5
votes
1answer
61 views

A question about Hartshorne III 12.2

In Hartshorne III 12.2, $X\to \text{Spec}\ A$ is a morphism, $\mathcal{F}$ is a coherent sheaf on $X$, flat over $\text{Spec}\ A$, $M$ any $A$ module, then we can construct the sheaf associated to the ...
1
vote
0answers
64 views

Colimit and push forward of quasi-coherent sheaves in the analytic setting

I'm currently working on my master thesis and have some unresolved questions about quasi-coherent sheaves. Since I'm new to algebraic geometry, they might be rather trivial. I'm working in the ...
0
votes
1answer
65 views

Hartshorne Exersice 1.17 Skyscraper sheaf Chapter II Schemes

I am able to verify the statements about the stalk. I want to see how the direct image of the the skyscraper sheaf can be thought of as the constant sheaf. Observation- If $P\notin U$, then $U\cap ...
5
votes
1answer
72 views

Is there a coherent sheaf which is not a quotient of locally free sheaf?

Suppose $X$ is an algebraic variety, is there a coherent sheaf $\mathcal{F}$ on $X$ which is not a quotient of locally free sheaf´╝č (Hartshorne II Cor 5.18 showed that on every projective variety, ...
9
votes
0answers
105 views

Hartshorne Theorem 8.17

I can't understand the proof of theorem 8.17 from Hartshorne's "Algebraic Geometry". Namely, he says that we have an exact sequence $$ 0 \to \mathcal J'/\mathcal J'^2 \to \Omega_{X/k} \otimes ...
1
vote
2answers
84 views

Soft sheaves adapted to $f_!$

I'm reading Gelfand-Manin, Homological Algebra. I understand that the class of soft sheaves is sufficiently large, because every injective sheaf is soft. Now to see that this class is adapted to ...
0
votes
1answer
49 views

What is a Presheaf (intuitively) and help with the technical machinery.

I have come across things such as that a Presheaf $\mathcal{F}$ associates data (such as rings, groups, other sets etc.) to open sets $U$ of $X$. That the Presheaf $\mathcal{F}$ becomes a Sheaf if ...
0
votes
1answer
64 views

Prove: $U \mapsto \mathrm{Hom}(U, Y)$

Rewording this problem via what Zhen Lin's notion of the original question is. For $X$ and $Y$ ringed spaces Prove: For each open $U \subset X$ the Presheaf $U \mapsto \mathrm{Hom}(U, Y)$ is a ...
4
votes
1answer
81 views

Exact sequence of sheaves of holomorphic functions

This is from Exercise 2.4.P. June 2013 version of Ravi Vakil's Math 216 notes. The idea is to show $\mathscr{O}_X \xrightarrow{\text{exp}} \mathscr{O}^*_X$ is an epimorphism. It seems ...
1
vote
1answer
49 views

Connected scheme but not quasicompact

I am searching for a connected scheme which is not quasi compact. My try: Suppose $A_i$=Spec$k[x]$, where $k$ is algebraically closed field are affine schmes . I am planning to glue $0$ of the ...
0
votes
0answers
43 views

Sheaf of ideals [duplicate]

Let $(X, \mathcal{O}_X)$ be a scheme and and $f \in \mathcal{O}_X(X)$. We define the sheaf of ideals by the assignment $$ U \longmapsto f|_U \mathcal{O}_X(U) $$ and denote this sheaf by ...
1
vote
0answers
48 views

Hom sheaf over a scheme in the case of quasi-coherent sheaf at first argument

Let $X$ be a scheme and $\mathcal{F},\mathcal{G}$ be two sheaves of $\mathcal{O}_X$-modules. I showed that the presheaf which assigns each open subset $U$ of $X$, $$ U \longmapsto ...
4
votes
0answers
31 views

Are acyclic coverings cofinal in the set of coverings?

I am interested by the following question in algebraic geometry. Recall that a covering $\mathfrak{U}$ of a topological space $X$ is acyclic for a sheaf $\mathscr{F}$ if we have $H^q(U_{i_0,\cdots, ...
1
vote
1answer
82 views

How to prove directly that, if $A$ is Noetherian and $X=\operatorname{Spec} A,$ then $\mathscr O_X$ is a coherent $\mathscr O_X$-module?

I use the following definition: Definition Let $(X,\mathscr O_X)$ be a locally ringed space. An $\mathscr O_X$-module $\mathscr F$ is coherent if (i) it is locally finitely generated. (ii) for every ...
1
vote
1answer
148 views

Is the cotangent sheaf quasi-coherent?

Allow me to reconstruct what is written here, in order for me to present the question. Let $(\mathscr{M},\mathscr{O}_{\mathscr{M}})$ be a smooth manifold, and let $\mathscr{M\times M}$ be the ...
6
votes
1answer
60 views

exact sequence induced by restriction to closed subscheme

I'm somewhat embarrassed to be confused about this issue after a while studying algebraic geometry, but here goes. Let $\iota: Y \hookrightarrow X$ be the inclusion of a closed subscheme, $U$ is the ...
2
votes
1answer
40 views

What is a sheaf of rings? (question regarding the definition)

I was reading some notes on Algebraic Geometry and it says, "Suppose $O_X$ is a sheaf on rings on a topological space $X$ (i.e., a sheaf on $X$ with values in the category of rings)." What does this ...
0
votes
1answer
35 views

Kernel,image of quasi-coherent sheaf is quasi-coherent for ringed spaces?

If $X$ is a ringed or locally ring space (not necessarily scheme), do we still have the kernel, image of quasi-coherent sheaves quasi-coherent?
0
votes
1answer
40 views

Pushforward of pullback of an etale sheaf

I hope this question is not too elementary, but I'm a bit lost : Let $S$ be a finite set of closed points of $\mathbb{P^1_{C}}$ and $j : U \longrightarrow \mathbb{P^1_{C}}$ be the inclusion of the ...
4
votes
3answers
86 views

What is the inverse image of a sheaf

Let $f : X \rightarrow Y$ be a continuous map of topological spaces and $\mathcal{G}$ a sheaf on $Y$. What exactly is $f^{-1}\mathcal{G}$? It seems like we should be able to describe the sections ...
0
votes
2answers
185 views

Non-cohomological proof that a quasi-coherent sheaf over an affine scheme is quasi-flasque

Let $\mathcal F$ be a quasi-coherent sheaf over an affine scheme $X$. Let $0 \rightarrow \mathcal F \rightarrow \mathcal G \rightarrow \mathcal H \rightarrow 0$ be an exact sequence of sheaves on $X$, ...
4
votes
1answer
49 views

Do I correctly understand the constructions involved in definition of Cartier divisor?

Let $(X,\mathcal O)$ be a ringed space, where $\mathcal O$ is a sheaf of unital commutative integrity domains. Let $\widetilde{\mathcal M}_U$ be the field of fractions of ring $\mathcal O_U$ for any ...
1
vote
0answers
38 views

Describing a locally free sheaf sitting between two locally free sheaves which are given as extensions

Assume $X$ is a two dimensional scheme with $Pic(X)=\mathbb{Z}$ such that every rank two locally free sheaf $\mathcal{E}$ is given by an exact sequence $0\rightarrow \mathcal{O}_X(n)\rightarrow ...
4
votes
2answers
105 views

Geometric interpretation and computation of the Normal bundle

My question concern the definition, geometric meaning, and usage of the normal bundle in algebraic geometry. Let $X$ be a nonsingular variety over an algebraically closed field $k$, and $Y\subseteq ...
3
votes
1answer
91 views

How to prove the sheafification is a sheaf?

I know that this question might be too easy for you, but I have to study on my own, so please explain for me. In the page 64, Hartshone defined the sheafification of a presheaf $\mathcal{F}$ by ...
5
votes
2answers
133 views

Sheafification of the Presheaf of continuous and bounded functions

Let $X$ be a topological space. $U\subset X$ open. $\mathfrak{B}(U) = \{f:U\to \mathbb{R}| f \textrm{ continuous and bounded}\}$ is a presheaf. I would like to see the sheafification of this ...
3
votes
1answer
62 views

Connection(gauge field) in Fubini-Study metric is pull back of a connection A of line bundle $\mathcal{O}(1)$ on $\mathbb{CP}^{N-1}$

One can describe a $\mathbb{CP}^{N-1}$ manifold with a Fubini-Study metric $g^{FS}$, and there is a connection one form $v$ on it. A is connection one form(gauge field) of a line ...
0
votes
1answer
33 views

$X=Spec(A)$. $X=\bigcup\limits_{i=1}^N D(f_i)\Rightarrow (f_1, …, f_N)=A$

Let $X=Spec(A)$ and note $D(f)\simeq Spec(A_f)$. $X=\bigcup\limits_{i=1}^N D(f_i)\Rightarrow (f_1, ..., f_N)=A$ We used this to proof a special case of $\mathscr{O}$ the sheaf of rings is a sheaf. I ...
1
vote
0answers
41 views

Existence of a long exact sequence for sheaf cohomology

Let $X$ be a normal variety over $\mathbb{C}$ , and let $U$ be a open subset of $X$, then there is an long exact sequence for singular or De Rham cohomology with compact support that relates the ...
0
votes
0answers
33 views

relation between quotient field of holomorphic functions and meromorphic functions

Let $U \in \mathbb{C}^n$ open connected. Is it true that the quotient field of $\mathcal{O}_{\mathbb{C}^n,z}$ for $z \in U$ is the stalk of the sheaf $\mathcal{K}$ where $\mathcal{K}(U)$ is the field ...
2
votes
0answers
64 views

Example: Push-Forward Sheaf

Let $f: X\to Y$ be a continuous map, $\mathscr{F}$ a sheaf on $X$. $f_*\mathscr{F}$ is the sheaf on $Y$ defined by $f_*\mathscr{F}(U)=\mathscr{F}(f^{-1}(U))$ Uand $\rho_{VU}=\rho_{f^{-1}(V)f^{-1}(U)}: ...
3
votes
1answer
79 views

Sheafification - Construction of a Sheaf

I tried different books and lecture notes to understand sheafification, but for instance in Hartshore or Shafarevich's book, but I found it hard to understand. The following is the approach my ...
2
votes
0answers
32 views

Cech cohomology [duplicate]

There are 2 complexes computing Cech cohomology. The difference between them is that in the second one we require skew symmetry when you change the order of indices. How to show that they are ...
5
votes
1answer
116 views

When is the global section functor exact?

Given sheaves $\mathcal{F}_1,\mathcal{F}_2, \mathcal{F}_3$ on some scheme $X$, and an exact sequence $$ 0\rightarrow \mathcal{F}_1\rightarrow \mathcal{F}_2\rightarrow \mathcal{F}_3\rightarrow 0 ...
4
votes
2answers
145 views

What is the zero subscheme of a section

Let $X$ be a scheme, $\mathcal F$ a locally free sheaf of rank $r$ and $s \in \Gamma(X, \mathcal F)$ a global section of $\mathcal F$. Question: What is the zero subscheme of $s$? I can't believe ...
3
votes
1answer
50 views

Coherent sheaves of finite length over $\mathbb{P}^n_k$

Let $k$ be an algebraically closed field. Are there any nonzero coherent sheaves on the projective space $\mathbb{P}^n_k$ that are supported at (only) finitely many closed points? If they don't exist, ...
1
vote
0answers
41 views

$\operatorname{Spec} (\cdot)$ is functorial [duplicate]

If $A$ is a ring (with unity) I'm trying to prove that the assignement $A\mapsto\operatorname{Spec}A$ defines a contravariant functor from the category of rings to the category of affine schemes. If ...
6
votes
1answer
74 views

Functor of points $h_X$ is an fpqc sheaf on $\operatorname{Spec} \Bbb{Z}$

I want to show the following. Let $X$ be any scheme (say over the terminal object $\operatorname{Spec} \Bbb{Z}$ in $\textbf{Sch}$) and $A \to B$ a faithfully flat ring homomorphism. Then $$h_X(A) \to ...
5
votes
1answer
78 views

Meromorphic functions a constant sheaf?

Is the sheaf of meromorphic functions on a (connected) compact Riemann surface constant? I am refering here to meromorphic functions in the sense of complex analysis and not to those of algebraic ...
4
votes
1answer
90 views

Using the cocycle condition to glue sheaves

Given a cover $\{U_i\}$ of a space $X$ and for each $U_i$ a sheaf $\mathcal{F}_i$ and isomorphisms $\phi_{ij}:\mathcal{F_j}|_{U_i \cap U_j} \rightarrow \mathcal{F_i}|_{U_i \cap U_j}$ satisfying the ...
1
vote
1answer
46 views

Linear systems and rational maps

I'm following Beauville's book on Complex Algebraic Surfaces. If $D$ is a divisor on a surface $S$, we write $|D|$ for the set of all effective divisors linear equivalent to $D$ and we call it a ...
4
votes
1answer
74 views

Universal property of quotient sheaves

Recently, I was doing exercise 2.3 (b) in chapter 2 of Hartshorne's book celebrated book on algebraic geometry. The exercise is as follows: Let $(X,\mathcal{O}_X)$ be a scheme. Define a presheaf by ...
4
votes
1answer
67 views

Restriction of a sheaf to a fibre

I have come across the notion of the restriction of a sheaf to a fibre, but I haven't been able to find a proper definition, could anyone perhaps supply one? Suppose that $f: X \to Y$ is a morphism ...
4
votes
0answers
81 views

Criterion of nonsingular varieties

It's well-known fact, that if $X$ is non-singular algebraic variety over algebraically closed field $k$ and $Y \subset X$ is its irreducible closed subscheme defined by sheaf of ideals $J$, then $Y$ ...
6
votes
2answers
72 views

Ideal sheaf on a surface

Let $S\subset\mathbb{P}^n$ a smooth complex projective surface. I consider the exact sequence $$0\rightarrow I_S\rightarrow\mathcal{O}_{\mathbb{P}^n}\rightarrow\mathcal{O}_S\rightarrow 0,$$ where ...
2
votes
1answer
54 views

Is the zero set of a global section closed?

Let $(X,\mathcal{O}_X)$ be a locally ringed space and let $\mathcal{F}$ be an $\mathcal{O}_X$-module. For a section $s \in \mathcal{F}(X)$ and a point $x$, we say $s(x)=0$ if the stalk $s_x$ is zero ...
2
votes
1answer
75 views

$X$ being locally closed is not equivalent to every extension by $0$ of a sheaf $F$ being unique?

In Tennison's book "Sheaf Theory", the author presents a proof that there is a unique extension by $0$ for a sheaf $F$ over $X$ iff $X \subset Y$ is locally closed . However, apparently in the proof ...
1
vote
2answers
48 views

In a quasicompact scheme every nonreduced point has a closed nonreduced point in its closure

Let $x\in X$ be a non reduced point. Then I can find a nilpotent $f\in \Gamma(U, \mathcal O_X)$, where $U$ is some open neighborhood of $x$. I also know that when $X$ is assumed to be quasicompact, ...
3
votes
0answers
57 views

How to apply “proper base change” here

Reading a book about curves I encoutered the following claim, which I don't understand. Let $X$ be a smooth projective curve, and $\nu:X\times \mathrm{Pic}(X)\to \mathrm{Pic}(X)$. Pick a universal ...
1
vote
0answers
67 views

Skyscrapers sheaf's global sections

I'm reading a book written by Serre and, even though he's one of the best math writer ever, there's a step I don't understand. This may imply that I'm one of the worst math reader ever! ...