For questions about the derived categories of various abelian categories and questions regarding the derived category construction itself.

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17
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0answers
862 views

Is there a Grothendieck-Riemann-Roch type of theorem generalizing Grothendieck's Lefschetz trace formula

Grothendieck deduced that the L-function of a (constructible) $\ell$-adic sheaf on a variety over $\mathbf{F}_p$ is rational from the generalized trace formula. My first question is based on the ...
13
votes
0answers
477 views

When is every “solid” perfect complex faithful?

Let $R$ be a noetherian commutative ring. Consider $D^{perf}(R)=K^b(R-proj)$ the category of bounded complexes of finitely generated projective $R$-modules, with maps of complexes up to homotopy. ...
12
votes
0answers
496 views

References for a certain generalization of Hochschild cohomology?

Let $C$ be an algebra. Let $E = C^{\otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = C\otimes C^{op}\otimes C\otimes C^{op}\cdots\otimes C ...
10
votes
0answers
368 views

The derived category of integral representations of a Dynkin quiver.

Let $Q$ be a Dynkin quiver. Let $\mathbb CQ$ be its complex path algebra. It is defined in a way such that modules over $\mathbb CQ$ are the same as representations of the quiver $Q$. Let's write ...
9
votes
0answers
313 views

McKay correspondence and tensor products

The theorem of Bridgeland-King-Reid says that if $M$ is a smooth quasi-projective complex variety of dimension at most $3$ on which a finite group $G$ acts such that the canonical sheaf $\omega_M$ is ...
9
votes
0answers
638 views

Generating the derived category with line bundles

The following lemma is useful and well-known: LEMMA If $L^{\pm 1}$ is ample on proper scheme over a field $k$, then some number of powers $\mathcal{O},L,...,L^{m}$ generate the unbounded derived ...
7
votes
0answers
255 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 ...
6
votes
0answers
166 views

Evens norm as a derived functor

In the theory of cohomology of groups there are several important maps between cohomologies: restriction, inflation, transfer and Evens norm. The first three of them can be clearly seen on the level ...
5
votes
0answers
133 views

Compact objects and ind-objects in triangulated categories

question : let $A$ be triangulated category compactly generated by subcategory $A^c$ of compact objects. Consider category of ind-objects $Ind(A^c)$. Is there relation between $A$ and $Ind(A^c)$? ...
5
votes
0answers
196 views

Not isomorphic varieties with isomorphic tilting algebras

Let $X$ be a smooth projective variety over a field, than tilting object $T$ on $X$ is a perfect complex that is a compact generator of the derived category $\operatorname{D}(QCoh(X))$ and satisfies ...
5
votes
0answers
399 views

On the derived category of constructible étale sheaves

The derived category $D^{\flat}_{c}(X,R)$ of constructible sheaves of $R$-modules on $X_{et}$ is defined as the full subcategory of $D^b(X,R)$ whose cohomology sheaves are all constructible. Clearly, ...
5
votes
0answers
466 views

functor before cat?

As i read the literature, derived functors were there several years before derive categories - right?
4
votes
0answers
115 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 ...
4
votes
0answers
163 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 ...
4
votes
0answers
136 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 ...
4
votes
0answers
197 views

What is the structure of the stack of complexes supported in dimension less than r?

Let $X$ be something. (smooth and projective variety over C are my assumptions) The stack $M$ parameterising coherent sheaves on $X$ splits as a disjoint union of open and closed substacks $M_\alpha$, ...
4
votes
0answers
191 views

Leray degeneration for smooth projective morphisms and formality of families of compact Kähler manifolds

Let $\pi \colon X \to S$ be a smooth projective morphism of algebraic varieties, say over $\mathbf C$. By Deligne's argument ("Théorème de Lefschetz...", 1968), there is for each $i$ an injection $$ ...
4
votes
0answers
284 views

Stability conditions for coherent sheaves and GIT

I am learning stability conditions for derived categories of coherent sheaves, following Bridgeland, and coming from a vector bundles background. $\mu$-stability for vector bundles has a clear GIT ...
4
votes
0answers
226 views

What is the equivariant derived category good for?

Given a topological group acting on a topological space, Bernstein and Lunts construct the equivariant derived category, which looks like the derived category of the quotient would, if action was free ...
3
votes
0answers
166 views

Does GKZ's reflexivity theorem imply the Plucker formula?

Let $S\subset\mathbb{P}^n$, Gelfand-Kapranov-Zelevinsky defined its dual variety $S^\vee\subset\mathbb{P}^{n^\ast}$. In this paper (http://arxiv.org/pdf/math/0111179v1.pdf), the author obtained the ...
3
votes
0answers
425 views

unique enhancement for derived categories

I have two questions concerning the existence and uniqueness of enhancements in the following cases: i.) Let $A$ be a finite dimensional $k$ algebra of finite global dimension. Does the triangulated ...
3
votes
0answers
200 views

Definition of derived category of a stack

In their book, Bernstein an Lunts define the equivariant derived category in several ways. One can be expressed as follows: Let $X$ be a say complex variety with an action by an algebraic group $G$. ...
3
votes
0answers
172 views

Closed Model Category Structure on Chain Complexes Related to A Left-exact Functor

Let $F:A \to B$ be an additive left-exact functor of abelian categories (Do not assume that they have enough injectives / projectives.) Suppose we are given a class of objects $R$ adapted to $F$ (see ...
3
votes
0answers
434 views

Sebastiani-Thom isomorphism for D-modules

Considering $f:X\to \mathbb{C}$, $g:X\to \mathbb{C}$ and $f\oplus g:(x,y)\mapsto f(x)+g(y)$. The Sebastiani-Thom isomorphism is an isomorphism $\Phi_{f\oplus g}(M\boxtimes N) = \Phi_{f}(M) \otimes ...
2
votes
0answers
183 views

Why care about Fourier-Mukai partners?

Two (smooth, projective, complex?) varieties are called Fourier-Mukai partners if they have equivalent derived categories of coherent sheaves. On the other hand, my general impression is that cool ...
2
votes
0answers
100 views

Do all full exceptional sequences of a triangulated category have the same length?

Let $\mathcal{D}$ be a $k$-linear triangulated category and $\langle E_n,E_{n-1},\ldots, E_0\rangle$ be a sequence of objects of $\mathcal{D}$. We call $\langle E_n,E_{n-1},\ldots, E_0\rangle$ an ...
2
votes
0answers
105 views

Unbounded derived category that is not left-complete

Let me first recall some definition: Let $A$ be a Grothendieck Abelian category. Then, then category $\mathrm{Ch}(A)$ (I am using homological indexing) admits a combinatorial model structure (see for ...
2
votes
0answers
176 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 ...
2
votes
0answers
146 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 ...
2
votes
0answers
75 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
0answers
193 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 ...
1
vote
0answers
73 views

Is there any numerical obstruction for all perfect complexes on a scheme being strictly perfect?

Let $X$ be a scheme and $E^{\cdot}$ be a cochain complex of sheaves of $\mathcal{O}_X$-modules. We call $E^{\cdot}$ a strictly perfect complex if $E^{\cdot}$ is a bounded (in both direction) complex ...
1
vote
0answers
110 views

Thickness of the category of perfect complexes with finite length homology

Let $R$ be a commutative Noetherian local ring and let $D(R)$ be the derived category of $R$-modules. Recall that a chain complex $C_\bullet$ of modules over $R$ is called perfect if it is isomorphic ...
1
vote
0answers
107 views

Why $D^b(S)\cong D^b_{\text{Car}}(\text{cosq}(X\rightarrow S))$?

Let $p: X\rightarrow S$ be a map between topological spaces and we can construct the simplicial space $\text{cosq}(X\rightarrow S)$ where $X_0=X$, $X_1=X\times_S X$ and $$ X_n=\underbrace{ X\times_S ...
1
vote
0answers
144 views

Derived category of product of complex manifolds

Let $X$ and $Y$ be a compact complex manifold. Is it possible to describe the derived category $D^b(X\times Y)$ of coherent sheaves in terms of $D^b(X)$ and $D^b(Y)$? I am particularly interested in ...
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.
1
vote
0answers
118 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 $MHM(X)$ ...
1
vote
0answers
149 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 ...
1
vote
0answers
310 views

Additive functors preserving quasi-isomorphism

Let $F: \mathcal{A} \rightarrow \mathcal{B}$ be an additive functor between abelian categories (with enough injectives and projectives) and $K^\cdot, L^\cdot$ objects of $\textrm{Ch}(\mathcal{A})$. ...
0
votes
0answers
34 views

Derived tensor product and restriction

Let $A$ be a commutative ring, and let $B$ be a commutative $A$-algebra. We have a restriction map $(-)_A : D(B) \to D(A)$ which takes an object of the derived category of $B$-modules $M$, and ...
0
votes
0answers
249 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 ...
0
votes
0answers
165 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 ...
0
votes
0answers
98 views

descent of a complex of sheaves

Let X a projective variety over an algebraically closed field on which an abelian variety $A$ acts freely. Then $X/A$ is a projective variety. Let $f:X\rightarrow X/A$ Let $K\in D_{c}^{\leq ...
0
votes
0answers
93 views

Periodicity Theorem for D(R)

For a derived category of a Noetherian ring (or perhaps more generally), can we talk about a Periodicity Theorem? We have Thick Subcategory Theorems and Nilpotence Theorems (HPS 91) for D(R), and in ...
0
votes
0answers
118 views

D(R) versus Ho(HR)?

Given an algebraic ring, how is its derived category related to the homotopy category of HR modules? Thanks. This is essentially a reference request, since I know there may be a lot (or nothing) to ...
0
votes
0answers
130 views

functors determined by associated bimodules

When is a Ch-enriched (enriched over the category of chain complexes) functor $F: A \to B$ determined by the bimodule $A^{op} \otimes B \to Ch_{dg}$, $(a,b) \mapsto Hom_B(F(a),b)$. Or in general how ...