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. (Def: http://en.m.wikipedia.org/wiki/Coherent_sheaf)

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Why did Serre choose coherent sheaves?

First thing - I don't know any algebraic geometry. I'm trying to understand a little bit about quasi-coherent sheaves but not for the sake of AG, so please rely on as little knowledge as possible. ...
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56 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 - ...
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27 views

subsheaf of free sheaves

Let $X$ be an irreducible nodal curve, $E:=\oplus_{i=1}^r \mathcal{O}_X$ be a free sheaf on $X$ and $F \subset E$ a (coherent) subsheaf. Is it possible to write $F$ as a direct sum of subsheaves ...
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27 views

Riemann-Roch for nodal curves

Let $X$ be an irreducible, nodal curve and $E$ a coherent subsheaf of a free sheaf $\oplus_{i=1}^r \mathcal{O}_X$ on $X$ of rank strictly less than $r$. Assume that $r \ge 2$. It follows that $H^0(E)$ ...
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79 views

K-theory of projective space

Is there any way to prove that the twisting sheaves $\mathcal{O}(K)$ generate the algebraic K-theory of projective space without actually using any K-theory machinery (e.g. Bott periodicity)? Like for ...
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27 views

A question on very ample line bundle and closed immersion

Let $X$ be a projective scheme, $i:X \hookrightarrow \mathbb{P}^n$ a closed immersion, $\mathcal{L}:= i^*\mathcal{O}_{\mathbb{P}^n}(1)$ a very ample line bundle. Let $j:{\mathbb{P}^n} \hookrightarrow ...
4
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1answer
74 views

Additivity of the first Chern class

I have a highly elementary question: is the first Chern class additive? More specifically, given a short exact sequence of coherent sheaves on a nonsingular curve $X$ $$ 0 \to \mathscr F'\to \mathscr ...
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22 views

Locally free sheaves on reducible curves and their subsheaves

Let $X$ be a reducible (but reduced), connected, projective curve with at worst nodal singularities and $\mathcal{F}$ be a locally free sheaf of rank $r$ on $X$. Suppose that $\mathcal{F}'$ is a depth ...
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38 views

Globally generated vector bundle

Could someone give me a definition of globally generated vector bundle? A rapid search gives me the definition of globally generated sheaves, but I am in the middle of a long work and don't really ...
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1answer
65 views

Computing the sheaf of 1-forms on a toric variety

Consider projective space $P^{2}$ and its corresponding fan. We have the affine opens defined by $U_{\sigma_{0}} = Spec(\mathbb{C}[x,y])$, $U_{\sigma_{1}} = Spec(\mathbb{C}[x^{-1},x^{-1}y])$ and ...
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1answer
67 views

Negative degree invertible sheaves on non-singular varieties have no global sections

Let $X$ be a non-singular projective complex variety and $\mathcal{L}$ be an invertible sheaf on $X$ with negative degree. Is it true that $\mathcal{L}$ has no global sections? If so, can someone ...
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47 views

Are trivial vector bundles on curves semistable?

Let $C$ be an irreducible projective curve with at worst nodal singularities. Let $E$ be the trivial locally free sheaf of rank $r$ i.e., $E$ is the direct sum of $r$ copies of the trivial line ...
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1answer
42 views

If $F$ a sheaf and $S\subset F$ a subfunctor, then $S$ is a subsheaf if and only if…

This is Proposition 1 from Maclane & Moerdijk's Sheaves in Geometry and Logic, part II, section 1. Proposition 1. Let $F$ be a sheaf on $X$ and $S\subset F$ a subfunctor. $S$ is a subsheaf if ...
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2answers
76 views

Duality and Serre's criterion

Let $X$ be a projective scheme, $\mathcal{F}$ a coherent sheaf on $X$ which is $S_2$. Then under what additional conditions is its dual, $\mathcal{H}om_X(\mathcal{F},\mathcal{O}_X)$ also $S_2$?
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24 views

Reflexive sheaves on stable curves-II

This is an extension of Reflexive sheaves on stable curves. Let $C$ be a stable curve and $\mathcal{F}$ a reflexive sheaf on $C$ supported on the whole of $C$. Is the projective dimension of ...
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1answer
31 views

Reflexive sheaf on normal surfaces

Let $X$ be a normal, projective scheme of pure dimension $2$ and $\mathcal{F}$ is a reflexive coherent sheaf on $X$. Is $\mathcal{F}$ locally free?
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Dévissage for complex manifolds

In algebraic geometry one has the following result: Let $X$ be a noetherian scheme and $\mathcal{F}$ a coherent sheaf with support $Z \neq X$. Then $\mathcal{F}$ has a finite filtration $\mathcal{F} ...
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Etale locally free sheaf is always locally free in Zarissky topology.

I'm trying to solve exercise III.10.5 from Hartshorne "Algebraic geometry". Let $\mathcal F$ be a coherent sheaf on a scheme $X$ locally free in 'etale topology, namely for any $x \in X$ there is an ...
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71 views

Hartshorne notation in section III.12

I am reading section III.12 in Hartshorne, the one about the Semicontinuity Theorem. For $f:X \rightarrow Y$, where $Y=\mathrm{Spec}A$ and $\mathcal{F}$ a coherent sheaf on $X$, he writes ...
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1answer
44 views

Action of $Aut(X)$ on $Coh(X)$

I was reading about Bondal and Orlov reconstruction theorem. In particular that for a smooth variety with ample or anti-ample canonical bundle $\mathrm{Aut}(D^b(X)) \cong \mathbb{Z} \times ...
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40 views

derived versions of natural isomorphisms

I have just recently started approaching the topic of derived categories in algebraic geometry, and I'm doing so reading Huybrechts "Fourier-Mukai transforms in algebraic geometry". I have a doubt ...
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Perfect complexes and the derived category of a smooth projective variety

I know that on a smooth projective variety any coherent sheaf has a finite locally free resolution. I read somewhere that this implies that any object in $D^b(X)$ for $X$ smooth projective is then ...
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1answer
52 views

Tensor product of coherent sheaf with the stalk of structure sheaf

If $A$ is a commutative ring, $M\in A\text{-mod}$ and $\mathfrak{p}$ is a prime ideal of $A$ then it is an easy fact from commutative algebra that $M\otimes_{A}A_{\mathfrak{p}}\cong M_{\mathfrak{p}}$. ...
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52 views

Definition of a $\mathcal{O}(a,b)$?

Can any one tell me what is the definition of this notation $\mathcal{O}(a,b)$. I know $\mathcal{O}(a)= \widetilde{S}(a)$ for some ring $S$. Can $\mathcal{O}(a,b)$ be defined in the same way. thanks ...
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48 views

Property of pullback of quasi-coherent sheaves

In Hasrtshorne the pullback $f^{*}\mathcal{F}$ of a sheaf $\mathcal{F}$ on $Y$ via a map $f:X \rightarrow Y$ is defined as $f^{-1}\mathcal{F}\otimes_{f^{-1}\mathcal{O}_Y}\mathcal{O}_X$. It is quite ...
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57 views

Projective dimension of a coherent sheaf in a short exact sequence

Let $X$ be a noetherian integral scheme. We define the projective/homological dimension of a torsion free coherent sheaf $E$ to be $\mathrm{dh}(E)= \sup\{dh(E_x)|x\in X\}$, here dh$(E_x)$ denotes the ...
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34 views

Basic algebraic geometry

Given an algebraic variety $X$ and two $Q$-Cartier divisors $D_1$ and $D_2$. Given $f \in H^0(X, \mathcal{O}_X(D_1))$ and $g\in H^0(X, \mathcal{O}_X(D_2))$. It is always true that $\frac{g}{f}$ is a ...
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1answer
122 views

Degree of a torsion-free subsheaf

Suppose that $R$ is a torsion-free subsheaf (of positive rank) in another torsion-free sheaf $S$, on a smooth complex projective variety $X$. If $S$ is (slope) semistable, is it true that the degree ...
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34 views

Relative flat vs flat direct image

Let $Y$ be a Noetherian scheme. Let $\mathscr{F}$ be a coherent sheaf on $\mathbb{P}^n \times Y$. Denote $\pi: \mathbb{P}^n \times Y \rightarrow Y $ canonical projection. We have two notions. $\pi_* ...
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69 views

Cohomology groups of coherent sheaves for very small and very big twists.

Let $\mathcal{F}$ be nonzero coherent sheaf over the projective space $\mathbb{P}_k^n$. The Serre vanishing Theorem says that $h^i \mathcal{F}(d)=0$ for $i>0$ and $d\gg 0$. I am wondering if it is ...
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50 views

Question about Tate resolution and cohomology groups of nonzero coherent sheaves.

Let $\mathcal{F}$ be a nonzero coherent sheaf on the projective space $\mathbb{P}_{k}^m$. I am trying to show that for every integer $d$ there is $j$ for which $h^j\mathcal{F}(d-j) \neq 0$. My ...
3
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71 views

Criterion of flatness for projective morphism

Let $f: X \to Y$ be a projective morphism, $\mathcal O(1)$ is a relatively very ample sheaf, then f is flat iff $f_*\mathcal O(m)$ is locally free for big $m$. I can prove flatness implies that ...
2
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68 views

Coherent sheaf on reduced scheme is free on dense open set

It should be well-known fact, but I couldn't find this in Hartshorne's "Algebraic Geometry", Mumford-Oda or Ravi Vakil's Lecture notes. Let $X$ be a reduced connected scheme and $\mathcal F$ is a ...
3
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1answer
42 views

Construct natural transformation $u^* R^i f_* \rightarrow R^i g_* v^*$ without assumption of quasi-coherence

I am reading Hartshorne Algebraic geometry. Chapter 3 Proposition 9.3 (in particular remark 9.3.1). It states that if we have commutative diagram in category of schemes (namely morphisms $f, g, h, u$ ...
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2answers
93 views

Stalks of the sheaf $\mathscr{H}om$?

The question is basically the title. What are the stalks of the sheaf $\mathscr{H}om_{\mathscr{O}_X}(\mathscr{F},\mathscr{G})$? If $X$ is a noetherian scheme and $\mathscr{F}$ is coherent, then ...
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118 views

Comparing two definitions of determinant of coherent sheaves

Let $f:X \to S$ be a smooth, projective morphism of $k$-schemes for some field $k$. Let $\mathcal{F}$ be a coherent sheaf on $X$ flat over $S$. We know (by Proposition $2.1.10$ of Huybrechts-Lehn, ...
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1answer
83 views

Extension of coherent sheaf from open subspace.

I'm trying to solve following problem: Let $X$ be a noetherian scheme, $U$ an open subspace of $X$, $\mathcal F \in Qcoh(X), \mathcal G\in Coh(U), \mathcal G \subset \mathcal F|_{U},$ then there ...
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1answer
80 views

Does the canonical morphism commute with direct image functor?

I am trying to prove the representability of the Quotient functor. I have the following problem. Let $\phi \colon T \to S$ be a morphism of noetherian schemes and let $F$ be a coherent sheaf on ...
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210 views

Rank, degree and slope of a general coherent sheaf

Let $(X,\mathcal O_X)$ be a ringed space and $\mathcal F$ be a coherent sheaf of $\mathcal O_X$-modules on $(X,\mathcal O_X)$. Are there the definitions of rank, degree and slope of $\mathcal F$ in ...
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39 views

G-equivariant invertible sheaves on affine curves

Let $A$ be a Noetherian integral domain, and $G$ a finite group of automorphisms acting on $A$. Let $B = A^G$, the ring of invariants. The inclusion $B \hookrightarrow A$ induces a surjective morphism ...
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1answer
73 views

Question on the definition of sheaves.

When defining a sheaf of $O_X $-modules, or sheaves in general, I have nearly always seen it given as a functor from the category of all the open sets to another category with the usual properties. ...
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1answer
82 views

Base-points and invertible sheaves

Once again I am confused after thinking too much about something I thought I already understood... Let $\mathcal{L}$ be an invertible sheaf on a smooth projective curve $X$ such that $\deg ...
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1answer
251 views

Torsion sheaves on a curve

This is probably a silly question, but I'm a bit confused. Regarding exercises 6.11 and 6.12 of Chapter II of Hartshorne: Let $X$ be a nonsingular projective curve over an algebraically closed field ...
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Book Suggestion - Complex algebraic surfaces

I am studying for an exam of algebraic geometry, in particular, I am dealing with ruled surfaces and numerical invariants, rational surfaces, Castelnuovo's Theorem and its application. I am reading ...
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1answer
58 views

Very ampleness of pullbacks of twisting sheaves

Suppose we are given $X =\mathbb{P}^1_k\times_{k}\mathbb{P}^1_k$ with its canonical projections $\pi_i$ to $\mathbb{P}^1_k$. I'd like to show that ...
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1answer
55 views

Vanishing of local cohomology groups

Let $k$ be a field and let $X$ be a smooth separated $k$-variety. Let $T$ be a closed integral subscheme of $X$ of generic point $\eta$. The object of interest here is the local cohomology group $$ ...
4
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1answer
178 views

Pullback of an ample line bundle through a projection

Assume $X_1$ and $X_2$ are two smooth projective curves and let $M_i$ be an ample line bundle on $X_i$, for $i=1,2$. Further, denote the natural projection map on the $i$-th factor by ...
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1answer
120 views

When is the morphism between global sections surjective

Let $C$ be a smooth projective irreducible curve. Let $Z$ be a closed subscheme of $C$ consisting of a finite set of points. Denote by $i:Z \to C$ the closed immersion. Note that this is a proper ...
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Extension of morphism of Coherent sheaves over the projective space

Let $\mathcal{F}_1, \mathcal{F}_2$ be coherent sheaves over $\mathbb{P}^n_{\mathbb{C}}$ for $n \ge 3$. Denote by $U_i$ the fundamental affine schemes defined by the non-vanishing of the coordinates ...
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237 views

Source: Coherent locally free sheaves and projective modules

What is a good and very quick and concise article for the proof of the equivalence of the categories of locally free sheaves on $\mathrm{Spec}(A)$ and finitely generated projective $A$-modules?