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 [coherent-sheaves]

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space.

1
vote
0answers
36 views

Hartshorne III.7.6(b) - Duality for a projective scheme

Let $X$ be a closed subscheme of $P=\mathbb{P}^N_k$ of dimension $n$. Theorem III.7.6(b) of Hartshorne states that: Suppose that for any $\mathcal{F}$ locally free on $X$, we have $H^i(X,\mathcal{F}...
1
vote
1answer
64 views

are the global sections of a flat sheaf over a discrete valuation ring a free module?

Let $f:X\to \operatorname{Spec}\mathbf{Z}_p$ be a smooth proper $\mathbf{Z}_p$-scheme and $F$ a coherent sheaf on $X$ which is flat over $\mathbf{Z}_p$. Further suppose that $H^1(X_p, F_p)=0$ where $...
1
vote
0answers
37 views

Inverse Image Sheaf and pullback of a Vector Bundle.

Let $M,N$ be smooth manifolds and $f:N\to M$ a smooth map. Denote by $\mathcal{O}_M,\mathcal{O}_N$ the corresponding sheaves of smooth functions. If we regard $\mathcal{O}_M$ as the sheaf of smooth ...
1
vote
1answer
47 views

Characterization of torsion free sheaves

In "The Geometry of Moduli Spaces of Sheaves" by Huybrechts and Lehn a torsion free sheaf is defined as coherent sheaf $E$ on an integral Noetherian scheme $X$ s.t. for every $x\in X$ and every non-...
1
vote
0answers
5 views

Can one obtain a generating set for a module of local sections of a coherent sheaf by finding generating sets at the stalks?

Question is possibly slightly different than posed if I am misunderstanding what coherency means. I do not know the correct term for "set of local sections corresponding to $F(U)$". Let $F$ be a ...
1
vote
0answers
37 views

Vakil 13.7.A(b): Proof verification

In FOAG by Vakil, exercise 13.7.A part (b) I am supposed to show the following: If $\mathcal{F,G}$ and $\mathcal{O}_X$ are coherent sheaves on a scheme $X$, then $\mathcal{Hom(F,G)}$ is also ...
1
vote
0answers
49 views

Confused about why the conormal exact sequence is what it is on a scheme

Consider a composition of morphisms of schemes, $$Z \stackrel{j}{\longrightarrow} X \stackrel{f}{\longrightarrow} Y,$$ where $j: Z \rightarrow X$ is a closed immersion with sheaf of ideals $\mathscr{I}...
0
votes
1answer
57 views

Confusion about Exercise II .5.15 in Hartshorne

I'm a bit confused about Exercise II 5.15 in Hartshorne's Algebraic Geometry, especially part (b) and (c) which are (b) Let $X$ be an affine noetherian scheme, $U$ an open subset, and $\mathscr{F}...
2
votes
0answers
47 views

Clarifying a step in an answer about exterior products of coherent sheaves

The question is about the accepted answer here. I decided not to ask in a comment there since the original asker is no longer active. In "Step 2" of the answer given there, Roland claims, "This is ...
1
vote
1answer
34 views

What is the module and sheaf of differentials (actually)?

Throughtout, assume all rings are commutative with identity, and all schemes are separated. If $ A \rightarrow B$ is an $A$-algebra, I am having a lot of trouble understanding precisely what is meant ...
1
vote
1answer
26 views

Computing degree of hom sheaf of coherent sheaves.

Let $C$ be complex projective curve (Cohen-Macaulay at least). Let $\mathcal{F}$ and $\mathcal{G}$ be coherent sheaves on $C$. Is there any way to express the degree of $\underline{Hom}_{\mathcal{O}...
0
votes
0answers
27 views

The open affine subsets of an algebraic variety $X$ form an open base for the topology of $X$

Let $K$ be an algebraically closed field. Define an algebraic variety to be a pair $(X,\mathscr{O}_X$) for a topological space $X$ together with a sheaf $\mathscr{O}_X$ that is a subsheaf of the ...
3
votes
1answer
91 views

So many different 'varieties', which one is this? Serre's algebraic variety

Anyone who has ever tried to study algebraic geometry has experienced the phenomenon of being burdened by countless types of varieties (variety, affine variety, projective variety, quasi-affine ...
0
votes
0answers
24 views

Inducing Sheaf of Local Rings to Locally Closed Subspace

In this English translation of Serres FAC [working on p.38], for $X = K^r$ in the Zariski topology, $K$ algebraically closed, we define a locally closed subspace $Y$ as usual: the intersection of ...
5
votes
2answers
113 views

Importance of Vanishing Cohomology

As part of my masters project I have been working through Serre's FAC. Below are three closely related results I will be presenting as part of my defense. These results are from n$^{°}$ 52, page 63 ...
0
votes
0answers
106 views

Is the pull-back of the structure sheaf the structure sheaf?

Maybe this is a stupid question, but I got irritated by it: Suppose $f: X \rightarrow Y$ is a morphism of schemes. That comes with a map of sheaves $f^\#: \mathcal{O}_Y \rightarrow f_* \mathcal{O}_X$. ...
1
vote
1answer
96 views

Serre says open covers do not form a set, why? Directed sets and limits.

The following selection is from Serre's FAC (Chapter 1, §3, n°22, page 26). The relation `$\mathfrak{U}$ is finer than $\mathfrak{V}$' (which we denote hencforth by $\mathfrak{U} \prec \mathfrak{...
1
vote
2answers
69 views

Dimension of an Open Covering - Serre's FAC

I am reading Serre's Faisceaux Algébriques Cohérents (Henceforth FAC) and he uses some terminology I have not seen. I have searched around a bit but can't get a clean and clear definition. Question:...
0
votes
0answers
93 views

On torsion sheaf of a coherent sheaf of $\dim X$

$\underline {Background}$:Let,$E$ be a coherent sheaf on a Noetherian,integral scheme $X$ and $\dim E$=$\dim X$. Then we have the unique torsion filtration of that coherent sheaf as $0\subset T_0(E)...
1
vote
1answer
59 views

Group Action on a Scheme

Let $X$ be scheme and $G \subset Aut(X)$ be a subgroup of automorphism group of $X$. By definition $G$ acts espectially on local sections $\mathcal{O}_X(U)$ for open $U$ and one can therefore define ...
2
votes
0answers
97 views

Morphisms between torsion sheaves

Let $X$ be a smooth projective variety over $\mathbb{C}$, and $Y,Z\subset X$ be closed subvarieties of $X$. Denote the embeddings of $Y$ and $Z$ into $X$ by $i_Y$ and $i_Z$ respectively. What is the ...
1
vote
0answers
51 views

Condition of pushward commutes with tensor product

Let $f$ be a morphism between schemes. Is there a sufficient and necessary condition on $f$ such that $f_*$ commutes with $\otimes$? i.e. $$f_*F\otimes f_*G\cong f_*(F\otimes G)$$ for all coherent ...
1
vote
0answers
71 views

sections of regular functions $\mathcal{O}_X$ of the sphere $X = \{x^2 + y^2 + z^2 - w^2 = 0\}$ in projective space $\mathbb{P}^3$ [closed]

I am trying to understand the sheaf of regular functions $\mathcal{O}_X$ in the case of the sphere. The machinery seems rather difficult to set up. Let't try using the following proposition from the ...
2
votes
0answers
38 views

What's the direct sum of infinite sheaves?

In the Lemma 5.1.3 of Liu Qing's book on algebraic geometry, he uses $O_{X}^{(I)}$ which the direct sum of $O_{X}$indexed by $I$. What's the global section of this sheaf? Is it $\bigoplus O_{X}(X)$? ...
3
votes
1answer
87 views

rational points on the quadrifolium $(x^2 + y^2)^3 = (x^2 - y^2)^2$

I have been reading the Wikipedia page on the Quadrifolium there are two of them: \begin{eqnarray*} r &=& \sin 2\theta \\ (x^2 + y^2)^3 &=& 4 x^2 y^2 \end{eqnarray*} and it's $45^\...
0
votes
0answers
63 views

Not understanding a proof about coherent sheaves on projective schemes in Hartshorne

I have been stuck on the proof of the following statement for a while now. Let $S$ be a graded noetherian domain which is finitely generated by $S_{1}$ as an $A$-algebra where $A = S_{0}$ is a ...
1
vote
0answers
55 views

In what sense is a constant sheaf of abelian groups with stalks G isomorphic to G?

In the English translation of Serre's FAC (link to the PDF can be found in this mathoverflow discussion in the top answer) Serre gives his first example of a sheaf of abelian groups, the constant ...
0
votes
0answers
48 views

First Isomorphism Theorem for Sheaves, Missing Step

The following excerpt is from an english translation of Serre's FAC (here). This is in Chapter I: Sheaves, $\S1:$ Operations on Sheaves, n$^{\circ}$ 7: Subsheaf and Quotient Sheaf and n$^{\circ}$ 8: ...
2
votes
1answer
58 views

Why are the noetherian objects in a category of quasicoherent sheaves just the coherent ones?

The question says it all really. Let $X$ be a noetherian scheme. Let $\mathcal{A}$ be the category of quasicoherent sheaves on $X$. I want to show that an object $\mathcal{F}$ in $\mathcal{A}$ is ...
3
votes
0answers
50 views

$f_* f^* \mathcal{G}= \mathcal{G}$ and $f^* f_* \mathcal{F}= \mathcal{F}$ for Quasicoherent Sheaves

Let $f: X \to Y$ a morphism between schemes, $\mathcal{F}$ a quasicoherent $\mathcal{O}_X$ module, $\mathcal{G}$ a quasicoherent $\mathcal{O}_Y$ module. My question is what are the weakest possible ...
1
vote
1answer
56 views

Monomorphisms of torsion-free sheaves induce monomorphims of their determinant line bundles.

I've been trying to understand the following proof in Kobayashi's book. Let me state some relevent definitions. Let $X$ be a complex manifold and $\mathscr{O}_X$ its structure sheaf. A coherent ...
2
votes
2answers
78 views

Not Following Serre's Argument: Extension/Restriction of a Sheaf, Continuity

I am working through the proof of proposition 5 in Section 5: Extension and restriction of a sheaf in FAC by Serre. FAC can be found in english here, and in particular this question arises on page 11,...
0
votes
0answers
10 views

Reference: working with coherent sheaves

I am about to study coherent sheaves and their derived category. I know something about the general theory of coherent sheaves, but I was looking for a book that focused more on "playing" with them, ...
1
vote
1answer
43 views

Linking the cohomology of a coherent sheaf on a curve with the cohomology of its restrictions to irreducible components

$\newcommand{\H}{\operatorname{H}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}}\newcommand{\O}{\mathcal{O}}\newcommand{\I}{\mathcal{I}}$ Let $X$ be a projective scheme of dimension one over ...
3
votes
1answer
65 views

Torsion free sheaf on $\Bbb P^2$

Let $F$ be a coherent torsion-free sheaf on $\Bbb P^2$ and $L \subset \Bbb P^2$ be a line. Assume that there is an isomorphism $f : F_{|L} \to \mathcal O_L^r$ for some $r \in \Bbb N$. Questions : 1)...
2
votes
1answer
77 views

Regular global sections of invertible sheaves

$\newcommand{\L}{\mathcal{L}}$ $\newcommand{\ox}{\mathcal{O}_X}$ Let $X$ be a projective scheme of dimension one over a field $k$ and let $\L$ be an invertible sheaf on $X$. What are sufficient ...
0
votes
0answers
33 views

Sufficiently large Euler-Characteristic provides zero first cohomology of coherent sheaf?

$\newcommand{\F}{\mathcal{F}}$ Let $X$ be a projective curve over $k$, $k$ a field. Let $\F$ be a coherent sheaf on $X$. Is there a bound $b(X,\F) \in \mathbb{Z}$ depending on $X$ and $\F$ such that ...
0
votes
1answer
41 views

Are $u_1,u_2,\cdots,u_n$ independent in $M$?

Suppose $X=\operatorname{Spec}A$ and $A$ is Noetherian, $M$ is a $A$-module and the $\mathcal O_X$-module $\widetilde M$ is coherent. For some $x\in X$, if $\widetilde{M}_{x}$ is free of rank $n$ on $\...
0
votes
0answers
48 views

tensoring invertible sheaf by function field

Let $X$ be an integral scheme with function field $K$. Let $\mathcal{L}$ be an invertible sheaf on $X$, and let $\mathcal{L}_K = \mathcal{L} \otimes_{\mathcal{O}_X} K$, where $K$ is the constant sheaf ...
3
votes
1answer
101 views

Counter-examples for quasi-coherent, coherent, locally free and invertible sheaves

I'm trying to find at least one counter-example for each of these concepts to feel more comfortable with understanding the ideas behind them but I cannot even get started :( Please help me find ...
2
votes
0answers
19 views

Inverse image of meromorphic connections

Let $X$ be a complex manifold and $D$ a divisor on $X$. Let $f : Z \rightarrow X$ be a morphism of complex manifolds and assume that $f^{-1}D$ is a divisor on $Z$. A meromorphic connection on $X$ ...
2
votes
0answers
16 views

Support of a section of a coherent module on a complex manifold

Let $X$ be a complex manifold of dimension $n$, and $M$ be a coherent $\mathcal{H}_X$-modules. Let $s \in M$ be a section, and $h \in \mathcal{H}_X$ be a holomorphic function on $X$. I want to prove ...
0
votes
0answers
36 views

Morphism of sheaves of $\mathcal{O}_X$-modules

Let $(X,\mathcal{O}_X)$ be a ringed space and $\mathcal{O}_X$ coherent over itself i.e a complex space or an algebraic variety. Let $M$ be a coherent $\mathcal{O}_X$ module and $N=\bigcup N_i$ a $\...
1
vote
1answer
113 views

Prove Projection Formula

$\newcommand{\H}{\operatorname{Hom}{}}$ $\newcommand{\HH}{\mathscr{H}}$ Let $f : X \to Y$ a quasi-compact separated morphism of schemes , F a quasi-coherent sheaf on X, $\mathcal{E}$ a locally free ...
1
vote
1answer
56 views

Tensoring locally free sheaves - global sections

Suppose you two locally free sheaves $L,M$ on an integral scheme $X$ over $k$. Is it true that the map $L(X) \otimes M(X) \to ( L \otimes M)(X)$ Is injective? The map is formally induced by the ...
0
votes
0answers
95 views

Stalk of the direct image of a locally free sheaf

Suppose $C$ is an irreducible algebraic curve, $U \subset C$ is an open subset, $M$ is a locally free $\mathcal{O}_X$-module of rank $\mathrm{n}$. Let $j : U \rightarrow C$ be the open embedding, I ...
3
votes
1answer
62 views

Rank of a locally free coherent sheaf on a not-necessarily connected scheme

If $X$ is a noetherian scheme and $\mathcal{F}$ is a coherent locally free sheaf on $X$, we define the rank of $\mathcal{F}$ over some trivialising open subset $U \subseteq X$ to be the smallest ...
2
votes
1answer
142 views

Extensions of free sheaves over a scheme

Let $X$ be scheme. Consider an extension $$ 0 \to \mathcal{O}_X^n \to E \to \mathcal{O}_X^m \to 0 $$ with $E$ a locally free $\mathcal{O}_X$-module. My question is : does $E$ have to be free ? I ...
0
votes
0answers
21 views

Coherent module over the sheaf of meromorphic functions

Let $X$ be a complex manifold with a complex hypersurface $D$. Let $\mathcal{O}_X$ be the structure sheaf of holomorphic functions of $X$ and $\mathcal{O}_X[D]$ the sheaf of meromorphic functions of $...
4
votes
1answer
91 views

Freeness of stalk Implies locally free

Let $ A $ be a Noetherian ring, and $ M $ a finitely generated $ A $ module. Suppose that $ \mathfrak { p } \in M $ such that $ M_{\mathfrak{p}} $ is free. Show that there is a $ f \in A \setminus \...