Questions tagged [coherent-sheaves]

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Finer classification of semistable sheaves

Usually in the moduli space of semistable sheaves, two semistable sheaves correspond to one point if and only if they are S-equvialent, i.e. the graded objects associated to their Jordan-Holder ...
Display Name's user avatar
1 vote
0 answers
111 views

Support of sections of flat coherent sheaf over a base

Suppose $\pi\colon M\times N\to N$ is the (flat) projection of complex analytic spaces with connected fibers and suppose that $\mathcal{F}$ is a coherent sheaf of modules on $N$. Is it true that ...
Thomas Kurbach's user avatar
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1 answer
199 views

Derived pushforward of a projection

Given two smooth projective varieties, $X,Y$, consider their derived categories $D^b(X), D^b(Y)$. Let $\mathcal{F}$ a complex of coherent sheaves in $D^b(X \times Y)$, why the derived pushforward of ...
Abel 's user avatar
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0 answers
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Are $\mathcal{O}_X$-modules "more actual" then quasicoherent sheaves in some sense?

In the Stacks project and in a book of Brian Conrad the "main" derived category of a scheme is the one of $\mathcal{O}_X$-modules. I would like to understand whether $D(\mathcal{O}_X)$ is ...
Mikhail Bondarko's user avatar
8 votes
1 answer
719 views

Is the functor from the unbounded derived category of coherent sheaves into the derived category of quasi-coherent sheaves fully faithful?

Let $X$ be a Noetherian scheme. Is the obvious functor $D(\operatorname{Coh}(X))\to D(\operatorname{QCoh}(X))$ fully faithful? If this is true then $D(\operatorname{Coh}(X))$ is equivalent to the full ...
Mikhail Bondarko's user avatar
1 vote
1 answer
142 views

flatness of restriction of structure sheaf over ring of global sections

Let $X$ be an affine scheme. $U \subseteq X$ open. Then I want to show that $\mathcal{O}_X(U)$ is flat over $\mathcal{O}_X(X)$. But I want to prove it only by knowing the definition of structure sheaf ...
Hamed Khalilian's user avatar
1 vote
1 answer
103 views

Families of torsion-free sheaves whose length jumps

For a long time, I had a false belief that the space/stack $\text{Coh}^{tf}_{c_1,c_2}S$ of torsion-free sheaves $\mathcal{E}$ on a smooth algebraic surface $S$ was not connected, since if you take its ...
Pulcinella's user avatar
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2 votes
1 answer
206 views

Maps from $\mathbb A^1/ \mathbb G_m$ to Coherent sheaves

I am reading this paper https://arxiv.org/abs/1608.04797 Let $\Theta$ be the stack $\mathbb A^1/{\mathbb G_m}$. Let $X$ be a smooth projective curve of genus $g$ over a field $k$. Let $Coh_P$ be the ...
angry_math_person's user avatar
1 vote
0 answers
59 views

Understanding coherent sheaf obtained via sheaf injections of holomorphic vector bundles on TCP^1

My problem involves holomorphic vector bundles $E,F$ of the same rank on $T\mathbb{C}P^1$. I have a short exact sequence of sheaves $$0\rightarrow E\rightarrow F\rightarrow Q\rightarrow 0.$$ I want to ...
AlgGeoNoob's user avatar
4 votes
2 answers
524 views

Basic question on projective bundles

Let $\mathcal{E}$ be a coherent sheaf on an irreducible scheme $S$ ($S$ can be pretty nice, say noetherian of finite type), and let $\mathbf{P}(\mathcal{E}):=\mathrm{Proj}(\mathrm{Sym}(\mathcal{E}))$ ...
rfauffar's user avatar
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1 answer
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Resolving complexes of coherent analytic sheaves

Background Throughout, let $X$ be a smooth complex manifold. It is a classical fact that a coherent analytic sheaf admits a local resolution by locally free sheaves (also known as a local syzygy). ...
Tim's user avatar
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2 votes
1 answer
133 views

Could certain closed covering determine a coherent sheaf?

We know that a coherent sheaf on a scheme is determined by its restriction on certain open coverings (satisfying compatibility condition). Now I wonder how about a closed covering. To do so I started ...
user avatar
3 votes
0 answers
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Explicit example of wall-crossing for sheaves

I would like to see an explicit example of a coherent sheaf $\mathcal{E}$ on a projective complex threefold $X$ which crosses the wall of stability. That is, I would like some $1$-parameter family $\...
Quaere Verum's user avatar
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0 answers
77 views

Analogous tensor product operation for reflexive sheaf

Suppose now $(X,\mathcal O_X)$ is a normal complex space, and $\mathcal F$ is a coherent analytic sheaf on it. Product the reflexive sheaf $$\mathcal F^{[p]}:=(\mathcal F^{\otimes p})^{**},$$ where $\...
Invariance's user avatar
1 vote
0 answers
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Chern class of rank one sheaves supported on subvarieties

Let $X$ be a smooth, quasi-projective variety of dimension $n$ and $\mathcal{F}$ be a globally generated coherent sheaf supported on a codimension two subvariety $V \subset X$. Is $c_2(\mathcal{F}) \...
Chen's user avatar
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1 answer
280 views

Are the tensor-invertible coherent sheaves on an algebraic space (Zariski) locally free of rank one?

On a scheme, the coherent sheaves that are invertible objects for the tensor product (monoid) operation are precisely the coherent sheaves that are (Zariski) locally free of rank one. Is the same ...
Jason Starr's user avatar
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0 answers
189 views

Sheaves of abelian groups over a smooth projective variety

Can someone point some good reference (books or lecture notes) for these topics: Let $X$ a smooth projective variety over an algebraically closed field Sheaves of abelian groups over $X$ Quasi-...
Abel 's user avatar
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3 votes
1 answer
257 views

Existence of rigid objects in the derived category of a smooth projective variety

Let $X$ be a smooth projective variety (say over $\mathbb{C}$). An object $F \in D^b(X)$ is said to be rigid if $\mathrm{Ext}^1(F,F)=0$. I was wondering if we can always find a rigid object on a ...
Libli's user avatar
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Stability of sheaves of non-constant rank

Let $E\to X$ be a coherent sheaf over a compact (projective) Kahler manifold. The definition of stability of sheaves as stated in Huybrechts-Lehn (Definition 1.2.12) says that $E$ is stable if for all ...
BinAcker's user avatar
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3 votes
0 answers
286 views

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} \...
Cranium Clamp's user avatar
2 votes
0 answers
162 views

Understanding spaces is the same as understanding (sheaves of) functions on the space

I'm trying to understand Ravi Vakil's FOAG. In chapter 2 it is written: [...] understanding spaces is the same as understanding (sheaves of) functions on the spaces, and understanding vector bundles (...
Abel 's user avatar
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3 votes
1 answer
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Is this a true weakening of the quasi-coherence property?

Let $R$ be a commutative Noetherian ring, and $X=$Spec$(R)$ the associated affine scheme. Let $F$ be a sheaf of $O_X$-modules. Consider the following condition (#) For all containments $V \subseteq ...
Neil Epstein's user avatar
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3 votes
1 answer
258 views

Can we recover the sheaf from the functor?

Let $k$ be an algebraically closed field of characteristic zero. Let $S$ be a scheme of finite type over $k$. Let $\mathrm{Sch}/S$ be the category of schemes of finite type over $S$. Let $\mathcal F$ ...
Display Name's user avatar
4 votes
0 answers
117 views

Connectedness of moduli spaces of semistable sheaves on K3

In 1987/88, Mukai described the moduli spaces of (semi-)stable sheaves on a K3 surface $X$, showing that they consist of smooth pieces $M(v)$ of dimension $\langle v , v \rangle + 2$, for every ...
Urs Schreiber's user avatar
1 vote
0 answers
154 views

Projectivization in the derived category of coherent sheaves

Let $X$ be a compact Kahler manifold. There exists a notion of projectivization of holomorphic vector bundles and coherent sheaves over $X$. Does that concept extend to objects in the derived category ...
BinAcker's user avatar
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3 votes
0 answers
172 views

For a family of short exact sequences of coherent sheaves, can we define the splitting subscheme?

This question has been asked in SE. 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 ...
Display Name's user avatar
4 votes
1 answer
170 views

Higher direct image of coherent sheaf and rigid analytification

Let $k$ be a non-archimedean field of characteristic zero. Then let $$f:X \rightarrow Y$$ be a (proper) morphism of smooth projective varieties over $k$. The GAGA functor (for rigid analytic spaces) ...
KKD's user avatar
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0 answers
180 views

Projectivization of a coherent sheaf using resolution by vector bundles

Let $\mathcal{F}\to X$ be a coherent sheaf over a compact Kahler manifold and let $E^{\bullet}\to \mathcal{F}$ be a resolution of $\mathcal{F}$ by holomorphic vector bundles. Is there a way to ...
BinAcker's user avatar
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3 votes
1 answer
245 views

When is a sheaf $\mathcal{L}_1 \subset \mathcal{F} \subset \mathcal{L}_2$ sandwiched between two line bundles also a line bundle?

This question is in the interest of answering one part of this question, but I think it is distinct enough to warrant a separate question. Let $X$ be a regular 2-dimensional Noetherian scheme, for ...
PrimeRibeyeDeal's user avatar
3 votes
1 answer
249 views

Is local freeness open for curves?

Let $X$ be a complete nonsingular curve and $S$ a scheme over $k$ algebraically closed, and $\cal{F}$ a coherent sheaf on $X \times S$, generated by finitely many global sections and flat over $S$ (...
nolatos's user avatar
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5 votes
1 answer
311 views

Which complexes of coherent sheaves can be presented as countable homotopy limits of perfect complexes?

Let $X$ be a noetherian scheme (actually, I need the case where $X$ is proper over an affine scheme), $C$ is an object of the derived category $D_{coh}(X)$ of coherent sheaves on $X$. Under which ...
Mikhail Bondarko's user avatar
2 votes
0 answers
186 views

Stability and simplicity of tangent sheaf of Grassmannian

Everything is over the complex numbers. Let $ X = \text{Gr}(k,n) $ be a Grassmannian variety and with tangent sheaf $ T_X $. (1) Is $ T_X $ simple, i.e. is $\text{Hom} ( T_X, T_X) = \mathbb{C} $? (2)...
Cranium Clamp's user avatar
2 votes
0 answers
148 views

Higgs quotient sheaf of a Higgs bundle

Let $X$ be a smooth complex projective variety of dimension $n\geq2$, let $\mathfrak{E}=(E,\varphi)$ be a Higgs bundle over $X$ of rank $r\geq2$. Does exists a Higgs quotient sheaf $\mathcal{Q}$ of $\...
Armando j18eos's user avatar
2 votes
1 answer
173 views

Semi-orthogonal decompositions over singular schemes

Where can I find any more or less explicit semi-orthogonal decompositions of derived categories of perfect complexes or of bounded derived categories for singular schemes that are proper over a ring R?...
Mikhail Bondarko's user avatar
3 votes
0 answers
138 views

Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts are categories of modules?

I am interested in $P$ that is smooth and proper over a field and such that the derived category of coherent sheaves $D^b(P)$ possesses a $t$-structure whose heart is the category of finitely ...
Mikhail Bondarko's user avatar
8 votes
1 answer
663 views

Relation between ProCoh and solid modules

There are two languages endow the theory of coherent sheaves with a six functor formalism (that I "know" of), one being formulated in $\text{ProCoh}(X)$ by Deligne and the other being $D(\...
curious math guy's user avatar
3 votes
2 answers
327 views

When are two resolutions of a coherent sheaf 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 ...
BinAcker's user avatar
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1 vote
0 answers
79 views

Is the resolution of a sub-sheaf into complex of holomorphic vector bundles a "sub-resolution"?

Let $F\rightarrow X$ be a coherent sheaf on a projective Kahler manifold. We can resolve it into a complex of holomorphic vector bundles $E^{\bullet}\rightarrow X$. Let $G\subset F$ be a subsheaf of $...
BinAcker's user avatar
  • 747
1 vote
0 answers
253 views

Stability of vector bundles and corresponding coherent sheaf

Let $j:Y\hookrightarrow X$ be an embedding of projective complex manifolds. Let $E\rightarrow Y$ be a vector bundle and $S=j_*E$ the corresponding coherent sheaf on $X$ (see Push forward of a Vector ...
BinAcker's user avatar
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9 votes
1 answer
307 views

Does $X\times Y$ have the resolution property if both $X$ and $Y$ have?

We say a complex manifold $X$ has the resolution property if every coherent sheaf $\mathcal{M}$ on $X$ admits a surjection $\mathcal{E}\twoheadrightarrow \mathcal{M}$ by some finite rank locally free ...
Zhaoting Wei's user avatar
  • 8,427
4 votes
0 answers
79 views

Derived category supported in a Serre subcategory of a locally noetherian category

This is a cross-post from math.stackexchange at https://math.stackexchange.com/questions/4251692/derived-category-supported-in-a-serre-subcategory-of-a-locally-noetherian-catego, since I didn't get ...
mnm's user avatar
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3 votes
1 answer
144 views

Dual family of torsion-free rank-1 sheaves on Gorenstein curves

Let $X$ be a Gorenstein curve over a field an consider the compactified Jacobian parametrizing torsion-free, rank-1 sheaves on $X$. Is there a chance that the dual functor $Hom(\_, \mathcal O_X)$ ...
Raffaele C's user avatar
8 votes
0 answers
240 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 ...
naf's user avatar
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2 votes
0 answers
212 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 $M$ ...
user127776's user avatar
  • 5,607
1 vote
0 answers
138 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)+...
Mohammad Farajzadeh-Tehrani's user avatar
0 votes
1 answer
145 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....
John117's user avatar
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4 votes
0 answers
153 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 $...
user127776's user avatar
  • 5,607
4 votes
0 answers
133 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? ...
Mikhail Bondarko's user avatar
7 votes
2 answers
892 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.,...
user45397's user avatar
  • 2,013
7 votes
1 answer
479 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 ...
user12344321's user avatar

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