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.

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Serre FAC book by McLennan

I am planning to read Serre's FAC paper. I came to know that there is a book/notes by Andy McLennan, containing the commutative algebra background as well as an english translation of the FAC (Ravi ...
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Examples of base-change of torsion-free sheaves that pick up torsion

Let $f:X \to Y$ be a flat, projective morphism between integral schemes over $\mathbb{C}$. Assume futher that for every $y \in Y$, the fiber $X_y:=f^{-1}(y)$ is normal and integral. Let $F$ be a ...
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An exact sequence for coherent sheaves on $\mathbf{P}^n_k$

Let $k$ be a field and $\mathscr{F}$ a coherent sheaf on $\mathbf{P}^n_k$. In paragraph $5.2$ of Fundamental Algebraic Geometry, it is claimed that if $H\subseteq\mathbf{P}^n_k$ is a hyperplane which ...
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Why is the dualizing sheaf $\omega_X$ locally free of rank 1 at generic points for a scheme $X$ satisfying $G_0$ and $S_2$?

In his paper Generalized Divisors and Biliaison Hartshorne states that if $X$ is a noetherian, embeddable (say projective) scheme of pure dimension which satisfies the conditions $S_2$ (Serre ...
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$M_x$ is free $\Rightarrow \widetilde{M}$ is locally free at $x$ [duplicate]

Let $X=\text{Spec}(A)$ where $A$ is noetherian. Suppose $M$ is a finitelly generated $A$-module and that $M_x$ is a free $A_x$-module with finite rank for some $x\in X$. Show that there exists an open ...
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Trying to understand the statement of Nakayama's lemma for coherent modules in Mumford' red book

Here is the statement of a version of Nakayama's lemma in Mumford's red book. Let $X$ be a noetherian scheme, $F$ a coherent $O_X$-module and $x \in X$. If $U$ is a neighbourhood of $x$ and $a_1, \...
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Question about the proof of “direct image sheaf of coherent sheaf is coherent”

Let $f:Y\rightarrow X$ be a finite morphism of noetherian schemes. Let $\mathcal{F}$ be a coherent $\mathcal{O}_{Y}$-module. Then $f_{*}\mathcal{F}$ is a coherent $\mathcal{O}_{X}$-module. Let $\{U_{...
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Representative in $Dcoh(X)$

I heard that any element in the derived category of coherent sheaves of a complex manifold $X$ can be "represented" by a complex of holomorphic vector bundles. If this is accurate where can I find a ...
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When two Algebraic vector bundles on a Noetherian quasi-affine scheme are equal in $K_0$ of the scheme

Let $X$ be a (connected) Noetherian scheme and $K_0(X)$ denote the Grothendieck group of the category of Algebraic vector bundles (coherent sheaves that are locally free and of constant rank ( as $X$ ...
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64 views

A bundle pull-back along itself

Let $X$ be a scheme, $\mathcal{E}$ a locally free $\mathcal{O}_X$-module of finite rank and $p: E\to X$ the corresponding geometric vector bundle (with global sections $\mathcal{E}$). Do we have an ...
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Twisted sheaves in Hartshorne

I have a few questions concerning twisted sheaves as defined in Hartshorne, II.15. 1) Let $X = \text{Proj}(S)$ for a graded ring $S$. I do understand how Hartshorne defines the sheaves $\mathcal O_X(...
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Deformation of extensions

I am trying to work on a deformation theory problem, but I don't have much experience in it, so any reference or insight would be much appreciated. Given two coherent sheaves $F$ and $G$ on a ...
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85 views

Compatibility of tensor product and external tensor product in algebraic K-theory

Let $X$ be a smooth, compact, quasi-projective, complex algebraic variety and let $K(X)=K_0(X)=K_0(\mathrm{Coh}(X))$ be the Grothendieck group of coherent sheaves on $X$. There are several notions of ...
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Pushforward as integral functor for Azumaya varieties

Consider $(X,\mathcal{A}_X)$ and $(Y,\mathcal{A}_Y)$ two Azumaya varieties over a field $k$. Recall that an Azumaya variety is the data of a variety and a sheaf of semisimple $\mathcal{O}_X$-algebra $\...
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Weak embedded resolution (from Resolution of Singularities by Kollar)

I have a question about a point in the proof of Theorem 1.52 from Janos Kolloar's Lecture on Resolution of Singularities (page 33): THEOREM 1.52 (Weak embedded resolution, I). Let $S_0$ be a smooth ...
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Sections of very ample line bundle

Let $f: C \to D$ a dominant morphism which is not an isomorphism between two irreducible, reduced, projective curves $C,D$ over an alg closed field $k$ (unsure if algebraically closedness is ...
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When is the image of a line bundle again a line bundle

Hello everybody Motivation of my question Let $X$ be a scheme. Given a morphism $\mathcal{L}\overset{\beta}\to\mathcal{O}_X$ of line bundles over $X$. I want to understand under what conditions the ...
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Blow up and morphism of locally free sheaves

I would like to know if what I say below makes sense. Let $X$, $Y$ be smooth projective varieties, $Y \subset X$ and $\pi: \widetilde{X} \longrightarrow X$ the blowing-up of $X$ along $Y$. We know ...
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73 views

Injective map on a coherent sheaf on a projective scheme must be an automorphism

Given a coherent sheaf $E$ on a projective scheme $X$ over a field and an endomorphism $f:E \rightarrow E$, show that if $f$ is injective then it is an isomorphism. Give a counterexample to this ...
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Tautological bundle: algebraic geometry vs topology

I'm going to compare the two construction of twisted sheaf/bundle $\mathcal{O}(1)$ from algebraic and topological viewpoint: 1) Algebraic construction (Hartshorne's Algebraic Geometry, p. 117): ...
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All Harder-Narasimhan factors of $E$ are semistable with slope $\leq B \in \mathbb{R}$ implies $E$ semistable

I'm trying to show the following claim: Let $E$ be a vector bundle on a surface $X$, of slope $\mu(E) = B$ such that it's Harder-Narasimhan factors are $\mu$-semistable of slope $\leq B \in \mathbb{...
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The existence of Hom fibration between families of coherent sheaves

Let $S$, $X$ be algebraic varieties, and suppose $X$ is a protective smooth curve. And let both $\mathcal{G}$, $\mathcal{G}'$ be families of coherent sheaves on $X$ parameterized by $S$, i.e. $\...
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How S.E.S of these coherent sheaves induces S.E.S of these vector spaces?

Let, $E$ be a coherent sheaf on a projective variety $X$ over $\mathbb C$ and $f \in Hom (E,E)$ be such that $kerf,Imf$ are nontrivial proper subsheaves of $E$.Also let $V \subset H^0(E)$ be a fixed ...
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Is pullback of ample sheaf by formal completion ample again?

Suppose $X$ is a Noetherian scheme and $I$ a quasi-coherent sheaf if ideals. We can formally complete the original scheme to get a formal scheme and a morphism of locally ringed spaces $k:\mathcal{X} \...
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Methods for resolutions of sheaves on projective schemes

$\newcommand{\QCoh}{\mathsf{QCoh}} \newcommand{\mod}{\text{-} \mathsf{Mod}} \newcommand{\oh}{\mathcal{O}}$If I want to find a resolution of sheaves on an affine scheme $(X = \mathrm{Spec}(R), \oh_X)$ (...
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About Castelnuovo-Mumford Regularity

I asked the following question here in our forum: (How to calculate this cohomology? ) Proposition (1): Let $\mathcal{F}$ be be coherent sheaf on $\mathbb{P}^{n}$ and let $E$ be a locally free sheaf ...
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Sheaves morphism on Blowing up Variety.

Definition: A codimension r distribution $\mathcal{F}$ on a smooth complex manifold X is given by an exact sequence: $$0 \longrightarrow T_{\mathcal{F}} \longrightarrow T_{X} \longrightarrow N_{\...
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Construct a coherent subsheaf of a quasi-coherent sheaf

I am trying to prove the following statement Given a Noetherian scheme $X$ and a surjective morphism of quasi-coherent sheaves on $X$: $$\mathcal{F}\xrightarrow{f} \mathcal{G} \to 0$$ where $\...
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Cohen-Macaulay sheaves in Picard group of Cohen-Macaulay schemes

Let $X$ be a Noetherian, integral, separated, CM (Cohen-Macaulay) scheme. Is it true that the set $\{ [L] \in Pic (X) : L$ is CM $ \}$ is finite ? If this is not true in general, then what if we ...
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More confusion about the definition of smooth morphisms of schemes

Let $f: X \rightarrow Y$ be a finite type morphism of noetherian schemes with $x \in X$ and $f(x) = y$. Then $f$ is smooth at $x$ if, 1) $f$ is flat at $x$; 2) $\Omega_{X/Y}$ is locally free of rank ...
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Lemma of devissage from Mumford's AGII

In Mumford's & Oda's Algebraic Geometry II, on page 81 the authors give a proof for 6.12: Lemma of devissage. after a carefully reading of the proof, I failed to understand an argument: Theorem ...
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219 views

How is the degree defined in this case?

By Daniel Huybrechts, we have: Definition: Let $E$ be a coherent sheaf of dimension $d = \text{dim}X$. The degree of $E$ is defined by: $$\text{deg}(E) = \alpha_{d-1}(E) - \text{rk}(E).\alpha_{d-1}(\...
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313 views

Blow up and relationship between tangents sheaves

Let $X = \mathbb{P}^{n}$ and $Y \subset X$ a smooth subvariety of $X$. Let us consider the blowup morphism of $X$ along of $Y$, denoted by $\pi : \widetilde{X} \longrightarrow X$ with exceptional ...
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Noetherian $R$-algebra corresponds to a coherent sheaf of rings on $\operatorname{Spec}(R)$

Let $R$ be a ring and $A$ a Noetherian $R$-Algebra. Let $\newcommand{\m}{\mathcal} \m{A} = \tilde{A}$ be the corresponding $\m{O}_X$-Module, where $(X, \m{O}_X) = \operatorname{Spec}(R)$. I would like ...
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Conormal and Tangent Sheaves of a Distribuition in $\mathbb{P}^{n}$ when $n = 2$ and $n = 3$

Definition 1): A codimension one distribution of degree $d \geq 0$ in $\mathbb{P}^{n}$ is given by an exact sequence: $$\mathscr{F}: 0 \longrightarrow T_{\mathscr{F}} \longrightarrow T\mathbb{P}^{n}\...
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Degree of a coherent sheaf after a blow up

Let $Y$ be a smooth projective scheme and $X$ a projective subscheme of $Y$. Let us consider the blowup morphism of $Y$ along $X$, denotated by $\pi_{X} : \widetilde{Y} \longrightarrow Y$. Let $\...
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Restriction and extension of scalars for cohomology groups on a projective scheme

I am going over the proof of Serre duality for a coherent sheaf on a projective k-scheme. First, I am trying to understand it for the case where $X = \mathbb{P}_{k}^{n}$. I am following Hartshorne III ...
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Euler-Poincare characteristic for relative schemes

Let $X\to S$ a morphism of Noetherian schemes. Assume that $\mathcal F$ is a coherent sheaf on $X$ with the following property: the support of $\mathcal F$ is proper over a subscheme of $S$ of ...
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101 views

Quasi-coherencity of the annihilator ideal sheaf of the sheaf associated to an A-module M

I am trying to find an example which shows that the annihilator ideal sheaf, denoted by $\mathrm{Ann}(\mathcal F)$, of a quasi-coherent sheaf $\mathcal F$ on a locally-noetherian scheme $X$, is not ...
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When does the direct image functor respect direct sums of sheaves?

Let $f: X \rightarrow Y$ be a morphism of schemes, where $Y = \text{spec}A$ is affine. Let $\mathcal{L}$ be an invertible sheaf on $X$. Is it true that the direct image functor $f_{*}$ respects ...
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Direct sums of invertible sheaves commuting with global sections

I am looking at the Stacks Project's treatment of the functor of points for projective space. Let's restrict to the case that $S$ is a graded ring, generated by $S_{1}$ as an $S_{0}$ algebra. The ...
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Is the inverse image exact on locally free sheaves?

Let $f : X \to Y$ be a morphism of noetherian $k$-schemes. Under what condition is the functor $f^* : \textbf{Coh}\ Y \to \textbf{Coh} \ X$ exact on locally free finite rank sheaves?
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Relating the sheaf associated to a cyclic module over an affine scheme to the structure sheaf

Let $(X,\mathcal O_X)$ be the affine scheme of a commutative Noetherian ring $R$. Let $J$ be an ideal of $R$. From the sheaf $\mathcal O_X$ and the closed subset $Z:=V(J)$ of $X$ , how do we recover ...
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Quot-like scheme for torsion sheaves

I am wondering if, as the Quot schemes parametrizes flat (quotients of) sheaves over schemes, there is anything similar for torsion sheaves. In first approximation, if $I$ is a sheaf of ideals over a ...
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316 views

Pullback of locally free sheaves is locally free

Lemma 17.4.3 states that if $f:X \rightarrow Y$ is a morphism of ringed space, $G$ is a locally free $O_Y$-module, then $f^*G$ is a locally free $O_X$ module. Suppose that $G$ is a locally free $O_Y$...
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Broad question on morphisms of stalks of quasi-coherent sheaves on schemes

This question was inspired by reading about a criterion for a morphism into projective space (over an algebraically closed field) to be a closed immersion based on local rings. It got me thinking ...
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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}...
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105 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 $...
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1answer
364 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-...
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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 ...

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