# Questions tagged [coherent-sheaves]

The coherent-sheaves tag has no usage guidance.

197
questions

**7**

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179 views

### Direct summands of a pushforward in the derived category of coherent sheaves

For a Noetherian scheme $X$, let $D^b(X)$ denote the bounded derived category of coherent sheaves on $X$.
Let $X$ be a Noetherian scheme, $i:Y \hookrightarrow X$ a closed subscheme and $F$ an object ...

**2**

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131 views

### Generic rank of proper pushforward of the trivial line bundle

Given a proper surjective morphism $f:X\rightarrow Y$ where $X$ and $Y$ are smooth projective varieties. The proper pushforward $f_!$ is the homomorphism that sends the class of a coherent sheaf $M$ ...

**1**

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86 views

### A question about self-intersecting normal crossing divisors

Let $D=D_1\cup D_2$ be a simple normal crossing (snc) divisor in a smooth complex projective variety $X$. Let $E=\mathcal{O}_X(V_1)\oplus \mathcal{O}_X(V_2)$. Then, obviousely,
$$
c(E)\equiv 1+c_1(E)+...

**0**

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**1**answer

93 views

### Support of a coherent sheaf over a fiber product scheme

I'm trying to prove the following fact which I don't know if it is true since I am not able to find a counterexample:
Let $X,S$ be two $K$-scheme of finite type with $K$ an algebraically closed field....

**4**

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143 views

### Coherent sheaves and space filling curves

This paper constructs smooth space filling curves for smooth varieties over finite fields. Let's say we are working in char $p$ on the variety $X$ then this means that there is smooth curve $C_i$ in $...

**4**

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80 views

### Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts have enough injectives?

For which smooth projective $P$ over a field there exists a bounded $t$-structure $t$ on the bounded derived category of coherent sheaves $D^b(P)$ such the heart $Ht$ of $t$ has enough injectives? ...

**7**

votes

**2**answers

714 views

### Grothendieck group generated by classes of invertible sheaves

Given a smooth, projective (complex) varieties $X$, is it true that the grothendieck group $K_0(X)$ of equivalence classes of coherent sheaves on $X$, is generated by clases of invertible sheaves i.e.,...

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135 views

### Associated point of coherent sheaf

$\DeclareMathOperator\Ann{Ann}\DeclareMathOperator\Ass{Ass}$That's a question about a proof I found in E. Sernesi's
Deformations of algebraic schemes on page 188:
The sheaf $F$ is assumes to be ...

**7**

votes

**1**answer

239 views

### Heart of a bounded $t$-structure on the derived category of coherent sheaves

Let $X$ be an elliptic curve and $D(X)$ the bounded derived category of $Coh(X)$, coherent sheaves on $X$. If $(D^{\leq 0}, D^{>0})$ is a bounded $t$-structure, then can we already say that the ...

**7**

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230 views

### What is the category of coherent sheaves on a logarithmic scheme?

I try to learn basic things on logarithmic geometry, and in particular I don't find much on the category of coherent sheaves on a logarithmic scheme: is it a notion that makes sense or differ from ...

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60 views

### Two questions regarding double short exact sequences

Two short exact sequences on the same objects is called double short exact sequence. The morphism of double short exact sequences is defined in the same way you'd expect, it is a morphism of the ...

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57 views

### Lifting section coherent sheaf restriction on $\mathbb{P}^1$

Let $\mathcal{F}$ be a rank $n$ sheaf on $\mathbb{P}^n$ given as the image of a square matrix with linear entries of size $> 2n$. In particular, we have two exact sequences:
$$ V \otimes \mathcal{O}...

**1**

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101 views

### Extending automorphism from an affine

Given a projective variety $X$ and an open affine $U$ in $X$. Is there a way to decide whether a given automorphism of a vector bundle $E$ on $U$, is the restriction of automorphism of some coherent ...

**3**

votes

**1**answer

165 views

### On locally 3-syzygy sheaves

This is a question that came up in the comments section of here. A reflexive sheaf $E$ is called "locally $3$-syzygy" if it fits into an exact sequence $0\rightarrow E \rightarrow F_1\...

**3**

votes

**1**answer

125 views

### Kernels of surjections from a vector bundle to a reflexive sheaf

Reflexive sheaves on a regular quasi-projective variety can be characterized by the following property that they are the kernel of a surjection from a vector bundle to a torsion-free sheaf. I wonder ...

**2**

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112 views

### When a reflexive sheaf is flat over its base

Let $X$ be a projective and smooth variety with a codimension 2 closed subvariety $Z$. Let $Y=X\times \mathbb{A}^2$ and $E$ a reflexive sheaf on $Y$ that is a vector bundle outside of $Z\times \mathbb{...

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90 views

### Finite resolution for non-regular schemes

Over varieties that satisfy resolution property, the fact that every coherent sheaf has a finite resolution by vector bundles is equivalent to the variety to be regular. I was wondering whether the ...

**4**

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162 views

### Which derived categories of coherent sheaves are equivalent (or “$t$-related”) to derived categories of rings?

As far as I understand, it was Beilinson who proved that the bounded derived category of coherent sheaves $D^b(\mathbb{P}^n)$ is equivalent to the bounded derived category of a certain (non-...

**4**

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89 views

### Cancellation property of vector bundles on non-proper varieties

Krull-Schmidt theorem for proper varieties over a field implies that given an isomorphism of vector bundles between $E\oplus F$ and $G\oplus F$ we can deduce that $E$ and $G$ are isomorphic. My ...

**2**

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120 views

### A question about extending vector bundles from formal neighborhood to a coherent sheaf

I have a question which probably is very straightforward but because of my lack of knowledge of formal schemes I'm asking it here. Let's assume we have a vector bundle $E$ on the formal completion $...

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56 views

### Map from the stack of coherent sheaves on a curve to the Grothendieck group

Let $X$ be a smooth, projective curve. We let $Coh(X)$ be the stack of coherent sheaves on $X$. Its Grothendieck group is $Pic(X)\times\mathbf{Z}$. Is the map
$$
Coh(X)\rightarrow Pic(X)\times \mathbf{...

**2**

votes

**1**answer

175 views

### Varieties satisfying the extension of vector bundles property

We know if we have a regular variety $X$ with $U$ an open sub-scheme such that $codim(X\setminus U)\geq 2$, then any reflexive sheaf has a unique extension from $U$ to $X$. My question is when a ...

**7**

votes

**3**answers

797 views

### Sheaf of relative Kähler differentials intuitively

Let $f: X \to Y$ be a separated morphism between $k$-varieties or more general schemes
of finite type. The most common way in standard literature on algebraic
geometry to define the sheaf of relative ...

**2**

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60 views

### Semicontinuity of length for coherent sheaves

Given a coherent sheaf F over a noetherian scheme Y, a classical result in algebraic geometry states the upper-semocontinuity of the function sending any point $y \in Y$ to $\mathrm{dim}_{k(y)}(F \...

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173 views

### On definition of stable vector/Higgs bundle

Recall that the slope of a holomorphic vector bundle $\mathcal{E}$ over a smooth projective variety (or rather a compact Kähler manifold) $X$ is defined as
$\mu(\mathcal{E}) :=\frac{\operatorname{deg}(...

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109 views

### Use of flattening stratification (from Nitsure's construction of Hilbert and Quot schemes)

I study Nitin Nitsures paper on the Construction of Hilbert and Quot Schemes and not understand the propetry (F) completely:
In previous chapter (Embedding Quot into Grassmanian) it was
proved that ...

**0**

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**0**answers

76 views

### Locally freeness of twisted direct images $\pi_* \mathcal{F}(i)$

I have a question about a step in the proof of the
Existence of Flattening Stratification I found in
Nitsure's paper here: https://arxiv.org/abs/math/0504590 This question is closely related to Local ...

**4**

votes

**1**answer

211 views

### Yoga on coherent flat sheaves $\mathcal{F}$ over projective space $\mathbb{P}^n$

I'm reading Mumfords's Lectures on Curves on an Algebraic Surface (jstor-link: https://www.jstor.org/stable/j.ctt1b9x2g3)
and I found in Lecture 7 (RESUME OF THE COHOMOLOGY OF COHERENT SHEAVES ON
$\...

**0**

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72 views

### Hyperplane which does not contain any associated point of qc sheaf $\mathcal{F}$

I have a question about an argument on $m$-regularity
from 'Fundamental Algebraic Geometry' by Fantechi on page 114, Chapter
5.2: Castelnovo-Mumford regularity. The statement is:
Let $k$ be a field ...

**1**

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98 views

### Local freeness of $\pi_*F(r)$ from flatness of $F$

In 'Fundamental Algebraic Geometry' by Fantechi there is a lemma in section 5.3.2, page 119:
LEMMA 5.5 Let $S$ be a noetherian scheme and let $F$ be a coherent sheaf on $\mathbb{P}^n_S$. Suppose there ...

**3**

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**1**answer

395 views

### Proper and flat over $\mathbb{P}^1_{\mathbb{Z}}$ implies locally free

Let $\pi:X\to \mathbb{P}^1_{\mathbb{Z}}$ be a proper flat morphism with $X$ an integral scheme. Is $\pi_*\mathcal{O}_X$ necessarily locally free?

**3**

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**1**answer

172 views

### Algebraic vector bundles on the punctured spectrum: an exact reference for a result

Let $(R, \mathfrak m)$ be a Noetherian local ring of depth at least $2$. Let $X=Spec(R)$ denote the affine- scheme with structure sheaf $\mathcal O_X$ and $U=Spec(R)\setminus \{\mathfrak m\}$ be the ...

**0**

votes

**1**answer

250 views

### Completed stalks of the pushforward of the structure sheaf

Let $\pi:X\to S$ be a proper morphism of Noetherian schemes. Is it possible that $\pi_*\mathcal{O}_X\neq \mathcal{O}_S$ but the natural map $\mathcal{O}_s^\wedge\to (\pi_*\mathcal{O}_X)_s^\wedge$ is ...

**2**

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56 views

### Does direct image via proper map preserve coherence of unbounded complexes?

As for the title, I'm considering a proper map $f : X \rightarrow Y$ of Noetherian schemes and I'm trying to understand whether the direct image $Rf_{\ast} : D_{qc}(X) \rightarrow D_{qc}(Y)$ sends the ...

**3**

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102 views

### Organizing mirror pairs

At a maximally vague and naive level, mirror symmetry asks the following question: given a complex manifold $(X, I)$, is there a symplectic manifold $(M, \omega)$ and an equivalence between the ...

**4**

votes

**2**answers

365 views

### Serre's theorem on global generations on stacks

Let $X$ be a quasi-projective scheme, the followings are quite useful.
Every coherent sheaf is globally generated after tensoring with a suitable line bundle.
Every coherent sheaf has trivial ...

**1**

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**0**answers

108 views

### Correct reference for a proposition in a paper of Kapranov-Vasserot

In the paper "Kleinian singularities, derived categories and Hall algebras" Math. Ann. 316 (2000) of Kapranov-Vasserot, the authors write in page 569 that the complex $\mathcal{L}'$ (defined in p.568) ...

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125 views

### Reference request: Singular curves

I'm interested in coherent sheaves on a singular curve.(For example, global dimension, Serre duality, Riemann-Roch's theorem for singular curves,etc....)
I find treatment of it only in Hartshorn's ...

**3**

votes

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147 views

### Bass theorem on non-affine scheme

A famous theorem of Bass tells that over a noetherian ring $A$, with $\operatorname{Spec}(A)$ connected, every projective module of infinite type is free.
Now, consider a connected noetherian scheme $...

**3**

votes

**1**answer

224 views

### A question on the proof of $D^b(coh(X))\simeq D^b_{coh}(Qcoh(X))$

Proposition 3.5 of "Fourier-Mukai Transforms in Algebraic Geometry" by Huybrechts claims that the is an equivalence of categories
$$
D^b(coh(X))\overset{\sim}{\to} D^b_{coh}(Qcoh(X))
$$
where $D^b(coh(...

**3**

votes

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155 views

### Do we have $D^b_{coh}(X)\simeq D^b(coh(X))$ for a compact complex manifold $X$?

Let $X$ be a compact complex manifold and $\mathcal{O}_X$ be the structure sheaf of holomorphic functions. We call a sheaf of $\mathcal{O}_X$-module $\mathcal{F}$ coherent if it satisfies the ...

**8**

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**1**answer

299 views

### Does the sheaf $\mathcal{O}^*$ on a complex manifold have an acyclic cover?

Let $X$ be a complex manifold and let $\mathcal{O}^*$ be the sheaf nonvanishing holomorphic functions on it. Does it have an acyclic cover? That is, a cover for which all open sets and all ...

**4**

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265 views

### Chern classes of torsion-free sheaves

Let $X$ be a smooth projective variety and $Z$ a closed subvariety of co-dimension $k$. The first $k-1$ chern classes of the ideal sheaf of $Z$ vanishes and the $k$-th chern class is given by ...

**4**

votes

**1**answer

511 views

### Injectivity of pullback composed with pushforward

Let $\phi:X \to Y$ be a projective/proper, birational morphism between complex algebraic varieties, with connected fibers and $\phi_*\mathcal{O}_X \cong \mathcal{O}_Y$. Suppose further that $X$ is a ...

**6**

votes

**1**answer

516 views

### Grothendieck-Verdier duality without the noetherian condition

The Grothendieck-Verdier duality:
$$
Rf_*\big(R\mathcal{H}\textit{om}_X^\bullet(\mathcal{E}^\bullet,f^!\mathcal{F}^\bullet)\big) \cong R\mathcal{H}\textit{om}^\bullet_Y(Rf_*\mathcal{E}^\bullet,\...

**1**

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191 views

### Devissage lemma (Mumford's & Oda's AG II)

This question is part II of my proof reading of Lemma of devissage
from Mumford's & Oda's Algebraic Geometry II, findable on page 81; Theorem 6.12:
Theorem 6.12 (“Lemma of devissage”). Let $K$...

**5**

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174 views

### Coherent cohomological dimension and affine morphisms

For simplicity, all varieties in this question are quasiprojective varieties over an algebraically closed field of characteristic $0$.
The coherent cohomological dimension $cd(X)$ of a variety $X$ is ...

**2**

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233 views

### Gluing for derived category of coherent sheaves

Let $X$ be a scheme and assume $X=U \cup V$ for two affine schemes $U_0$ and $U_1$. If $\mathcal F'$ and $\mathcal F''$ are some (coherent) sheaves on $U$ and $V$ respectively such that $\mathcal F'|_{...

**2**

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167 views

### Derived category of coherent sheaves with a codimension $\geq$ 1 support

Let $X$ be some smooth algebraic variety. I would like to understand the relation between the following two categories:
$D^b_{cd,1}\text{Coh}(X) \subset D^b\text{Coh}(X)$: the full subcategory of the ...

**2**

votes

**1**answer

329 views

### Push-forward of flat module under a finite, flat morphism

Let $f:X \to Y$ be a finite, faithfully flat morphism of noetherian, affine $\mathbb{C}$-schemes. One can assume $Y$ is non-singular. Let $A$ be a local artinian $\mathbb{C}$-algebra and $f_A:X_A \to ...