves used?

Non-quasi-coherent sheaves of $\mathcal O_X$ modules on a scheme seem like a wild concept to me; are they actually used for something?
3
votes
2answers
324 views

Projectivity of free O_X modules with respect to the sheafy hom?

I've heard that given a ringed topos $(X,\mathcal{O}_X)$, the functor $Hom_{\mathcal{O}_X-\operatorname{Mod}}(\mathcal{O}_X, -)$ often fails to be exact. Is this only the case for the unenriched hom ...
2
votes
2answers
290 views

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 ...
4
votes
5answers
2k views

sheaves and cosheaves

I am struggling hard to understand the pushforwards and pullbacks of cosheaves. Are they also cosheaves? And what are quasicoherent cosheaves? Is there anything like coquasicoherent cosheaves? Please ...
8
votes
4answers
2k views

Derived categories of coherent sheaves: suggested references?

I am interested in learning about the derived categories of coherent sheaves, the work of Bondal/Orlov and T. Bridgeland. Can someone suggest a reference for this, very introductory one with least ...
2
votes
2answers
259 views

Is a submodule of the sheaf of sections of a smooth vector bundle necessarily finitely generated?

Let $X$ be a finite-dimensional smooth manifold, $\mathcal C^\infty(X)$ its algebra of smooth functions, $V\to X$ a finite-dimensional smooth vector bundle, and $\Gamma(V)$ the space of smooth ...
1
vote
1answer
775 views

Quasi coherent sheaf

One really stupid, trivial question: A Quasi coherent sheaf $F$ on an affine group scheme(Spec R) is simply an R-module. What happens in case R is a Hopf algebra? Will the Q.coherent sheaf $F$ be an ...
7
votes
0answers
324 views

Applications of sheaf theory to the computation of invariants of LS-category type

I would like to know if sheaf theory can be applied to a particular class of questions in topology. The Schwarz genus (also known as sectional category) of a continuous map $p\colon\thinspace E\to ...
22
votes
6answers
3k views

What is the inverse image sheaf necessary for in algebraic geometry?

Given a continuous map $f \colon X \to Y$ of topological spaces, and a sheaf $\mathcal{F}$ on $Y$, the inverse image sheaf $f^{-1}\mathcal{F}$ on $X$ is the sheafification of the presheaf $$U \mapsto ...
1
vote
1answer
255 views

Realizing a restriction as direct/inverse image of sheaves

Consider the inclusion $j$ of ${\mathbb{R}}$ as the real axis of ${\mathbb{C}}$. On ${\mathbb{C}}$ I have a real polynomial algebra ${\mathbb{R}}[x,\bar{x}]$, where $\bar{x}$ denotes conjugation. ...
43
votes
5answers
6k views

What is sheaf cohomology intuitively?

What is sheaf cohomology intuitively? For local systems it is ordinary cohomology with twisted coefficients. But what if the sheaf in question is far from being constant? Can one still understand ...
4
votes
0answers
333 views

Monomorphisms of sheaves

The following is surely pretty standard, but I have been unable to prove it or find a proof in the literature (many sources assert it without proof). Let $\phi : \mathcal{F} \rightarrow \mathcal{G}$ ...
1
vote
1answer
770 views

Ringed and locally ringed spaces

A pair $(X,O_X)$ is a ringed space if $X$ is a topological space and $O_X$ is a sheaf of rings. If every stalk $O_{X,x}$ is a local ring, then we say that $(X,O_X)$ is a locally ringed space. In the ...
0
votes
0answers
117 views

Are sieves in locally small categories still sets?

In "Sheaves in Geometry and Logic", M&M define a sieve of an object $C$ as a downward-closed set of arrows $S$ with codomain $C$. They go on to say that for a locally small category, a sieve of ...
7
votes
2answers
412 views

Restriction of Ext sheaves

Let $f \colon X \to Y$ be a map of schemes, $\mathcal{F}, \mathcal{G}$ two coherent sheaves on $Y$. I'm interested in conditions which guarantee an isomorphism $$f^{*} \mathcal{E}xt^i(\mathcal{F}, ...
9
votes
4answers
717 views

Is there an easy way to describe the sheaf of smooth functions on a product manifold?

A smooth structure on a manifold $M$ can be given in the form of a sheaf of functions $\mathcal{F}$ such that there is an open cover $\mathcal{U}$ of $M$ with every $U\in \mathcal{U}$ isomorphic ...
7
votes
3answers
1k views

Do we have non-abelian sheaf cohomology?

Lets $X$ be a complex manifold (algebraic variety), $N$ an integer, and consider the sheaf $F$ defined by: $F(U)$ ={ holomorphic maps $f: U\rightarrow GL(N,\mathbb{C})$ } with multiplicative ...
10
votes
1answer
712 views

Understanding the etale space construction from a formal viewpoint

Suppose I have a topological space $X$. Let $\mathcal{O}(X)$ denote the poset of open subsets. There is a canonical functor $\mathcal{O}(X) \to Top/X$ which sends an open $U \in \mathcal{O}(X)$ to $U ...
3
votes
1answer
258 views

The upper semi-continuous rank of a module sheaf

The article "Intuitionistic algebra and representations of rings" is fantastic; it develops the language, logic etc. for the topos $Sh(X)$, where $X$ is a fixed topological space, from scratch and ...
2
votes
1answer
308 views

How to find the smallest flabby sheaf containing a given sheaf ?

None of the spaces C^k (\mathbb{R}^n), with 0 \leq k \leq \infty, is a flabby sheaf. However, they are respectively contained in the smallest flabby sheaves C^k_{nd} (\mathbb{R}^n) of functions f : ...
1
vote
1answer
662 views

How is the right adjoint $f_*$ to the inverse image functor $f^*$ described for functor categories $Set^C$, $Set^D$ and $f : C \to D$

For $C,D$ small categories, and $f : C \to D$ a functor between them, there is a precomposition, or "inverse image", functor $f^* = (-) \circ f : Set^D \to Set^C$. It has a left and a right adjoint. ...
9
votes
1answer
805 views

Mitchell's embedding theorem

Mitchell's embedding theorem http://en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem tells us that every small abelian category ${\cal A}$ has a full, faithful and exact embedding $V : {\cal A} ...
8
votes
6answers
1k views

Is the dual notion of a presheaf useful?

It seems that there is a common theme in mathematics where, if we want to find out about a category C, then we look at $\hat{C}$ (the category of contravariant functors from $C$ to $Set$). There are ...
3
votes
2answers
642 views

Describing global sections of sheafifications

Recently on glancing through Hartshorne's description of Cartier divisors I started pondering the definition of sheafification which led me to a question I can't answer. Neither can I find a ...
1
vote
1answer
411 views

The fiber of the sheaf of invariants

Let us suppose the the group $G:=\mathbb{Z}/2\mathbb{Z}=(1,i)$ freely act on a smooth projective variety/$k$ $X$ and denote by $Y$ the G.I.T. quotient $X/G$. Let $\pi:X\longrightarrow Y$ the quotinet ...
2
votes
1answer
707 views

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 ...
11
votes
0answers
1k views

Idea of presheaf cohomology vs. sheaf cohomology

Let $X$ be a topological space and $U$ an open cover of $X$. In this thread Angelo explained beautifully how presheaf cohomology (Cech cohomology) relates to sheaf cohomology: The zeroth Cech ...
3
votes
1answer
1k views

Grothendieck spectral sequence and Mayer-Vietoris sequence

Suppose $U'\cup U''=X$ is an open cover $U$ of a topological space $X$ and $F$ is a sheaf on $X$ with values in abelian groups. There is a special instance of the Grothendieck spectral sequence ...
3
votes
0answers
203 views

Descent of singular cohomology

When proving that singular cohomology of an appropriate space $X$ equals sheaf cohomology of $X$ with "values" (does one say that?) in the sheaf $\mathbb{Z}_X$ of locally constant functions, the ...
16
votes
1answer
842 views

Galois Group as a Sheaf

I've noticed that the Galois groups associated to Galois field extensions $L$ of a given field $K$ seem remarkably like a sheaf, with the field extensions taking the place of open set, and the Galois ...
2
votes
1answer
339 views

vanishing theorems

I would be glad to know about possible generalizations of the following results: 1) (Grothendieck) Let $X$ be a noetherian topological space of dimension $n$. Then for all $i>n$ and all sheaves of ...
6
votes
2answers
2k views

Sheaf cohomology question

For a topological space $X$ and a sheaf of abelian groups $F$ on it sheaf cohomology $H^n(X,F)$ is defined. Singular cohomology of $X$ can be expressed as sheaf cohomology if $X$ is locally ...
4
votes
1answer
360 views

Does the concept of a basis for a topology on a category exist?

If we want to define a sheaf F on a topological space X and we have a basis B for the topology of X, what we can do is to define objects and restrictions for guys in B, check that they satisfy the ...
3
votes
1answer
384 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 ...
5
votes
1answer
773 views

Natural morphism appearing in Grothendieck spectral sequence

Assume we are in the setting of the Grothendieck spectral sequence (Weibel, 5.8): $G : A \to B, F : B \to C$ are left exact functors such that $G$ sends injective objects to $F$-acyclic objects. Now ...
11
votes
2answers
2k views

De Rham decomposition theorem, generalisations and good references

De Rham decomposition theorem states that every simply-connected Riemannian manifold $M$ that admits complementary sub-bundles $T'(M)$ and $T''(M)$ of its tangent bundle parallel with respect to the ...
2
votes
3answers
2k views

Is Bredon's Topology a sufficient prelude to Bredon's Sheaf Theory?

I intend to try working through Bredon's seminal sheaf theory text prior to graduating (I am currently a second year undergraduate), but it is at a level which is far beyond my own (friends of mine ...
7
votes
2answers
359 views

Sheaf Cohomology on a Stone Space

Let $X$ be a stone space, i.e. a compact, totally disconnected hausdorff space. Then $H^1(X,\mathbb{Z}/2)=0$. Here's one way of proving this: $X$ with $\mathbb{Z}/2$ (the constant sheaf) is an affine ...
7
votes
0answers
180 views

sheaves on thickened nodal cubics

Suppose F is an algebraically closed field (of any characteristic) and that h in F[x,y,z] is an irreducible cubic form defining a plane curve C with a node. A lot is known about sheaves on C; for ...
12
votes
1answer
279 views

References regarding a connection between recursion theory and sheaves

In Manin's A Course in Mathematical Logic for Mathematicians, he defines (p.201) a structure $(\mathcal{E},R)$ given an enumerable set $E \subset (\mathbb{Z}^+)^n$ by: $\mathcal{E}$ is the set of ...