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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Does a torsion-free coherent sheaf embed into a locally free sheaf?

Let $ X $ be a Noetherian integral regular scheme and $ \mathcal{F} $ be a torsion-free coherent sheaf. (One definition of torsion-free is that the natural map $ \mathcal{F} \rightarrow \mathcal{F} \...
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Rank of a coherent sheaf using resolution by vector bundles

The rank of a coherent sheaf is defined in terms of the Hilbert polynomial (See Huybrechts-Lehn 1.2.2 or Rank of a coherent sheaf in terms of coefficients of the Hilbert polynomial). Now let $\mathcal{...
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How can I prove this interpretation of right derived functor of the composition of internal hom followed by direct image?

The question comes from the following paper Lange, Herbert, Universal families of extensions, J. Algebra 83, 101-112 (1983). ZBL0518.14008. Let $f:X\to Y$ be a flat projective morphism of Noetherian ...
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“Usual argument for a bicomplex” in order to extend a cocycle

I was reading this article and I got stuck in the proof of Proposition 2.1. In particular I don't get the last paragraph of the proof when it talks about extending to a cocycle and using a similar ...
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Integral closure of the ideal sheaf is coherent

Demailly asserts on page 9 of Analytical Methods in Algebraic Geometry that the integral closure of an ideal sheaf is coherent. I don't know how to prove this. Is there any reference? We introduce ...
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Deformations of finite schemes

I am reading some texts about the tangent space to the Hilbert scheme. Apparently, $T_{[Z]}(Hilb (X)) = H^0(Z, N_{Z/X})$ for any $Z$ is a consensus which I agree with, but then regarding the hilbert ...
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For a family of short exact sequences of coherent sheaves, can we define the splitting subscheme?

Let $k$ be an algebraically closed field of characteristic zero. Let $X$ be a projective scheme over $k$. We can talk about short exact sequences of coherent sheaves on $X$. Suppose we have a family ...
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The splitting locus in a Quot scheme, is it closed or locally closed?

Assume we work over $\mathbb C$. Let $X$ be a projective scheme over $\mathbb C$ with an ample line bundle $\mathcal L$. Let $\mathcal F_1,\mathcal F_2$ be coherent sheaves on $X$. Let $P_1,P_2\in\...
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When is the reflexive hull of the conormal sheaf locally free?

Let $X\subseteq Y$ be a (singular) complex analytic subspace or closed subscheme with defining ideal $I$, where $Y$ is smooth. Stalks of the structure sheaf of $X$ are assumed to be integral ...
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Coherent sheaves of an Abelian variety

I am currently studying the very basics of Abelian varieties over a field $k$. Besides this, I am trying to understand Fourier-Mukai transforms in Algebraic Geometry. My current aim is to study the ...
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Valuative criterion for flatness of sheaves

Let $X \rightarrow S$ a morphism of schemes with $S$ reduced and Noetherian over a field. Let $\mathcal{F}$ be a coherent sheaf on $X$. To show $\mathcal{F}$ is flat over $S$ does it suffice to show ...
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Pushforwards from a projective bundle corresponding to a coherent sheaf

Consider a normal Noetherian scheme $X$. In the case I am most interested in, it's even Cohen-Macaulay. For a locally free coherent sheaf $E$ its projective bundle $\mathbb{P}(E)$ is usually defined ...
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Is local freeness open for curves?

Let $X$ be a nonsingular curve and $S$ a scheme over $k$ algebraically closed, and $\cal{F}$ a coherent sheaf on $X \times S$, flat over $S$ (via the projection). So my question is the following: is ...
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Modules arising as global sections of line bundles

All schemes and rings are Noetherian. Let $ f : X \rightarrow \operatorname{Spec} A $ be proper and $ M $ a finitely generated $ A$-module. Are there any criteria on $ A,X,f $ for there to exist a ...
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On a smooth surface, a subsheaf of a locally free sheaf with torsion free quotient is locally free.

I am reading D. Gieseker's On the moduli of vector bundles on an algebraic surface. In Lemma 4.1, the author seems to use the following result If $0\to G_2\to G_1\to F\to0$ is an exact sequence of ...
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Locally free sheaf and locally constant system

Suppose that $(M,\mathcal{O})$ is a $k$-ringed space, where $k$ is a field. By $k$-ringed I mean the structure sheaf is a sheaf of $k$-algebras with local stalks and residue field $k$. Suppose that $V$...
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Classification of coherent sheaves on $\Bbb P^1$ with a doubled point (nonseparated)

Let $X$ be the scheme obtained by gluing two copies of $\Bbb P^1_k$ along $D(x)$ (basically the projective version of the line with two origins). Is there a good classification of coherent sheaves on $...
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A condition for the validity of an identity in the derived category of bounded complexes of modules.

Let $\mathcal{R}$ be a sheaf of (not necessarily commutative) rings on a topological space $X$ and let $M$ and $N$ be $\mathcal{R}-$modules such that $M$ is quasi-isomorphic to a finite complex of ...
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For a coherent sheaf on a scheme, given an associated point, is there a subsheaf supported in the closure?

Let $X$ be a Noetherian scheme, and let $\mathcal F$ be a non-zero coherent sheaf on $X$. Let $x\in X$ be an associated point of $\mathcal F$. Can we always find a coherent subsheaf $\mathcal F'\...
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Hartshorne Exercise II.6.10 Proving existence of a filtration for coherent sheaves supported on a closed subscheme.

Fix $X$ a Noetherian scheme and $\mathscr{F}$ a coherent sheaf on $X$ whose support is contained inside some closed subscheme $Y\subseteq X$. I would like to show that there is a filtration $$ 0=\...
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Equivalence of two definitions for very ampleness

Hartshorne says an invertible sheaf $\mathcal{L}$ on $X$ is very ample relative to a field $k$ if there is an immersion $i:X\rightarrow \mathbb{P}_k^n$ for some $n$ such that $\mathcal{L}\simeq i^*\...
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Dual of locally free sheaves commute with direct sums?

For any locally free sheaf $\mathcal F$ on a scheme $(X,\mathcal O_X)$ of finite rank, its dual is defined as $\mathcal F^{\vee}:=\mathscr Hom(\mathcal F,\mathcal O_X)$. So, if $\mathcal F,\mathcal ...
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Tangent fiber space of cone and vector fibered spaces

Suppose that $M$ is a complex analytic space. In the category of analytic spaces over $M$ one can consider cone and vector objects. Which I will call cones or linear fiber spaces over $M$. Every cone ...
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Étale local universal sheaves

Moduli functors for (flat families of) semistable coherent sheaves over a projective scheme are not representable functors, which is to say that no global universal family exists. However one can ...
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Vanishing of Ext groups of Coherent sheaves over Noetherian regular scheme

Let $(X,\mathcal O_X)$ be a Noetherian regular scheme of dimension $1$. Then, for any coherent sheaf $\mathcal F$ and any quasi-coherent sheaf $\mathcal G$, it holds that $\mathcal Ext^i(\mathcal F, \...
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How is the exterior power of a coherent sheaf is coherent?

So, I was reading Hans Grauert's and Reinhold Remmert's book "Theory of Stein Spaces" and in page 13, they define the exterior power $\bigwedge^{p} \mathcal{F}$ of a sheaf $\mathcal{F}$, and ...
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How "many" non-reduced spaces with smooth reduction are there?

Suppose $(M,\mathcal{O}_M)$ is an analytic subspace of $\mathbb{C}^n$, such that its reduction is simply some $\mathbb{C}^k\subseteq \mathbb{C}^n$. How "many" such structure sheaves are ...
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Finiteness of cohomology of coherent sheaf for proper morphisms

It has been stated and proved that the Cohomology of a coherent sheaf $F$ on a closed projective subscheme X of $\mathbb P^n_A$ where A is Noetherian is finite dimensional A-module. If $f: X \to Spec ...
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Proving that a rational function is a section of an invertible sheaf

Let $S$ be a surface, $E$ be a curve on $S$, and $H$ be a hyperplane section of $S$. Let $a\in H^0(S,\mathcal O_S(H+(k-1)E))$, and $b\in H^0(S,\mathcal O_S(H+kE))$. Let $U$ be an open subset of $S$ on ...
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Embeddings of a surface that are given by linear systems of divisors

I am reading the proof of Castelnuovo's contractibility criterion in Beauville's Complex Algebraic Surfaces. I would like to clarify a paragraph. We have a hypeplane section $H$ of a surface $S$, a ...
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Hartshorne Exercise II 6.11 (c): Grothendieck group of a nonsingular curve

Exercise: Let $X$ be a nonsingular curve over an algebraically closed field $k$. (c) If ${\mathscr{F}}$ is any coherent sheaf of rank $r$(means that its stalk at the generic point has dimension $r$ ...
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Homomorphism between coherent sheaves concentrated in one point

I am reading the book “Fourier-Mukai transforms in algebraic geometry" by Daniel Huybrechts. In the proof of Lemma 4.5 in page 92, he uses Lemma: If $M$ is a finite module over a local ring $(A,m)...
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When two resolutions of coherent sheaves are homotopic

Let $\mathcal{F}$ be a coherent sheaf on a projective manifold $X$. It is well known that one can construct a resolution of $\mathcal{F}$ by holomorphic vector bundles (locally free sheaves). Are two ...
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Hartshorne problem III.4.10

I am stuck on this problem from Hartshorne and to be honest I don't really know where to start. The problem is the following: $X$ is a non-singular variety over an algebraically closed field $k$ and $\...
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Relative Riemann-Hilbert correspondence with singular fibers?

Suppose that $\phi\colon M \to N$ is a submersion of complex analytic spaces. Then the relative Riemann-Hilbert correspondence states that the category of coherent sheaves on $M$ with integrable ...
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3 votes
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Ringed space but not locally ringed space

Give an example of a ringed space which is not locally ringed space.Why we need to go to locally ringed space if all ringed spaces are locally ringed spaces before defining schemes in Hartshone?Please ...
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Is $f^*E|_{f^{-1}(y)}$ a trivial sheaf for a holomorphic map $f: X\to Y$?

Let $f: X\to Y$ be a holomorphic map between two complex manifolds, and $\mathcal{E}$ a finite rank complex holomorphic vector bundle (or even a general sheaf) over $Y$. Let $E$ be the corresponding ...
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Quartic in the projective space and exact sequence of sheaves

I am trying to understand a few basics about the twisting sheaves. I read that, given a smooth quartic $S$ in $\mathbb P^3$, we have an exact sequence $0\rightarrow\mathcal O_{\mathbb P^3}(-4)\...
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natural bijection between set of morphisms of varieties and homomorphims of k-algebras

In harshorne, proposition 3.5 we are trying to establish a bijection between the set of morphisms of varieties and homomorphims of k-algebras. $$ \alpha: \operatorname{Hom}(X, Y) \simeq \operatorname{...
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Very Ample sheaves

Let $X$ be the non singular cubic curve $y^2z=x^3-xz^2$ in projective space of dimension 2. Let $L$ be the invertible sheaf $L(P_0)$. How does $L(P_0)$ not being generated by global sections imply ...
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Dual of universal quotient bundle globally generated

We are consistent with the notation in the book of Hartshorne. Let $X=G(\mathbb P^k, \mathbb P^n)$ be the Grassmannian parametrizing $\mathbb P^k$ contained in $\mathbb P^n$. We have the so-called ...
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injective sheaves on a projective scheme cannot be coherent [duplicate]

Let $X$ be a projective scheme. If it helps (e.g. gives way to a short/elegant answer) fix a base field $k$ and assume smoothness. It is often said that the category of coherent sheaves over $X$ does ...
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2 votes
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Algebraic Geometry notation of poles

Let $X$ be a scheme (for instance an algebraic curve) over $Spec(k)$ being $k$ an algebraically closed field of characteristic $0$. Let $p\in X$ be a closed point, and let $E$ be a vector bundle of ...
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Derived category supported in a Serre subcategory of a locally noetherian category

This has now been cross-posted: https://mathoverflow.net/questions/404902/derived-category-supported-in-a-serre-subcategory-of-a-locally-noetherian-catego It is known that $Coh(\mathcal O_X)\subseteq ...
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What is the minimal number of generators for $\mathcal{O}_{\mathbb{P}^n}(1)$?

With generators for $\mathcal{O}_{\mathbb{P}^n}(1)$ I mean a set $a_0,...,a_m\in \mathcal{O}_{\mathbb{P}^n}(1)(\mathbb{P}^n)$ such that $a_0,...,a_m$ generate $\mathcal{O}_{\mathbb{P}^n}(1)_P$ at all ...
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Explicit correpsondence between ${\cal O}_{{\Bbb P}_k^1}(1) = {\cal O}_{{\Bbb P}_k^1}(x)$.

For a projective curve ${\Bbb P}_k^1$, we have a very ample line bundle ${\cal O}_{{\Bbb P}_k^1}(1)$. Geometrically, I prefer to consider ${\cal O}_{{\Bbb P}_k^1}(1)$ as the fibration \begin{equation*}...
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Linear fiber space of ideal sheaf and normal fiber space

Suppose that $\left(M, \mathcal{O}\right)$ is an analytic space and $A$ is a subspace with defining coherent ideal sheaf $J$. One can construct from any coherent sheaf $\mathcal{F}$ on $M$ the linear ...
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Chern character of a coherent sheaf over a projective manifold [closed]

I am looking for a reference describing the definition/construction of the Chern character of a coherent sheaf over a projective manifold (or variety) using a resolution by vector bundle.
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2 votes
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Invertible sheaf on a scheme is coherent

If we define invertible sheaf as a locally free sheaf of rank 1, which is the most common definition. I saw it is true that an invertible sheaf must be quasi-coherent. But why is it also coherent? ...
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Sections of Serre's twisting sheaves

My understanding of Serre's twisting sheaves is quite shaky. If $B = \bigoplus_{d\geq 0}B_d$ is a graded module, $X = \operatorname{Proj}B$, and $f\in B_1$, then I know that $\mathcal{O}_X(n)|_{D_+(f)}...
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