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5
votes
2answers
236 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 ...
3
votes
1answer
147 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 ...
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 ...
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 ...
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
160 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
365 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 ...
5
votes
2answers
632 views

Topological homotopy category as derived category

In the Introduction of his Derived Categories for the working mathematician Richard Thomas mentions the following theorem of Whitehead. Suppose that $X,Y$ are simplicial complexes, then the ...
8
votes
1answer
328 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 ...
6
votes
1answer
244 views

Are constructible derived categories invariant up to weak homotopy equivalence?

Let $X$ and $Y$ be two topological spaces and $R$ be commutative ring. Let $D_c^b(X, R)$ and $D_c^b(Y,R)$ be their respective bounded derived categories of constructible sheaves of $R$-modules. I ...
3
votes
1answer
252 views

Inverse of a tilting module

Let $k$ be a field, $A$ an associative unital $k$-algebra, $\operatorname{\mathsf{Mod}} A$ the category of left $A$-modules and $D^b(\operatorname{\mathsf{Mod}} A)$ the bounded derived category. Let ...
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 ...
5
votes
2answers
687 views

Higher vanishing cycles

The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's an issue, Morava mentioned a connection with Bousfield localization. I find the Morava's remarks ...
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 ...
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 ...
10
votes
1answer
623 views

Fiber functors to derived categories

Suppose that $G$ is an algebraic group over a field $k$. Then for any $k$-algebra $R$, a fiber functor from $\text{Rep}_k(G)$ to the category of projective modules over $R$ is the same as 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
230 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 ...
4
votes
1answer
384 views

(Co)localization of the derived category

Let me start saying that a similar question can be stated for general locally Noetherian Grothendieck categories but I state it for categories of modules as it is simpler. So we fix a right Noetherian ...
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 ...
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
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 ...
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.
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 ...
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 $MHM(X)$ ...
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 ...
3
votes
1answer
158 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 ...
16
votes
2answers
437 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
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
297 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 ...
31
votes
5answers
3k views

Intuition about the cotangent complex?

Does anyone have an answer to the question "What does the cotangent complex measure?" Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as ...
2
votes
1answer
200 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 ...
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 ...
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
499 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 ...
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 ...
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 ...
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
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 ...