Questions tagged [coherent-sheaves]
The coherent-sheaves tag has no usage guidance.
252
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is the category of coherent sheaves some kind of abelian envelope of the category of vector bundles?
This might be obvious to experts, but I'm not sure where to look for the answer. On a reasonably nice, at least noetherian, scheme (or variety, algebraic space, stack), can the category of coherent ...
- 1,152
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3
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Are schemes that "have enough locally frees" necessarily separated
Let me motivate my question a bit.
Thm. Let $X$ be a locally noetherian finite-dimensional regular scheme. If $X$ has enough locally frees, then the natural homomorphism $K^0(X)\longrightarrow K_0(X)...
- 9,417
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1
answer
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Do I know what "coherent sheaf" means if I know what it means on locally Noetherian schemes?
I've been trying to convince myself that "coherent sheaf" is a natural definition. One way I might be satisfied is the following: for modules over a Noetherian ring $A$, coherent and finitely ...
- 115k
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votes
2
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Does module Hom commute with tensor product in the second variable?
Let $A$ be a commutative ring, and $L, M, N$ be $A$-modules. Then is it true that
$$\text{Hom}_A (L, M)\otimes_A N \cong \text{Hom}_A (L, M\otimes_A N)$$
as $A$-modules?
(Note that there is a ...
- 1,856
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1
answer
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When do real analytic functions form a coherent sheaf?
It is known that, in general, the sheaf of real analytic functions on a real analytic manifold is not coherent. However, there are some examples, where we have coherence: for example, if $X$ is a ...
- 1,233
17
votes
1
answer
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Cohomology of real analytic coherent sheaves
Let $M$ be a real analytic variety
(if someone is concerned about distinction between
"real analytic spaces" and "real analytic varieties"
in real analytic geometry, let's assume that $M$
is both "...
- 9,095
17
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0
answers
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Is there an approach to Gabber's theorem from the singular support of coherent sheaves?
David BZ told me that the old theory of singular support of $D$-modules fits into the new theory of singular support of coherent sheaves, via the derived loop space. I wonder how to reconcile that ...
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2
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What are the merits of the different finiteness conditions on quasi-coherent sheaves?
It's my understanding that there's no disagreement about the right way to define a quasi-coherence for a sheaf $F$ of $O_X$-algebras (over a scheme, locally ringed space, or even locally ringed topos)....
- 9,288
14
votes
1
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Which category of sheaves on a manifold remembers the manifold?
Given a not too nasty topological space $X$, the category of sheaves of sets on $X$ remembers $X$.
Given a scheme $S$, the category of quasicoherent sheaves on $S$ remembers $S$.
Given a smooth ...
- 42.5k
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2
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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 ...
- 6,000
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0
answers
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Why do people study unbounded derived category of quasi-coherent sheaves rather than focus on bounded derived category of coherent sheaves?
Let $X$ be a scheme and let $D_{qoch}(X)$ and $D^b_{coh}(X)$ be the unbounded derived category of quasi-coherent sheaves and bounded derived category of coherent sheaves on $X$, respectively.
$D^b_{...
- 8,707
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votes
3
answers
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Derived categories of (coherent) sheaves of modules: exceptional images, gluing, and proper descent?
I am interested in the properties of (the derived categories) of various categories of (coherent) sheaves of modules (over varieties). I would like to understand in what extent these properties are ...
- 16.5k
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1
answer
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got any tricks to build up t-structures on derived categories?
Are there any good tricks to construct a heart of a t-structure? (I'm thinking on the derived category of coherent sheaves of some variety)
I'll start with the only one I know. If $(T,F)$ is a ...
- 1,265
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votes
3
answers
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Is it true that if the pushforward of a coherent sheaf is locally free, then the original sheaf is locally free?
I think the title says it all. If I have a finite map $p:X\to Y$ between schemes, and $F$ is a coherent sheaf on $X$ such that $p_*F$ is locally free, can I conclude that $F$ is locally free?
...
- 44k
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3
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Is the pushforward of a line bundle on the smooth locus of a terminal singularity again a line bundle?
In algebraic geometry, it is a sad fact of life that pushforward doesn't preserve being a coherent sheaf; for example, the pushforward by the complement of a divisor of the structure sheaf (or more ...
- 44k
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2
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Lemma 1 from Beilinson's "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", intuition?
Consider Lemma 1 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows.
Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, $F: \mathcal{...
user61522
10
votes
1
answer
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Characterization of schemes whose dualizing complex is perfect
I'm wondering if there is a characterization of schemes over a a field $k$ whose dualizing complex is a perfect complex in terms of singularities. E.g. on a proper Cohen-Macauley scheme over a field, ...
- 4,982
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1
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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 ...
- 16.5k
9
votes
1
answer
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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 ...
- 4,091
9
votes
1
answer
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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 ...
- 8,707
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1
answer
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Reference request: tangent space to moduli space of coherent sheaves is $\operatorname{Ext}^1(E, E)$
Is there a standard reference for the fact that, in an appropriate algebraic-geometrical context, the tangent space at the point $[E]$ to the moduli space $\mathcal M$ is something like $\operatorname{...
- 1,990
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votes
2
answers
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modular forms, invertible sheaves, and quotients
I'm very confused about some contradicatory statements, and I hope someone can help me clarify this.
Let $\Gamma$ be a congruence subgroup. It is well known that modular forms of weight $k$ for $\...
- 597
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votes
2
answers
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coherent sheaves on affine formal schemes
Let $\hat{X} = \text{Spf} \hat{A}$ be obtained as the formal completion of an affine scheme $X = \text{Spec} A$ where $A$ is an adic noetherian ring. Given a coherent sheaf $\mathfrak{F}$ on $\hat{X}$,...
- 81
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1
answer
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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,\...
- 914
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votes
1
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Progress on Bondal–Orlov derived equivalence conjecture
In their 1995 paper, Bondal and Orlov posed the following conjecture:
If two smooth $n$-dimensional varieties $X$ and $Y$ are related by a flop, then their bounded derived categories of coherent ...
- 305
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1
answer
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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(\...
- 4,246
8
votes
1
answer
969
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Proper morphism sending coherent to coherent
Hello,
Is there a proof that the push forward by a proper morphism of Noetherian schemes sends coherent sheaves to coherent ones, without passing in the argument through projective morphisms?
Thank ...
- 5,492
8
votes
1
answer
483
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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 ...
- 1,188
8
votes
0
answers
258
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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 ...
- 10.5k
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votes
0
answers
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Def-Obs theory of sheaves with fixed determinant on CY3.
Let $\mathcal{E}$ be a stable sheaf on a smooth complex projective threefold $X$ and $Ext^k_0(\mathcal{E},\mathcal{E})$ be the traceless Ext groups, defined by the kernel of the trace map
$$
Ext^k(\...
- 81
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votes
2
answers
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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.,...
- 2,195
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2
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English reference for the Grauert–Riemenschneider vanishing theorem
What is the best reference in English for the following theorem of Grauert–Riemenschneider:
Theorem:
Let $\phi:X \to Y$ be a proper bi-rational morphism of algebraic varieties over characteristic $0$...
- 2,581
7
votes
2
answers
2k
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Coherent sheaves and holomorphic vector bundles
For a complex manifold $M$, I'm trying to understand (from a differential geometry point of view) what its category of coherent sheaves is. If I understand correctly, then the sheaf of holomorphic ...
- 1,442
7
votes
1
answer
591
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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 ...
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Can one prove vanishing of higher direct images fiber-wise?
Let $\pi:X\to Y$ be a proper map of algebraic varieties (over $\mathbb C$) which is a bi-rational equivalence.
are the following statements equivalent?
The derived direct image of $O_X$ is $O_Y$.
...
- 2,581
7
votes
1
answer
851
views
Is the bounded derived category of coherent sheaves of a variety a small category?
The question is in the title.
I am trying to apply the Mitchell (Freyd-Mitchell?) embedding theorem, which states that for every small abelian category $A$, there exists a ring $R$ such that A ...
- 2,198
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votes
1
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When is a general sheaf (on the projective plane) globally generated?
Let $v$ be a chern character on $\mathbb P^2$ so that the moduli of sheaves of chern character $v$ is non-empty of the expected dimension. When is it true that the general sheaf in moduli is globally ...
- 1,469
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0
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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 ...
- 323
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2
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Deformations of sheaves via automorphisms. How to express $Ext^1$?
Let $X$ be a complex manifold (for example $\mathbb CP^n$), let $v$ be a holomorphic vector field on $X$, and let $F$ be a coherent sheaf (for example a vector bundle or a structure sheaf of a point). ...
- 28.8k
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Question on condition for a sheaf to be locally free in Orlov 2004
In "Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models", Orlov twice mentions the following criterion for a sheaf $P_1$ to be locally free:
If for all closed points $t:x ...
- 841
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2
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506
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Global sections of coherent sheaves on determinantal hypersurfaces in $\mathbb{P}^n$
Let us consider the short exact sequence of coherent sheaves on $\mathbb{P}^n$ $$0 \to \mathcal{O}_{\mathbb P^n}(-1)^{r} \stackrel{N}{\longrightarrow} \mathcal{O}_{\mathbb P^n}^{r} \longrightarrow \...
- 65.1k
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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 ...
- 2,126
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1
answer
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Algebraicity of the stack of coherent sheaves
I am trying to understand the proof of Theorem 4.6.2.1 in the book on algebraic stacks by Laumon and Moret-Bailly. The setting is this: $S$ is a Noetherian scheme, $f\colon X \rightarrow S$ is a ...
- 825
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votes
1
answer
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When is a sheaf coherent if its image under a Fourier-Mukai transform is coherent?
Let X and Y be to varieties and $F\colon D\mathrm{QCoh}(X) \to D\mathrm{QCoh}(Y)$ a continuous functor between the corresponding unbounded derived categories of quasi-coherent sheaves (given by a ...
6
votes
1
answer
452
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Is the Quot-scheme over non-singular curve reduced
Let $k$ be an algebraically closed field, $C$ a non-singular projective curve over $k$ of genus at least $2$ and $\mathcal{F}$ a locally free sheaf on $C$. Let $r,d$ be two integers satisfying $\...
- 1,949
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Do general sheaves on P^2 have cohomology governed by their Euler characteristic?
Suppose $\xi$ is chern character on $\mathbb P^2$. Then there is a moduli space $M(\xi)$ of semistable sheaves of chern character $\xi$.
If $\xi$ has Euler characteristic 0, then apparently there is ...
- 1,469
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votes
0
answers
277
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Is there a sheaf of categories $\text{QCoh}_X(1)$ analogous to $\mathcal{O}_X(1)$?
Given a scheme $X$ and sum of divisors $D$, you can take the line bundle
$$\mathcal{O}_X(D)\ =\ \{\text{functions }f\text{ with [conditions on zeroes/poles]}\}\ \subseteq\ j_*\mathcal{O}_\eta\ =\ \...
- 5,555
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votes
0
answers
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Obstruction to the existence of global resolution of coherent sheaf
It is well known that any coherent sheaf on a complex manifold (or more generally on some complex spaces) admits locally a resolution with locally free sheaves. It is also well known that for non-...
- 830
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2
answers
763
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Is there a presentation of the cohomology of the moduli stack of torsion sheaves on an elliptic curve?
Let $E$ be your favorite elliptic curve, and let $Tor^m$ be the moduli stack of torsion sheaves of degree $m$ on $E$. This sounds horrible, but it's not so bad; it's a global quotient of a smooth ...
- 44k
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1
answer
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Pushforward maps for cohomology of coherent sheaves
Let $X$ be a smooth projective algebraic variety over a field $k$, of dimension $n$, and let $Z$ be a smooth closed subvariety of dimension $m$, with $i: Z \hookrightarrow X$ the inclusion map.
For ...
- 36.1k