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

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18
votes
3answers
2k views

Why are derived categories natural places to do deformation theory?

It seems to me that a lot of people do deformation theory (of schemes, sheaves, maps etc) in derived category (of an appropriate abelian category). For example, the cotangent complex of a morphism ...
5
votes
1answer
531 views

Kodaira-Spencer map as a “differential”

Using the laguage of derived category, the Kodaira-Spencer map $\kappa(x) : Ext^1_X(k(x), k(x)) \rightarrow Ext^1_X(\mathcal F_x, \mathcal F_x)$ can be described as a Fourier-Mukai transformation ...
0
votes
0answers
123 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 ...
1
vote
2answers
500 views

Theorem on composition of derived functors, question about proof

I got a question about a proof I found in Gelfand-Manin's "Methods of homological algebra" (Page 200): Theorem 1. Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be three abelian categories, $F: ...
1
vote
0answers
322 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 $\textrm{Ch}(\mathcal{A})$. ...
2
votes
1answer
294 views

Showing morphism of sheaves is zero

I work in derived category $D^b(X)$ of constructible sheaves on a reasonable space $X$. Let $j\colon U\to X$ be an open inclusion and $i\colon Y\to X$ the closed complement. Let $M,N\in D^b(X)$ and ...
3
votes
1answer
405 views

Unbounded complexes, resolutions and computation of derived functors

Hey guys, let $F: \mathcal{A} \rightarrow \mathcal{B}$ be a left exact functor between abelian categories with enough injectives, let $K \in Kom(\mathcal{A})$ be an unbounded complex, I've heard that ...
2
votes
2answers
278 views

Obtaining derived functors from derived functors of similar complexes or “bluntly truncated” unbounded complexes (without adding 0's to the left)

I don't know if I'm actually using the right terminology here, to be clear I'm going to state explicitly what I'm trying to figure out to see if I can be pointed in the right direction: Let $F: ...
4
votes
0answers
195 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 $$ ...
3
votes
1answer
397 views

Semiorthogonal decompositions for Fano 3-folds and 4folds

Let $X$ be a projective Fano 3-fold or 4-fold and let $D^b(X)$ be the bounded derived category of coherent sheaves on $X$. For what $X$ is it known a semi orthogonal decomposition into indecomposable ...
4
votes
0answers
291 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
1answer
412 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 ...
7
votes
1answer
568 views

got any tricks to build up t-structures on derived categories?

Are there any good tricks to construct a heart of a t-structure? (I'm thinking on the derived category of coherent sheaves of some variety) I'll start with the only one I know. If $(T,F)$ is a ...
4
votes
2answers
304 views

Cone of a morphism in an abelian category when considered as a morphism in derived category. Connection between 4-term exact sequences and distinguished triangles.

Let $\mathcal{A}$ be an abelian category and let $$ 0 \rightarrow E \rightarrow F \rightarrow G \rightarrow 0 $$ be a short exact sequence. Then in $D(\mathcal{A})$, the derived category of ...
10
votes
1answer
399 views

Fullness of pullback functor in algebraic geometry

Given $f:X\to Y$ a morphism of schemes (or stacks if it's not harder), I am interested in a geometric reformulation of the condition that the functor $f^*:D^b(Coh(Y))\to D^b(Coh(X))$ is full. I can ...
7
votes
1answer
397 views

Lifting isomorphisms between derived categories

(Remark: I first asked this question at math.stackexchange. As it received no answer, I'm posting it here). Suppose $A$ and $B$ are commutative rings. Let $A\to B$ be a surjective ring homomorphism. ...
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 ...
1
vote
1answer
199 views

When do dg-lifts exist?

Let $\mathcal A$ and $\mathcal B$ be abelian categories (with enough injectives and countable products) and $$F:D^+(\cal A) \rightarrow D^+(\cal B)$$ be a triangulated functor. I am interested ...
11
votes
0answers
373 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 ...
4
votes
0answers
229 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 ...
6
votes
0answers
468 views

functor before cat?

As i read the literature, derived functors were there several years before derive categories - right?
5
votes
1answer
388 views

Generators of the derived category

For a ring $R$, which is a finite-dimensional algebra over a field, the category of finite-dimensional, projective, right $R$-modules, $\mathcal{P}_R$ is generated by the indecomposable projective ...
5
votes
1answer
495 views

Is the bounded derived category of coherent sheaves of a variety a small category?

The question is in the title. I am trying to apply the Mitchell (Freyd-Mitchell?) embedding theorem, which states that for every small abelian category $A$, there exists a ring $R$ such that A ...
3
votes
2answers
283 views

Earliest/most standard reference for derived categories of hereditary algebras

Let $A$ be a hereditary algebra and let $\mathcal{D}$ be the derived category of bounded complex of finitely generated $A$-modules. Then, for any complex $C_{\bullet}$ in $\mathcal{D}$, we have ...
2
votes
1answer
372 views

algebraic de Rham cohomology functoriality

Suppose that $Y/k$ is a an algebraic variety over a field $k$ of characteristic zero and that $Y\subseteq X$ is a closed embedding into a smooth variety over $k$. Then the completion of the de Rham ...
11
votes
1answer
661 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 ...
0
votes
1answer
733 views

Terminology - subcategories of Abelian categories

Hello, I have terminological question. Consider the following properties of a full subcategory $B \subset A$, where $A$ is an abelian category, and we assume $B$ to be closed under finite direct ...
2
votes
1answer
275 views

Equivalent forms of the proper base change isomorphism

$\DeclareMathOperator{\Nat}{Nat}$In a current project, I am trying to "commute" $!$ and $*$ functors that are both upper or both lower. (Sheaf-theoretic context: constructible étale sheaves.) ...
7
votes
2answers
761 views

The composition of derived functors - commutation fails hazardly?

Hello, When we have left exact functors $F: A \to B , G: B \to C$ (between abelian categories), we would like sometimes to state that $D(GF)=D(G)D(F)$ (functors between bounded below derived ...
1
vote
1answer
648 views

Adjunctions between derived functors

Given an adjunction $F\dashv G$ between functors between Abelian categories, we know that $F$ is right exact and $G$ is left exact so there are derived functors $LF$ and $RG$ between (bounded above, ...
0
votes
1answer
372 views

Recollements and global dimension

Let $A, B, C$ be algebras. Suppose that $D^b(A)$ (the bounded derived category of $A$) admits a recollement relative to $D^b(B)$ and $D^b(C)$. Then, by a result of Alfred Wiedemann's paper "On ...
0
votes
1answer
555 views

derived functors and triangulated categories

If you derive a right exact functor $F$ you get a functor normally denoted by $RF$ on the derived category. Similarly, if you start with a left exact functor $G$ you get a functor normally denoted by ...
7
votes
1answer
439 views

l-adic vs complex Perverse Sheaves

Let $X$ be a scheme of finite type over $Spec(\mathbb{C})$. Let $X_{an}$ denote the associated complex analytic space. After fixing an isomorphism $\overline{\mathbb{Q}}_l\cong \mathbb{C}$, by ...
3
votes
0answers
174 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 ...
5
votes
1answer
481 views

More questions about Verdier duality (and related math)

The first set of questions can be found here: Understanding (the wiki page on) Verdier duality I'm fairly confident that I understand something wrong, so I'll write down here clearly what my set of ...
6
votes
2answers
1k views

Understanding (the wiki page on) Verdier duality

My familiarity with concepts related to derived categories is only tangential, and little by little I intend to get more comfortable with them. I was playing around with Caldararu's introduction to ...
1
vote
1answer
281 views

dual modules in the derived category

Hello, This question is a follow up to Sasha's comment in Duals and Tensor products. In the comment there, it is claimed that the given a ring $A$ and modules $M$, $N$, there is an isomorphism $$ ...
2
votes
1answer
262 views

For an l-adic sheaf (F_n), why is the complex F_n of finite Tor dimension?

Let $X$ be a variety and let $\mathcal{F}=(\mathcal{F}_n)_{n\geq 0}$ be a (constructible) $\ell$-adic sheaf on $X$. Let $K_n$ be the object in the derived category $D(X,\mathbf{Z}/\ell^{n+1})$ of ...
3
votes
1answer
537 views

Perfect complexes and RGamma(X,F) without mentioning derived categories

Let $A$ be a commutative noetherian ring. Let $K_{parf}(A)$ be the full subcategory of the homotopy category $K(A)$ of $A$-modules whose objects are bounded complexes of finitely generated ...
9
votes
3answers
1k views

Derived categories of (coherent) sheaves of modules: exceptional images, gluing, and proper descent?

I am interested in the properties of (the derived categories) of various categories of (coherent) sheaves of modules (over varieties). I would like to understand in what extent these properties are ...
6
votes
1answer
597 views

Do I need to know what an infinity-Gerstenhaber algebra is, and if so, what is it?

I am in the following situation. I have two (rather explicit and specific) dg commutative algebras $R,S$ over a field of characteristic $0$. In fact, $S$ is an $R$-algebra, in that I have a map $R ...
11
votes
1answer
654 views

Analytic Torsion in the Derived Category

I recently learned about analytic torsion and about the amazing Cheeger-Muller theorem identifying analytic and Reidemeister torsion for compact Riemannian manifolds. Now analytic torsion is defined ...
2
votes
3answers
714 views

Exceptional collections with many Exts

Background definitions: Let $D$ be a triangulated category arising in nature (for instance as the cohomology category of a dg category). An object $E$ in $D$ is called exceptional if $RHom(E,E)$ is ...
6
votes
2answers
580 views

t-structures on the derived category of finitely generated abelian groups

is it possible to explicitly parametrise all the t-structures on the derived category of finitely generated abelian groups?
8
votes
1answer
943 views

What is the Hochschild cohomology of the dg category of perfect complexes on a variety?

Let $X$ be a quasi-projective variety over a field $k$. Let $D_{qcoh}$ be a dg enhancement of the unbounded derived category of quasi-coherent sheaves over $X$, and $D_{perf}$ its full subcategory of ...
3
votes
1answer
266 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 ...
18
votes
0answers
884 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 ...
10
votes
5answers
2k views

Sheaves without global sections

The line bundle $O(-1)$ on a projective space or $O(-\rho)$ on a flag variety has a property that all its cohomology vanish. Is there a story behind such sheaves? Here are more precise questions. Let ...
7
votes
4answers
755 views

Intuition about the triangulation of a homotopy category K(A)

Let $\cal{A}$ be an additive category. Given a morphism of (cochain) complexes $f:X\rightarrow Y$ we can form the mapping cone $C_f$, which is the complex $X[1]\oplus Y$ with differential given by ...
6
votes
4answers
2k views

group of Yoneda extensions and the EXT groups defined via derived category

Given an abelian category C, we can form the Yoneda extensions $YExt^i(X,Y)$ to the equivalent classes of $i$-extensions of X by Y. Given any abelian category C, we can always formulate the derived ...