**4**

votes

**0**answers

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
$$ ...

**3**

votes

**1**answer

386 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

**0**answers

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

**1**answer

408 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

**1**answer

549 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

**2**answers

295 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

**1**answer

388 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

**1**answer

392 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

**0**answers

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

**1**answer

197 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 ...

**10**

votes

**0**answers

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 ...

**4**

votes

**0**answers

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 ...

**5**

votes

**0**answers

466 views

### functor before cat?

As i read the literature, derived functors were there several years before derive categories - right?

**5**

votes

**1**answer

387 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

**1**answer

489 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

**2**answers

278 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

**1**answer

369 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

**1**answer

654 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

**1**answer

705 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

**1**answer

272 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

**2**answers

754 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

**1**answer

631 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

**1**answer

365 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

**1**answer

543 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

**1**answer

429 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

**0**answers

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 ...

**5**

votes

**1**answer

463 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

**2**answers

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

**1**answer

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

**1**answer

260 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

**1**answer

526 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

**3**answers

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

**1**answer

594 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

**1**answer

645 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

**3**answers

697 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

**2**answers

574 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

**1**answer

904 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

**1**answer

263 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
...

**17**

votes

**0**answers

861 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

**5**answers

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

**4**answers

732 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

**4**answers

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 ...

**27**

votes

**7**answers

4k views

### A down-to-earth introduction to the uses of derived categories

When I was learning about spectral sequences, one of the most helpful sources I found was Ravi Vakil's notes here. These notes are very down-to-earth and give a kind of minimum knowledge needed about ...

**9**

votes

**0**answers

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

**1**answer

223 views

### Classification of t-structures in derived category of R-mod?

I am looking for a reference talking about the complete(or not)description of t-structures in bounded derived category of $R-mod$, i.e. $D^b(R-mod)$.where $R$ is commutative ring, in particular, ...

**12**

votes

**2**answers

968 views

### What is a flop (and when are they conjectured to give derived equivalences)?

(1) Is the definition of flop given by Wikipedia the industry standard?
(2) Regardless of the answer to (1), when is it expected that a birational transformation gives rise to a derived ...

**8**

votes

**2**answers

1k views

### Derived Physics

Hello to all,
This question will probably be closed down as being off-topic faster than one can say "string theory", but here it goes: I've noticed that the problems I'm working on -the structure of ...

**5**

votes

**3**answers

1k views

### Is this a definition of equivariant derived category?

Let $X$ be a topological space and $G$ be a topological group acting on $X$, both locally compact Hausdorff. Denote by $D^b(X)$ the derived category of sheaves (say of abelian groups) on $X$. We ...

**7**

votes

**3**answers

724 views

### Are the underlying undirected graphs of two mutation-equivalent acylic quivers isomorphic?

Quiver mutation, defined by Fomin and Zelevinsky, is a combinatorial process. It is important in the representation theory of quivers, in the theory of cluster algebras, and in physics.
We consider ...

**5**

votes

**2**answers

666 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 ...