Questions tagged [coherent-sheaves]

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7
votes
0answers
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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0answers
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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0answers
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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1answer
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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0answers
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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0answers
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
2answers
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.,...
0
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0answers
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
1answer
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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0answers
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 ...
1
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0answers
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 ...
1
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0answers
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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0answers
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
1answer
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
1answer
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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0answers
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{...
1
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0answers
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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0answers
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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0answers
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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0answers
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 $...
2
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0answers
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
1answer
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
3answers
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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0answers
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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0answers
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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0answers
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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0answers
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
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1answer
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 $\...
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0answers
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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0answers
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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1answer
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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1answer
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
1answer
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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0answers
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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0answers
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
2answers
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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0answers
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) ...
2
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0answers
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
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0answers
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
1answer
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
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0answers
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
votes
1answer
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
votes
0answers
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
1answer
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
1answer
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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0answers
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
votes
0answers
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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0answers
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
votes
0answers
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
1answer
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 ...