# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $H^i(X,\mathcal{F}... 1answer 105 views ### are the global sections of a flat sheaf over a discrete valuation ring a free module? Let$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$where$...
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-...
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 ...