20
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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?
No, not always.
In
Positselski, Leonid; Schnürer, Olaf M., Unbounded derived categories of small and big modules: is the natural functor fully faithful?, J. Pure Appl. Algebra 225, No. 11, Article ID ...
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17
votes
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When do real analytic functions form a coherent sheaf?
For real-analytic manifolds, coherence of the structure sheaf always holds. The 1-sentence reason is that one can pass to real and imaginary parts on Oka's coherence theorem in several complex ...
14
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Different definition of sheaf cohomology
Taking global sections is the same thing as computing the hom from $O_X$. In other words, there is an isomorphism of functors $\Gamma(X,-)\cong\hom(O_X,-)$, so both functors have the same derived ...
- 47.3k
12
votes
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Grothendieck group generated by classes of invertible sheaves
Here is a different, perhaps more elementary example.
Consider the Grassmannian $\mathrm{Gr}(2,4)$ of lines in $\mathbf{P}^3$, and let $\mathscr{Q}$ be the universal quotient bundle. The line bundle $\...
- 2,808
10
votes
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Cohomology of real analytic coherent sheaves
In a smooth case, the reference is Proposition 2.3 in Atiyah and Hirzebruch's Analytic cycles on complex manifolds.
For a non-smooth case, I don't know the general reference, but Theoreme 3 in Henri ...
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10
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Grothendieck group generated by classes of invertible sheaves
The answer is "no" in general, even if one asks if the classes of invertible sheaves generated $K_0(X)$ as a ring; K3 surfaces provide a counterexample. For a K3 surface $X$ over $\mathbb{C}...
- 23k
10
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Converses to Cartan's Theorem B
For (4), there are finite CW complexes $X$ which are not contractible, but such that $H^q(X, \mathcal{F})$ vanishes for all (finite rank) locally constant sheaves $\mathcal{F}$. Specifically, take $X$ ...
- 156k
9
votes
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Grothendieck-Verdier duality without the noetherian condition
Does one have the duality in this setting?
Yes, we do have duality in a very general setting. Your question is equivalent to asking for the existence of a right adjoint to the derived pushforward ...
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9
votes
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Is there a non-semistable simple sheaf?
Let $C$ be a smooth curve of genus at least $3$. Then $C$ has a degree $1$ line bundle $L$ which is not effective, and by Riemann-Roch $$h^1(L) = -\chi(L) = g-2 > 0.$$ We have $$Ext^1(O_C,L) \cong ...
- 5,857
8
votes
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Question on condition for a sheaf to be locally free in Orlov 2004
The question is local, so it is enough to show that if $A$ is a Noetherian local ring with maximal ideal $\mathfrak{m}$ and $M$ is a finitely generated module such that $Ext^i(M,A/\mathfrak{m}) = 0$ ...
- 39.3k
8
votes
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A non-rational variety with a full exceptional collection?
Rationality of a variety with a full exceptional collection is a well-know folklore conjecture. In some form a similar open question is mentioned in the paper of Brown and Shipman "The McKay ...
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8
votes
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Locally ringed space with noetherian stalks and a non-coherent structural sheaf
Welcome new contributor. It is best to think about these kinds of examples on one's own. Thus, please try this for yourself before reading the following.
Let $(Y,Z)$ be any pair of a Noetherian ...
8
votes
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Does $X\times Y$ have the resolution property if both $X$ and $Y$ have?
Please find below a short argument in the case of schemes. The answer is positive for algebraic spaces too; in that case it can be proven using the characterization: $X$ has the resolution property $\...
- 1,144
8
votes
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When are two resolutions of a coherent sheaf homotopic
If this were true, then any short exact sequence of vector bundles would split. Indeed, if $0 \to \mathscr E_1 \to \mathscr E_2 \to \mathscr E_3 \to 0$ is a short exact sequence of vector bundles, ...
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8
votes
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Is local freeness open for curves?
To elaborate on Jason's comment: consider a morphism of schemes $f:\mathrm{X}\rightarrow\mathrm{S}$, and an $f$-flat coherent sheaf $\mathscr{F}$ on $\mathrm{X}$. Then
(1) the singular locus $\mathrm{...
- 2,808
8
votes
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Are the tensor-invertible coherent sheaves on an algebraic space (Zariski) locally free of rank one?
There are counterexamples by Stefan Schröer. One of them is not locally separated (a bug-eyed cover, as Kollár calls it), another is a (non-normal) proper surface. See the paper here.
About the link: ...
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8
votes
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Basic question on projective bundles
This can fail when $\mathscr E$ has torsion:
Example. Let $k$ be a field, let $R = k[t]$ with maximal ideal $\mathfrak m = (t)$, and let $S = \operatorname{Spec} R$. Take $\mathscr E = R\oplus R/\...
- 28.7k
7
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Heart of a bounded $t$-structure on the derived category of coherent sheaves
One can construct t-structures on the bounded derived category of coherent sheaves on a smooth projective curve (or higher-dimensional variety) by tilting, see Bayer's notes, Prop. 3.6.1, and the ...
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7
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When are two resolutions of a coherent sheaf homotopic
No. The simplest example is given by the following two resolutions of the structure sheaf of a point $P \in \mathbb{P}^1$:
$$
0 \to \mathcal{O}_{\mathbb{P}^1}(-1)
\to \mathcal{O}_{\mathbb{P}^1}
\to \...
- 39.3k
7
votes
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Smooth analogue of Cartan's Theorem B
It seems like Gro-Tsen's comment pretty much answers your main question, doesn't it? If "simple" = $C^\infty$ manifold, and "nice" = sheaf of modules over $C^\infty_X$. Then such a ...
- 35.2k
6
votes
Progress on Bondal–Orlov derived equivalence conjecture
A lot of work has been done to prove the Bondal-Orlov conjecture for (stratified)-Mukai flops. To quote a few names : Namikawa (Mukai flops), Kawamata (one stratified Mukai flop), Cautis and collab (...
- 7,300
6
votes
Pushforward maps for cohomology of coherent sheaves
The map is induced by the right adjoint $i^!$ of the pushforward functor and the adjunction morphism $i_*i^! \to \mathrm{id}$, in view of the formula $i^!(F) \cong i^*(F) \otimes \omega_{Z/X}[\dim Z/X]...
- 39.3k
6
votes
Accepted
Zero locus of a family of morphisms of vector bundles
I am just writing my comment above as an answer. This follows from Corollaire 7.7.8, EGA III.
Grothendieck, Alexander
Éléments de géométrie algébrique (rédigés avec la collaboration de Jean ...
6
votes
Which category of sheaves on a manifold remembers the manifold?
Assuming the manifold is Hausdorff, the category of sheaves of modules over the sheaf of smooth functions do the trick.
This category is equivalent to the category of non-degenerate module over the ...
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6
votes
Accepted
Reference request: tangent space to moduli space of coherent sheaves is $\operatorname{Ext}^1(E, E)$
Theorem 2.6 (page 9) from Hartshorne Lectures on Deformation Theory (it seems that Hartshorne uses the same notation both for an affine scheme $D$ and its function algebra):
Let $X$ be a scheme over $...
- 1,980
6
votes
Serre's theorem on global generations on stacks
Tame Artin stacks (in the sense of Abramovich, Olsson and Vistoli, https://math.berkeley.edu/~molsson/tame.pdf) with quasi-projective moduli spaces will have property 2: the line bundle is the ...
- 27k
5
votes
Vanishing of some Ext groups of coherent sheaves
I am posting my comment as an answer. Perhaps the result that you are looking for is Ischebeck's Theorem.
Ischebeck's Theorem. For finitely generated modules $F$ and $G$ over a Noetherian local ...
5
votes
Different definition of sheaf cohomology
There are two separate issues that are getting mixed up here, as Tyler's comment above indicates:
The point addressed in Mariano's answer: on the category of coherent sheaves, the functor of ...
- 44.7k
5
votes
Accepted
Injectivity of pullback composed with pushforward
The answer to both questions (injectivity ans surjectivity) is no without further hypothesis.
Surjectivity : Let $Y$ be a smooth projective variety and let $\phi : X \longrightarrow Y$ be the blow-up ...
- 7,300
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