All Questions
Tagged with ag.algebraic-geometry derived-categories
402
questions
1
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0
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121
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How does the Torelli theorem behave with respect to cyclic covering?
Let $Y\xrightarrow{2:1}\mathbb{P}^3$ be the double cover, branched over a quartic K3 surface $S$, known as quartic double solid. Assume $S$ is generic, we know that there is a Torelli theorem for $Y$ ...
- 1,962
7
votes
0
answers
315
views
Bondal-Orlov' theorem for noncommutative projective schemes
My question is very simple.
Is Bondal-Orlov's theorem known for noncommutative projective schemes in the sense of Artin and Zhang?
The commutative version is the following :
Let $X, Y$ be smooth ...
- 869
4
votes
0
answers
175
views
Relation between exotic sheaves in Achar's notes and in Bezrukavnikov-Mirkovic
I am trying to calculate explicitly a certain simple exotic sheaf (a simple object of the heart of the exotic t-structure on the Springer resolution, which is defined in Theorem 1.5.1 of Bezrukavnikov-...
- 2,540
5
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1
answer
232
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On the bounded derived category of sheaves with coherent cohomology
Let $(X,\mathcal{O}_X)$ be a locally ringed space such that $\mathcal{O}_X$ is locally notherian, and let $\operatorname{Coh}(\mathcal{O}_X)$ be the category of coherent $\mathcal{O}_X$-modules. The ...
5
votes
1
answer
178
views
A pushout diagram of derived categories coming from an open cover of schemes
Suppose $X=U\cup V$ is the standard open cover of $X=\mathbb{P}^1$ by two affine lines. The descent theorems say that the diagram (with all arrows restriction maps)
$\require{AMScd}$
\begin{CD}
D(X) @&...
- 53
3
votes
0
answers
110
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What is the correct definition of intermediate Jacobian for this singular threefold?
I am considering blow up of $\mathcal{C}\subset(\mathbb{P}^1)^3$, $X=\operatorname{Bl}_{\mathcal{C}}(\mathbb{P}^1)^3$, where $\mathcal{C}$ is a curve given by $$\{s^2u=0\}\subset\mathbb{P}^1_{s:t}\...
- 1,962
2
votes
1
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146
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Pullback morphism of a hyperplane inclusion is zero in the derived category
Let $L \subset \mathbb{C}^n$ be a hyperplane and let $i:L \to \mathbb{C}^n$ be the inclusion. Since $i$ is proper, we have induced maps $i^*: H^k_c(\mathbb{C}^n) \to H^k_c(L)$, and these maps are zero ...
6
votes
1
answer
565
views
Canonical comparison between $\infty$ and ordinary derived categories
This question is a follow-up to a previous question I asked.
If $\mathcal{D}(\mathsf{A})$ is the derived $\infty$-category of an (ordinary) abelian category $\mathsf{A},$ then the homotopy category $h\...
- 1,179
3
votes
1
answer
158
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Pushforward of exceptional vector bundle is spherical for local P^2
I've been reading through a bit of the literature on stability conditions, and one of the models that has come up is the 'local projective plane'. Explicitly, this is the total space of the canonical ...
- 275
3
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0
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88
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Projective resolution of a quiver with relations
How do we compute the projective resolution of a representation of a quiver with relations.
For example consider the Beilinson quiver $B_4$
$.
with the relations $\{\alpha_j^k\alpha_i^{k-1}=\alpha_i^...
- 169
2
votes
1
answer
190
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Find stratification to decompose constructible sheaf to constant parts (example from Wikipedia)
I have a question about techniques used in determining the stratification over which a constructible sheaf falls into even constant pieces demonstrated on this example from Wikipedia.
Let $f:X = \text{...
- 5,780
3
votes
1
answer
176
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Image, upto direct summands, of derived push-forward of resolution of singularities
Let $\mathcal C$ be a full subcategory (closed under isomorphism also) of an additive category $\mathcal A$. Then, $\text{add}(\mathcal C)$ is the full subcategory of $\mathcal A$ consisting of all ...
- 481
4
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1
answer
190
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Some questions on derived pull-back and push-forward functors of proper birational morphism of Noetherian quasi-separated schemes
Let $f: X \to Y$ be a proper birational morphism of Noetherian quasi-separated schemes. We have the derived pull-back $Lf^*: D(QCoh(Y))\to D(QCoh(X))$ (https://stacks.math.columbia.edu/tag/06YI) and ...
- 351
15
votes
1
answer
667
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Why do we say IndCoh(X) is analogous to the set of distributions on X?
$\DeclareMathOperator\IndCoh{IndCoh}\DeclareMathOperator\QCoh{QCoh}$I've seen it written (for example, in Gaitsgory–Rozenblyum) that for a scheme $X$, the category $\IndCoh(X)$ is to be thought of as ...
2
votes
1
answer
196
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Perfect complexes of plane nodal cubic curve
Let $C\subset\mathbb{P}^2$ be a plane nodal cubic curve with a unique singular point $O$ at the origin. Then I consider its normalization, denoted by $\widetilde{C}$ and let $\pi:\widetilde{C}\...
- 1,962
2
votes
0
answers
154
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When is $D^+(QC(X))$ not the same as $D_{qc}(X)^+$ for schemes?
Let $QC(X)$ be the abelian category of quasicoherent sheaves on a scheme $X$. There is a functor
$$D^+(QC(X)) \to D_{qc}(X)^+$$
which is an isomorphism if $X$ is Noetherian or quasi-compact with ...
- 2,035
3
votes
0
answers
270
views
Algebraic Fukaya categories and mirror symmetry
Dominic Joyce and collaborators have outlined a programme to construct algebraic Fukaya categories on an algebraic symplectic manifold (“Fukaya categories” of complex Lagrangians in complex symplectic ...
- 163
2
votes
1
answer
189
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How to conclude the quasi-projective case of the derived McKay correspondence from the projective case?
I am currently trying to understand the paper "Mukai implies McKay" from Bridgeland, King and Reid (cf. here). Let me sum up the setting we find ourselves in:
Let $M$ be a smooth quasi-...
- 41
4
votes
2
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269
views
Does there exist a faithful exact embedding of $D^b(\dim(N)) \to D^b(\dim(N-1))$
$\DeclareMathOperator\Hom{Hom}$I am trying to show that if $X,Y$ are nice schemes with $\dim(X) > \dim(Y)$ there is no faithful FM transform $\Phi_{K}: D^b(X) \to D^b(Y)$.
Does someone have a proof ...
- 293
4
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258
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Has anyone studied the derived category of Higgs sheaves?
Let $X$ be a complex manifold and $\Omega^1_X$ be the sheaf of holomorphic $1$-forms on $X$. A Higgs bundle on $X$ is a holomorphic vector bundle $E$ together with a morphism of $\mathcal{O}_X$-...
- 8,767
2
votes
1
answer
175
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liftability of isomorphism of curves in $P^3$
It is well known that the isomorphism between smooth curves $C$ and $C'$ in $\mathbb{P}^2$ can be lifted to an automorphism of $\mathbb{P}^2$ if degree of $C$ and $C'\geq 4$. Now I am considering an ...
- 1,962
5
votes
1
answer
384
views
What is the k-linear structure on the derived infinity category of quasi-coherent sheaves?
Let $f : X \overset{f}{\rightarrow} Y \overset{g}{\rightarrow} \mathrm{Spec} (k)$ be morphisms of schemes (feel free to add any hypothesis necessary). Let $\mathrm{QCoh}(Y)$ denote the derived (stable)...
- 53
3
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0
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169
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Relations between some categories of étale sheaves
I asked this question on math.stackexchange but nobody answers, so I try here even if I'm not sure my question is a research level one..
Let $X$ be a scheme over a number field $k$. Feel free to add ...
- 1,125
1
vote
1
answer
341
views
Tensor product and semisimplicity of perverse sheaves
Let $X/\mathbb{C}$ be a smooth algebraic variety. Let $D_c^b(X,\bar{\mathbb{Q}}_{\ell})$ be the category defined in 2.2.18, p.74 of "Faisceaux pervers" (by Beilinson, Bernstein and Deligne). ...
- 545
1
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0
answers
99
views
Computing Grothendieck group of (unnodal) Enriques surface
Let $X$ be an unnodal Enriques surface together with an isotropic 10-sequence $\{ f_1, \dots, f_{10}\} \subset \operatorname{Num}(X)$, and let $F_i^\pm \in \operatorname{NS}(X)$ denote the two ...
- 275
3
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1
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321
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resolution property and perfect stacks
Recall that for a scheme $X$, it has the resolution property if every coherent sheaf $E$ on $X$, is the quotient of a finite locally free $\mathcal{O}_X$-module.
On the other hand, Ben-Zvi-Nadler-...
- 3,452
1
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1
answer
214
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Derived McKay correspondence between a weighted projective plane and a Hirzebruch surface
Let $k$ be an algebraically closed field of $\text{ch}(k) =0$.
Let $\mathbb{P}(1,1,2)$ be the weighted projective plane of weight $(1,1,2)$ as a stack.
Let $\mathbf{P}(1,1,2)$ be the weighted ...
- 13
1
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0
answers
140
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Intermediate Jacobian for small resolution of a singular Fano threefold?
I am mainly interested in the nodal Gushel-Mukai threefold. Let $X$ be a Gushel-Mukai threefold with one node, then by page 21 of the paper https://arxiv.org/pdf/1004.4724.pdf there is a short exact ...
- 1,962
1
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1
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173
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Reference for localization distinguished triangles in the derived category of $\ell$-adic sheaves
Let us consider a variety $X$ over a field $k$ which is a finite field or an algebraic closure thereof. Let $\ell$ be a prime number different from the characteristic of $k$, and let $\Lambda = \...
- 717
1
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0
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215
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Characterization of morphisms of finite Tor-dimension by the hypertor functor
I am trying to have a concrete understanding of morphisms of finite Tor-dimension between arbitrary schemes.
In Hartshorne’s book Residues and Duality,a morphism of schemes $f:X\rightarrow Y$ is said ...
- 569