The tag has no wiki summary.

learn more… | top users | synonyms

3
votes
1answer
145 views

flat descent for perverse sheaves

Let $E \in D^{b}_{c}(X,\overline{\mathbb{Q}}_{l})$ where $X$ is a $k$ scheme of finite type for a field $k$. Let $Y\rightarrow X$ a finite flat surjective morphism such that $f^{*}E$ is perverse and ...
5
votes
1answer
163 views

Equivariant Formality

Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}_{dg}(\mathcal{A})$. Assume further there exists a ...
3
votes
0answers
93 views

Formal DG-algebras

Sorry for this question but I really have difficulties with model categories. Usually a $dg$-algebra $A$ is called formal, if there exists a $dg$-algebra $B$ and quasi-isomorphisms $$A\leftarrow B\to ...
0
votes
2answers
159 views

Injective resolution for right derived functor

This question is base on my previous question, and I repeat it here: Suppose $X$ is a projective variety and $D^{+}(X)$ is the derived cateogry of bounded below complexes of sheaf of ...
4
votes
2answers
364 views

Why is the derived tensor product only defined for bounded above derived categories?

In "Residues and Duality" by Hartshorne, the derived tensor $\otimes$ only defined for the bounded above categories (see Chapter I, section 4), that is one has $$\otimes: D^{-}(X) \times D^{-}(X) \to ...
8
votes
1answer
318 views

Are $D^b_{coh}(X)$ and $D^b(Coh(X))$ derived equivalent?

Let $X$ be a variety. Let $D^b(Coh(X))$ be the derived category of bounded complexes of coherent sheaves on $X$, and $D^b_{coh}(X)$ be the derived category of bounded complexes of sheaves of ...
5
votes
2answers
235 views

Equivariant derived category and invariant divisor

I'm looking for a reference of the following (folklore?) result. Let $X$ be a smooth projective variety equipped with a $G=\mathbb{Z}/2\mathbb{Z}$ action (we consider the simplest case, everything ...
0
votes
0answers
216 views

generators for derived category

Let $G$ be a algabraic group $G$ over a field $k$. We denote by $D^b(\mathrm{Repr}(G))$ the derived category of finite dimensional representations. Under what kind of assmumptions one has a generating ...
4
votes
0answers
152 views

Grothendieck group of intersection of quadrics

Conjecturally, for a complete intersection $X$ the group $K_0(X)$ is finitely generated iff $H^{p,q}(X)=0$ for $p≠q$. This holds if and only if $X$ is a quadric, a cubic surface, or an ...
0
votes
0answers
155 views

Hodge structure of abelian surfaces

In my case, I have an abelian surface $A$ of (2,8)-polarization, and I have some finite group (may not be abelian group) $G$ acting on $A$ without fixed point. I want to understand when there is a ...
6
votes
0answers
219 views

Reconstructing the Chow ring from the derived category

Let $X$ be a smooth projective variety and write $\mathbf{D}(X)$ for its triangulated category of perfect complexes of quasi-coherent sheaves. Recall that $\mathbf{D}(X)$ determines the Grothendieck ...
4
votes
1answer
252 views

Fourier-Mukai transform for abelian varieties

Let $A$ be an abelian variety over $\mathbb{C}$, $L$ be a very ample line bundle on $A$, then the dual abelian variety is $\hat{A} \cong A/K(L)$ with $K(L)$ the kernel of surjective morphism $A \to ...
-2
votes
1answer
282 views

Derived Category.

Question 1: Let $X$ be a scheme. Then generally for the complex $C^{\bullet}$ in $D^b(X)$, we define $R\Gamma(C^{\bullet})\colon$ = Complex obtained by applying $\Gamma$ to the injective resolution ...
2
votes
0answers
160 views

what is the zero locus of a morphism of the derived category?

For a morphism of quasi-coherent sheaves $v\colon E \to F$, on a scheme $X$, one can ask about the locus where $v =0$. When $F$ is a vector bundle, it's easy to see that this locus is closed.* Is ...
5
votes
1answer
228 views

Derived categories of singular varieties

Given my limited knowledge on derived categories, all the results on derived categories of complex of bounded sheaves are build upon smooth varieties, and people literally avoid singular case (as in ...
4
votes
1answer
129 views

Is the injective structure on unbounded chain complexes simplicial?

In Mark Hovey's article Model category structures on chain complexes of sheaves (arXiv:math/9909024) a model structure on the category $Ch(A)$ of unbounded chain complexes for a Grothendieck abelian ...
2
votes
0answers
106 views

semiorthogonal decomposition

We consider $\mathbb{P}^1$ and the semiorthogonal decomposition $<\mathcal{O},\mathcal{O}(1)>=D^b(\mathbb{P}^1)$. Let $x$ be a closed point and $k(x)$ the corresponding skyscraper sheaf. Every ...
3
votes
1answer
132 views

Is there a blow-up formula for the derived category of a singular ambient variety?

For a nonsingular variety sitting inside a nonsingular ambient variety there is a semi-orthogonal decomposition of the derived category of the blow-up (with center that subvariety). What can be said ...
0
votes
1answer
176 views

Could we construct the Jacobian variety of a smooth curve $C$ with genus $>2$ from its derived category $D(C)$?

Let's consider a smooth curve $C$ over $\mathbb{C}$. We know that the Jacobian variety $Jac(C)$ of $C$ is the moduli space of the degree $0$ line bundles on $C$. $Jac(C)$ is an abelian variety of ...
1
vote
0answers
49 views

injective dimension of dualizing complex

Let $R$ be a balanced dualizing complex of a Noetherian connected graded algebras $A$. Dose one always have $\text{id}_A R = \text{id}_{A^{op}} R$? Thanks a lot.
2
votes
0answers
89 views

The associated graded of a mixed Hodge module

Unfortunately this question will be a bit vague, since the question revolves around a memory of something I may have heard in a talk (long time ago). Let $X$ be a smooth complex variety. Let $MHM(X)$ ...
3
votes
1answer
158 views

Formality of classifying spaces (for not necessarily connected groups)

As should be evident from the title this question has a similar flavor to: Formality of classifying spaces However, unlike Geordie's question, I will be working with torsion free coefficients (say ...
1
vote
1answer
152 views

Could we extend the Atiyah class to the sheaf of poly-vector fields to get a Poisson bracket?

Let $X$ be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
4
votes
1answer
293 views

Can we define the tensor product in the derived category $D^b_{\text{coh}}(X)$ just from $D^b_{\text{coh}}(X)$ in certain cases?

This question arise from the comparision of the reconstruction theorems of Bondal-Orlov and Balmer and is inspired by Shizhuo Zhang's mathoverflow question: How to unify various reconstruction ...
10
votes
4answers
552 views

What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
1
vote
1answer
195 views

Balanced dualizing complexes according to A. Yekutieli

I am reading A. Yekutieli's original article on dualizing complexes for noncommutative algebras and I found a problem I cannot solve. First, some background. We start with a field $k$ and a ...
3
votes
1answer
157 views

The Hochschild cohomology of a variety “with coefficient” in a vector bundle

This question is related to one of my previous question Do we have the following isomorphism for $\mathcal{Ext}$? Let $X$ be a smooth variety (over $\mathbb{C}$) and $\Delta: X \rightarrow X \times ...
2
votes
0answers
72 views

Continuity of constructible derived category

Let $X_0$ be a variety over $\mathbb F_q$. Denote by $X_n$ its basechange to $\mathbb F_{q^n}$ and let $X=\lim X_n$ be its basechange to the algebraic closure $\overline{\mathbb F}_q$. Let ...
2
votes
1answer
137 views

Does Wolbert's derived equivalence between $E_*^R$-local $R$-modules and $R_E$-modules come from a Quillen equivalence?

Let $R$ be a ring spectrum (in the world of EKMM $S$-modules) and let $E$ be a smashing $R$-module. Denote by $R_E$ the $E_*$-localization of $R$. By a theorem of Wolbert (Theorem 2 in Classifying ...
5
votes
1answer
296 views

A naive question on eigensheaves for group actions on derived categories

In this Mathoverflow question, Examples of Eigensheaves outside of langlands, David Ben-Zvi says " Given a G -space X you can recover quasicoherent sheaves on X from sheaves on X/G (ie equivariant ...
9
votes
1answer
415 views

Why is proving fully-faithfulness of an integral functor locally analytically sufficient?

More than once I've come across a statement in a paper about derived categories in which it says something to the effect of "in order to prove that $\Phi:D^b(X)\rightarrow D^b(Y)$ is fully-faithful we ...
16
votes
2answers
436 views

Homotopy-theoretic derived Morita equivalences

Recall that two $k$-algebras $A, B$ are Morita equivalent iff their categories of left modules are equivalent. However, this relation turns out to be rather fine and one introduces a coarser ...
2
votes
1answer
199 views

why is the homological projective dual of this Lefschetz decomposition non-commutative?

I am reading these notes of an excellent course by Kuznetsov on Homological Projective Duality. On page 10 there is Example 1''. One starts with projective space, endowed with the identity embedding ...
5
votes
1answer
107 views

Is this a description of the $\aleph_1$-localizing subcategory generated by a compact generator?

This should be obvious but I'm not seeing it: The $\mathfrak T$ be a triangulated category with coproducts and with a compact generator $A$ (that is, the functor $\mathfrak T(A,\_)$ preserves ...
13
votes
1answer
498 views

How can one interpret homology and Stokes' Theorem via derived categories?

I am very far removed from being an expert on derived categories. Every few months, however, I read a different introductory text with the hope that eventually I will have some basic grasp on this ...
3
votes
3answers
260 views

When an exact embedding of abelian categories induces a full embedding of their derived categories?

Let $F:A\to A'$ be a (full) exact embedding of abelian categories. When $D(F):D(A)\to D(A')$ (or its bounded version) is a full embedding also? I would be interested in any necessary or sufficient ...
6
votes
1answer
224 views

Derived categories of toric varieties and convex geometry

Toric varieties and convex polyhedra are intimately connected. Some of this can be found in standard text books (the connection between divisors and mixed volumes seems to be a popular example). One ...
4
votes
0answers
127 views

Derived category of toroidal varieties

This question comes from the first reduction step of Theorem 4.2 of Kawamata's paper on K-equivalent implies D-equivalent on toroidal varieties. But my question has little to do with this theorem. A ...
1
vote
0answers
140 views

Model structure on the category of chain complexes in an abelian category gives rise to the derived category

Let $\mathcal A$ be an abelian category with enough projectives. Consider a morphism $f \colon X^\bullet \longrightarrow Y^\bullet$ in the category $C(\cal A)$ of chain complexes in $\mathcal A$. Is ...
2
votes
1answer
142 views

admissible subcategories over non algebraically closed fields

Let $X$ be a smooth projective variety over a field $k$ and $D^b(X)$ its bounded derived category. Let $\bar{X}$ the base change to $\bar{k}$. Let $A$ be a triangulated subcategory of $D^b(X)$ that ...
2
votes
0answers
187 views

How do I find abelian subcategories of periodic triangulated categories?

If $T$ is a triangulated category, then the formalism of $t$-structures gives a way to find abelian subcategories inside. You're supposed to find two strictly full subcategories, $T^{\le 0}, T^{\ge ...
2
votes
1answer
110 views

Bondal counter example to the Jordan-Holder property in derived categories

Can anybody give me the reference where this counter-example is explained in detail?. Consists on the following Bondal considered a quiver $Q$ with some relations and proved that $D(Q)$ has two ...
7
votes
2answers
408 views

What is a good reference (preferably thorough) for the Derived Category of a scheme/orbifold/stack?

I've sort of circled around the idea of derived categories a few times, read a few introductory papers ("Derived Categories for the working mathematician", e.g.), and feel now that this is something ...
1
vote
1answer
172 views

Uniqueness of the canonical etale coverings

This is a construction [Definition 6.1] given in the paper D-equivalence and K-equivalence by Kawamata. Let $X$ be a normal quasiprojective variety such that the canonical divisor $K_X$ is a ...
1
vote
1answer
202 views

Derived equivalence of two varieties which are isomorphism over certain open subvarieties

Let $X,Y$ be varieties over $\mathbb{C}$, and $D(X),D(Y)$ be the derived categories of bounded complex of coherent sheaves. Let $U \subset X, V \subset Y$ be open subvarieties, and let $X-U, Y-V$ have ...
3
votes
1answer
234 views

Commutativity of Tor

Let $A$ be a commutative ring with $1$ and $M,N$ be $A$-modules. Can you give a quick proof that $\textrm{Tor}_i(M,N) \cong \textrm{Tor}_i(N, M)$ using derived categories? In his Homological algebra ...
4
votes
1answer
242 views

Additive functors and Derived Categories

I have been learning about derived categories from Hartshorne's "Residues and Duality". One of the main theorems, as it is presented there seems to be the canonical isomorphism $RF \circ RG \cong R(F ...
1
vote
1answer
71 views

is the orthogonal complement of a saturated sequence saturated?

Suppose I have a smooth projective variety $X$, and a semi-orthogonal decomposition of its bounded derived category: $$D^b(X)= < A, E_1, E_2, ... , E_n >$$ where the $E_i$ are fully faithful, ...
2
votes
1answer
136 views

morphism of injective objects

Let $A,B$ be two bounded below complexes in module category, and $A \longrightarrow I$ (resp. $B \longrightarrow J$) a injective resolution. If $f: A \longrightarrow B$ is a morphism of complexes. My ...
2
votes
5answers
370 views

A statement for a triangulated category generated by a subset

Let $D$ be a triangulated category (the triangulated category in my mind is $D^{b}(X)$, that is the derived category of bounded complex of coherent sheaves on a smooth projective variety), $A \subset ...