All Questions
Tagged with ag.algebraic-geometry derived-categories
402
questions
3
votes
0
answers
166
views
Orlov's theorem on fully faithful functors between derived categories
According to an important result of Orlov, a $k$-linear exact fully faithful functor $\Phi\colon D^b(X)\rightarrow D^b(Y)$ for smooth projective varieties $X$ and $Y$ is isomorphic to a Fourier-Mukai ...
user500434
-1
votes
1
answer
171
views
When morphism of complexes is homotopic to 0?
Let $f\colon A^\bullet\to I^\bullet$ be a morphism of bounded below complexes in an abelian category. Assume all $I^i$ are injective objects. Assume also that $f$ induces the zero map on cohomology.
...
- 21.3k
4
votes
0
answers
289
views
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 ...
- 16.7k
9
votes
1
answer
903
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 ...
- 16.7k
3
votes
0
answers
381
views
Mapping cone is a functor
It is a well-known general fact that in a triangulated category, the cone $Z$ of a morphism $X \longrightarrow Y$ (that means there exists a distinguished triangle $X \longrightarrow Y \longrightarrow ...
- 823
1
vote
0
answers
115
views
Computing the equivariant Chern character
Suppose I know the Chern character of an object $F \in D^b(X)$, where $X$ is some smooth complex projective variety with a finite group $G = \mathbb{Z}/m$ acting on it. In $D^b([X/G]) \simeq D^b(X)^G$ ...
- 61
2
votes
1
answer
162
views
Homomorphism between Ext induced by the left mutation functor
$\DeclareMathOperator\Coh{Coh}\DeclareMathOperator\ev{ev}\DeclareMathOperator\cone{cone}\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\Ext{Ext}$This is a specific question concerning a statement in ...
user499214
0
votes
0
answers
152
views
Computing RHom of skyscraper sheaves / sheaves of subvarieties
Let $ X $ be a smooth projective variety (over $ \mathbb{C} $) of dimension $ n $ and $ x : \operatorname{Spec} \mathbb{C} \rightarrow X $ a point. How can I compute the complex $ \operatorname{\...
- 896
1
vote
0
answers
109
views
Kunneth formula for hypercohomology
Let $A_{\bullet}$ and $B_{\bullet}$ be two bounded complexes of sheaves over a variety $X$. Is there a Kunneth-like formula for the hypercohomology of the tensor product $A_{\bullet}\otimes B_{\bullet}...
- 492
0
votes
0
answers
167
views
Cone of morphism induced by Serre duality
For a smooth projective variety $X$, Serre duality gives an exact autoequivalence on the derived category :
$$
S_X : D^\flat(X) \to D^\flat(X), \hspace{3em} S_X(-) = - \otimes \omega_X[\dim X]
$$
...
- 275
2
votes
0
answers
113
views
dg-natural transformation between FM functors and Hom between kernels
The question is related to Morphism between Fourier-Mukai functors implies the morphism between kernels?
Consider a complex smooth projective variety $X$ and the bounded derived category $D^b(X)$, it ...
anon
1
vote
1
answer
160
views
There are only one type of Verra fourfold?
A Verra fourfold is a Fano fourfold which is defined as double cover of $\mathbb{P}^2\times\mathbb{P}^2$ with branch divisor to be $(2,2)$-hypersurface of $\mathbb{P}^2\times\mathbb{P}^2$, which is an ...
- 1,962
2
votes
2
answers
331
views
Moduli space of Bridgeland semistable objects: what is it?
I usually meet this kind of moduli space in recent papers on Bridgeland stability conditions:
the moduli space $M_{\sigma}(v)$ of $\sigma$-semistable objects of $\mathcal{T}$ with certain numerical ...
user493108
3
votes
1
answer
187
views
Autoequivalence group from semiorthogonal decomposition
Suppose we have a semiorthogonal decomposition $\mathcal{D} = \langle \mathcal{A}, \mathcal{B} \rangle$, and suppose we know fully the autoequivalence groups $\mathrm{Aut}(\mathcal{A})$ and $\mathrm{...
- 305
3
votes
1
answer
137
views
What do we know about a sheaf $M$ if we know its derived fibers $\mathsf{L}x^* M$, for $x\in X(k)$?
Let $X$ be a scheme over a field $k$. (Feel free to assume that $X$ is an algebraic variety, if needed.) Also, let $M^\bullet$ be a complex in the derived category of quasi-coherent sheaves $\mathsf{D}...
- 1,139