Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [sheaf-theory]

A Sheaf $\mathcal F$ on a topological space $X$ captures local data $\mathcal F(U)$ given on open sets $U\subseteq X$ and how such data can be restricted to smaller open sets or glued together. In typical cases, $\mathcal F(U)$ is a set of functions defined on $U$ and an element of $\mathcal F(V)$, $V\subseteq U$ is obtained by restricting the domain and not all elements of $\mathcal F(U)$ can be obtained by restricting a global section $\in\mathcal F(X)$.

67
votes
0answers
5k views

Pullback and Pushforward Isomorphism of Sheaves

Suppose we have two schemes $X, Y$ and a map $f\colon X\to Y$. Then we know that $\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where $\...
20
votes
0answers
566 views

Sheaves of categories

I recently read an answer on this MO post explaining that one reason people are interested in higher category theory is to make reasonable sense of something like a "sheaf of categories" on a ...
15
votes
0answers
337 views

Quasicoherent sheaves as smallest abelian category containing locally free sheaves

On page 362 of Ravi Vakil's notes, the author says "It turns out that the main obstruction to vector bundles to be an abelian category is the failure of cokernels of maps of locally free sheaves - ...
12
votes
0answers
462 views

Cup/cap product: sheaf cohomology vs singular cohomology

Is anyone aware of a good resource which deals with how the cup/cap products of sheaf cohomology classes are a generalization of those in singular cohomology? I would say that I already understand the ...
12
votes
0answers
177 views

Applications of $Ext^n$ in algebraic geometry

I have been doing a project about $\operatorname{Ext}^n$ functors for my commutative algebra class. I used the approach via extensions of degree n. Basically I have shown the long exact sequence ...
10
votes
0answers
219 views

Why is the sheaf $\mathcal{O}_X(n)$ called the “twisting sheaf” (where $X=\operatorname{Proj}(S)$ for a graded ring $S$)?

Basically my question is why the sheaf $\mathcal{O}_X(n)$ is called the twisting sheaf, here $X=\operatorname{Proj}(S)$ and $S$ any graded ring.
10
votes
0answers
661 views

Composition of derived functors and comparison between hypercohomology and sheaf cohomology

I had a few questions about compositions of derived functors, the comparison between hypercohomology, and sheaf cohomology and the following theorem from the Gelfand, Manin homological algebra book: ...
9
votes
0answers
198 views

Products of homology groups

In the topology course I attend we work with the following, rather unusual definition of homology groups: For topological spaces $X,Y$ define the presheaf $\mathcal{F}_Y$ on $X$ as follows: Let $\...
9
votes
0answers
375 views

Why do these two constructions of a sheaf associated to a module on a projective scheme agree?

Let $B$ be a graded ring an $M$ be a $\mathbb Z$-graded $B$-module. We can associate to $M$ a sheaf of modules on $\operatorname{Proj} B$ by defining the sheaf on the principal open sets $D_+(f)$ to ...
9
votes
0answers
494 views

Injective Resolutions in $\mathfrak{Ab}(X)$

Using right derived functors of the global sections functor, I'd like to calculate the first cohomology group of the constant sheaf $\mathbf{Z}$ on $S^1$ with its usual topology, $H^1(S^1,\mathbf{Z})$....
8
votes
0answers
175 views

Flabby sheaf comonad

If one Googles sufficiently hard one finds the statement that Roger Godement gives the first example of a comonad, used to compute flabby resolutions of sheaves, in his monograph "Topologie algébrique ...
7
votes
0answers
106 views

Set theoretic issues in the definition of a site in Stacks Project

I've been learning about sites from the Stacks Project, which is generally very precise in its terminology, but I've found some of their conventions very confusing in this part. Their definition of a ...
7
votes
0answers
236 views

What's the intuition behind “representable morphisms”?

A central notion in many algebro-geometrical stuff appears to be so-called "representable morphisms". A general (read: hand-wavy) definition could be the following, as far as I can tell: Let $\...
7
votes
0answers
177 views

Why don't we use the constant sheaf as the dualizing sheaf in Verdier Duality?

The Verdier dual of a complex of sheaves $F$ on a complex manifold $X$ is defined in the derived category $D_b(X)$ as $\mathcal{RHom}(F,\mathbb{D}_X)$, where $\mathbb{D}_X$ is the dualizing "sheaf" (...
7
votes
0answers
879 views

Prove extension by zero is a special case of lower shriek?

The lower shriek functor is defined by $$f_{!}F(U)=\{s\in\Gamma(f^{-1}(U),F)\;:\; f|_{\mathrm{supp}(s)}:\mathrm{supp}(s)\rightarrow U\text{ is proper}\}$$ On the other hand, if $j:V\subset X$ is the ...
6
votes
0answers
173 views

How to see this result of restricting and pushing forward a sheaf?

Let $X$ be a complex variety with trivial canonical bundle $K_B=\mathcal{O}_X$, with a projection morphism $f$ onto a complex codimension one subspace $s$. Consider the sheaf $$ \mathcal{F} := f_*\...
6
votes
0answers
223 views

Katz-Mazur chapter 1 AG questions

I was reading through Katz-Mazur "Arithmetic Moduli of Elliptic Curves", Chapter 1, and ran into some small issues (which might have a lot to do actually with notation). I think most of them are due ...
6
votes
0answers
239 views

Is there a universal statement for the construction of global Proj?

Let $X$ be a scheme and $\mathcal{A}$ be a sheaf of $\mathbb{Z}_{\ge 0}$-graded $\mathcal{O}_X$-algebras. From the data above there is a construction which gives the "global Proj" $Proj \mathcal{ A} ...
6
votes
0answers
593 views

Homotopy-invariance of sheaf cohomology for locally constant sheaves

Suppose we have a homotopy equivalence $f: X \to Y$ (with homotopy inverse $g: Y \to X$) and a local system (i.e. a locally constant sheaf) $\mathscr{S}$ on $Y$. Is the homomorphism $$f^*: H^k(Y,...
6
votes
0answers
175 views

Vanishing of Tor sheaf on a union of subschemes with vanishing Tor.

Suppose $X$ is a scheme and $Y$ and $Z$ are closed subschemes such that $Y$ is a finite union of closed reduced irreducible subschemes $C$ that satisfy $$\mathscr{Tor}_i^{\mathcal{O}_X}(\mathcal{O}_C,\...
6
votes
0answers
615 views

Leray's theorem for cech and derived sheaf cohomology.

My question is about the hypothesis of Leray's theorem. This theorem says that if $\mathcal{U}$ is an open cover of a topological space $X$, and $\mathcal{F}$ is a sheaf over $X$ and if $\check{H}^q(...
6
votes
0answers
424 views

Čech cohomology with values in a presheaf

Let $X$ be a topological space, $\mathcal F$ a sheaf of abelian groups on $X$. Let $\breve{H}^n(X, \mathcal F)$ denote the $n$-th Čech cohomology of $X$ with coefficients in $\mathcal F$. Thus $\...
6
votes
0answers
256 views

The theorem on formal functions

Recall the Theorem on Formal Functions [Hartshorne, III.11.1] Let $f:X \to Y$ be a projective morphism of noetherian schemes, let $\mathcal{F}$ be a coherent sheaf on $X$ and let $y\in Y$. Then the ...
6
votes
0answers
446 views

Mistake in contradiction argument to show that sheafification commutes with cokernel

The sheafification functor is a left-adjoint to the forgetful functor. Hence it commutes with colimits. The cokernel is a colimit. Hence the cokernel of a sheafified morphism is the same as the ...
6
votes
0answers
695 views

Why didn't Cartan-Eilenberg develop homological algebra on sheaf theory?

Cartan-Eilenberg created homological algebra on modules over rings. I wonder why they didn't develop it also on sheaves over ringed spaces. Grothendieck and Godement did that soon after(or almost at ...
5
votes
0answers
152 views

Alternative definition of “sheaf”

Let $(X,\tau)$ denote a topological space and $\mathcal{O}$ denote a presheaf on this space with codomain $\mathbf{Set}$. We can take the category of elements of $\mathcal{O}$, which consists of a ...
5
votes
0answers
129 views

epimorphism of fppf sheaves is an fppf morphism

Suppose $0\to F\to G \to H\to 0$ is an exact sequence of group schemes (over some base scheme $S$) by which I mean that the corresponding sequence of fppf-sheaves is exact. I read somewhere that the ...
5
votes
0answers
141 views

Reference for computing sheaf cohomology of constructible sheaves by \v{C}ech cohomology?

Let $X$ be a topological space. A stratification of $X$ is a decomposition into a collection of subsets $X = \amalg_i \ X_i$ (called strata) such that $X_i$'s are open in its closure, and $\partial ...
5
votes
0answers
913 views

When is the direct image functor exact?

Consider a morphism of topological spaces $f:X\to Y$. The direct image functor takes a sheaf $\mathcal{F}$ on $X$ to the sheaf defined by $f_*\mathcal{F}(U)=\mathcal{F}(f^{-1}(U))$. It's a right ...
5
votes
0answers
163 views

On the definition of ugly manifolds

Let $X$ be a subset of a (finite dimensional) Banach space $E$. Let's call a function $f\colon X\longrightarrow \mathbb{K}$ smooth (in whatever sense we are interested in) if there is an open ...
5
votes
0answers
69 views

Relating evaluations of Functors and polynomials

I came across the following two constructions recently, which both have in common of relating: on the one hand the evaluation of something resembling a function, and on the other hand a more ...
5
votes
0answers
160 views

Hartshorne II.9.1

I have queries relating to Hartshorne Ex. II.9.1. Take a connected, non-singular, positive-dimensional subvariety $i: Y \to X = \mathbb{P}_k^n$, where $k$ is algebraically closed, with ideal sheaf $\...
5
votes
0answers
143 views

Sheaves with surjective map from the ring of global sections to stalks

Let $M$ be a smooth manifold, $\mathscr O_M$ the sheaf of vector fields. Then, by an argument involving partition of unity, we can show that the map $$\mathscr O_M(M)\to\mathscr O_{M,x}$$ is ...
5
votes
0answers
785 views

Possible mistake in Gortz-Wedhorn's algebraic geometry book

I'm trying to solve exercise 2.14c in Gortz-Wedhorn's book on algebraic geometry, and it looks to me like it's wrong. Here's the statement. Let $X$ be a topological space and $i:Z \rightarrow X$ the ...
5
votes
0answers
203 views

Is there an explicit description for injective sheaves?

I want to find a criterion for sheaves of modules to be injective. It would be great if one can such a criterion for sheaves of modules over a ringed space. But an answer for sheaves of abelian groups ...
5
votes
0answers
69 views

Is there a name for a sheaf where every germ extends to a global section?

This is strictly weaker than being flasque -- consider, for instance, the case of the sheaf of smooth functions on a smooth manifold, which has this property but is not flasque.
5
votes
0answers
310 views

Two definitions of equivariant sheaves

Let $G$ be a topological group. Here are two definitions of $G$-equivariant sheaves on a $G$-space $X$. (a) Define an $G$-equivariant sheaf by a sheaf $F$ (étalé space) equipped with a $G$-action ...
5
votes
0answers
246 views

Which local ringed spaces are schemes?

This paper gives a proof that the underlying topological spaces of affine schemes are precisely the spectral spaces (compact, T0, sober, and the compact open subsets are closed under finite ...
5
votes
0answers
518 views

Computing the hypercohomology of a complex of acyclic sheaves

Let $K^{\bullet}$ be a cochain complex of sheaves of finite-dimensional vector spaces, I wanted to compute $\mathbb{H}^{\bullet}(X,K^{\bullet})$ = the hypercohomology of the complex $K^{\bullet}$, the ...
5
votes
0answers
107 views

local system attached to a finite morphism

Imagine you have a finite proper morphism between smooth projective complex curves, say $f: X \to Y$. Denote by $S$ the image of ramification points and by $d$ the degree of $f$. Then $f_\ast \mathbb{...
5
votes
0answers
281 views

When does a sheaf exist with prescribed stalks?

I think the intended question is clear, but let me attempt to formulate it precisely: If $X$ is a topological space, and $f$ is a function on $X$ such that $f(x)$ is an abelian group for every $x \in ...
5
votes
0answers
306 views

Locally free sheaves on locally ringed spaces

One can define the notion of a locally free sheaf (of finite rank) on any locally ringed space. If you restrict to the category of (noetherian?) schemes, this category is equivalent to the category ...
5
votes
0answers
521 views

Identifying isomorphic schemes

Suppose $(X, \mathcal{O}_X)$ and $(Y, \mathcal{O}_Y)$ are isomorphic as schemes. Then by definition there is an isomorphism of locally ringed spaces $(\psi, \psi^{\sharp}): (X, \mathcal{O}_X) \to (Y, \...
4
votes
0answers
40 views

A function being “finite” over a point on non-normal schemes?

I recently came across a remark about Cartier divisors in a textbook on algebraic geometry. I'm not sure how to interpret the remark. I've attached the previous paragraph as well for context. The ...
4
votes
0answers
63 views

How to prove $\text{Sh}_G(X)\simeq \text{Sh}(G\backslash X)$ when $X$ is a free $G$-space?

Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$. Let $G$ be a topological group which act on $X$ continuously from the left....
4
votes
0answers
80 views

Stalk at a point of the sheaf of open sets

Fix a base topological space $X$. For every open subset $U$ of $X$, define $\mathcal F(U)$ to be the set of open subsets of $U$. The restriction map $\mathcal F(V) \to \mathcal F(U)$ sends $W$ to $W ...
4
votes
0answers
81 views

Section of pullback bundle isomorphic to the sheaf pullback of sections

We have $p: E \to X$ an holomorphic vector bundle,where $X$ is a complex manifold and $f: Y \to X$ where $Y$ is anothet complex manifold. We can build up $f^* E$ pullback bundle on $Y$. Now,is it true ...
4
votes
0answers
120 views

Obstruction for nonzero section of nonorientable vector bundle

Suppose we have real $n$-vector bundle $p:E\to X$ over some $CW$-complex $X$. It is well-known that if $E$ is orientable (i.e. $\pi_1X$ acts on orientation of fibers trivially) then the obstruction to ...
4
votes
0answers
111 views

Equivalent Definitions of Twisted Sheaf $ \mathcal {O}(1)$

Let $\mathcal {O}(-1)$ be the tautological line bundle $X$ of $ \Bbb CP^1$, where $X=\{(z,l) \in \Bbb C^2 \times \Bbb CP^1 : z \in l \}$ together with canonical projection $X \to \Bbb CP^1$ (line ...
4
votes
0answers
77 views

Moral justification for “sheaf=continuously variable set” and local injectivity

From the topos theory perspective it is a general motto that sheaves are continuously variable sets. The internal logic of sheaf toposes justifies this motto, but I would like an additional "local" ...