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20 views

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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19 views

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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0answers
59 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
39 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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0answers
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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2answers
87 views

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
41 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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1answer
49 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 ...
3
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0answers
35 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 ...
3
votes
1answer
51 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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0answers
32 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 ...
3
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1answer
105 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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0answers
27 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_* ...
4
votes
0answers
63 views

Help understanding the proof of a theorem about Cohomology of Vector Bundles

I am trying to understand a paper called Betti tables of graded modules and cohomology of vector bundles, but i am stuck in Proposition 6.8 which states: Let $\mathcal{E}$ be a vector bundle on ...
2
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2answers
68 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 ...
2
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0answers
46 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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0answers
69 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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0answers
59 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
37 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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22 views

Reference request: Order of a vector bundle

Please could you link me to an accessible reference/set of lecture notes on the definition of the order of an algebraic vector bundle? Google just shows some very complicated definition which is ...
2
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2answers
86 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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0answers
88 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
61 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
73 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 ...
3
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0answers
140 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 ...
2
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0answers
38 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
69 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. ...
3
votes
1answer
75 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 ...
3
votes
1answer
180 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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0answers
60 views

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 ...
2
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1answer
54 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
50 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
141 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
99 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 ...
4
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0answers
48 views

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 ...
2
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2answers
201 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?
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1answer
58 views

Associated sheaf to a $\mathbb{C}$-module

In section II.5 Hartshorne describes the notion of a sheaf $\tilde{M}$ associated to a module M over a ring A. Now in the case $A:= \mathbb{C}$, we have $\operatorname{Spec}(\mathbb{C}) = {0}$. ...
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0answers
66 views

$\operatorname{Hom}(f^*F, G) = \operatorname{Hom}(F, f_*G)$

For topological spaces $X, Y$ and the continuous morphism $f \colon Y \to X$, consider the sheaf $F$ on $X$ and $G$ on $Y$. A very famous formula says that ${\mathrm{Hom}_{\mathrm{sh}}}(f^*F,G) = ...
2
votes
1answer
286 views

Reflexive sheaves and determinants

Let $X$ be a complex manifold. I just recall some notions, for my question below to be clear. Let $F\in\textrm{Coh}(X)$ be coherent of rank $r$. If it is torsionfree, it has a well-defined ...
2
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1answer
229 views

Pushforward commutes with external tensor product?

Let $f:X\rightarrow X'$ and $g:Y\rightarrow Y'$ be morphisms of varieties. Let $\mathcal F$ be a coherent sheaf on $X$ and $\mathcal G$ be a coherent sheaf on $Y$. Is it true that $$(f\times g)_* ...
3
votes
0answers
46 views

When does one have $f_\ast(Im(a))=Im(f_\ast a)$?

Let $f:X\to Y$ be a morphism of schemes, and $a:F\to G$ a morphism of coherent sheaves on $X$, with image $Im(a)=E\subseteq G$. I am trying to understand under which conditions one has ...
3
votes
0answers
62 views

Coherent sheaves on $\mathbb P^r$ of very high and very low dimension

Let $\mathcal F$ be a coherent sheaf on $\mathbb P^r$, the $r$-dimensional projective space over an algebraically closed field $k$. The support of $\mathcal F$, namely $$\textrm{Supp }\mathcal ...
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0answers
50 views

Questions about the details in the construction of virtual fundamental class

Let $\pi :D \subset \mathcal{X} \to S$ be a flat family of stable curves of genus $g$ with marked points $D$. Let $\mathcal{X} \to X$ be a flat family of stable morphism in the sense of Kontsevich ...
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0answers
202 views

Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both Möbius strips?

This question follows up this previous question, which has an accepted answer that I am having trouble believing. (Edit: the answer has now been fixed; thanks Georges!) I have been working on giving ...
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0answers
106 views

Example of a coherent sheaf on an open subset where extension isn't trivial?

Motivation: I am working on problem II.5.15 in Hartshorne's Algebraic Geometry, which is to prove that, given a noetherian scheme $X$, an open subset $U\subset X$, and a coherent sheaf $\mathscr{F}$ ...
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2answers
279 views

Serre duality for curves, the other statement.

Here's a question from someone who's just found out what Serre duality (in the case of curves) is. It occurs to me that the popular statement which can also be interpreted as the Riemann-Roch theorem ...
1
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1answer
194 views

Homological dimension of skyscraper sheaf

Let $X$ be a smooth variety, $x$ a closed point, and $\mathcal{O}_x$ the skyscraper sheaf of the residue field $k(x)$ at $x$. Recall that the homological dimension of a complex of coherent sheaves ...
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1answer
50 views

Why is any rank $1$ sheaf always stable?

Let $X$ be a projective scheme over $\mathbb{C}$. For a sheaf on $X$, $$ p_E(d)=\chi(X,E(d)) $$ be the Hilbert polynomial of $E$. A sheaf $E$ on $X$ is siad to be stable if for every proper subsheaf ...
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0answers
89 views

Characterization of Hilbert schemes of points

Let $X=\operatorname{Proj}(R)$ be a projective scheme with a $k$-algebra $R$. Let $\mathcal{M} \in \operatorname{Coh}(X)$ be a coherent sheaf on $X$ whose Hilbert polynomial $\chi(\mathcal{M})$ is a ...
5
votes
1answer
50 views

Testing local freeness on curves

Let $X$ be a smooth variety (over an algebraically closed field, if it makes a difference), and $\mathscr{F}$ a coherent sheaf on $X$. I have heard it claimed that $\mathscr{F}$ is locally free if and ...