**2**

votes

**0**answers

40 views

### Are there any interesting examples of geometric triangulated categories with the Jordan-Holder property?

In this paper, Kuznetsov mentions that the following triangulated categories have the Jordan-Holder properties
$\mathbf{D}(\Bbb P^1)$ and $\mathbf{D}(\Bbb P^1/\Gamma)$
connected Calabi-Yau ...

**5**

votes

**0**answers

82 views

### Constructible sheaves on general stratified spaces

I am not an expert in the field, so my question might be rather standard. Let $X$ be a compact metric space. Assume that $X=\cup_{i=1}^NS_i$ is a finite disjoint union of locally closed topological ...

**3**

votes

**2**answers

144 views

### Is there a compact generated triangulated category which does not have a compact generator?

Let $\mathcal{T}$ be a triangulated category which has arbitraty direct sums. An object $E\in \mathcal{T}$ is called compact if the functor Hom$(E,-)$ commutes with arbitrary direct sums.
A ...

**0**

votes

**0**answers

69 views

### Quotient of triangulated category? (quiver)

This maybe a stupid question, but I really want to know the answer:
Background: Given a quiver with potential, one can consider the derived category of the complete Ginzburg algebra of it, then ...

**4**

votes

**0**answers

226 views

### The derived version of the Grothendieck spectral sequence

Consider the (very well known) Grothendieck spectral sequence for composition of functors $\mathcal F: \mathcal A \to \mathcal B$ and $\mathcal G: \mathcal B \to \mathcal C$ between abelian categories ...

**6**

votes

**1**answer

248 views

### The naive approach to deriving profunctors - What's wrong with it?

Let $(\mathcal{C,W})$ be relative category (equipped with a wide subcategory of weak equivalences satisfying 2 out of 3 property). Consider a profunctor $F: \mathcal{C \times D^{op}} \to \mathsf{Set}$ ...

**1**

vote

**0**answers

100 views

### On the category of $D$-modules

Let $X$ be a smooth variety over an algebraically closed field $k$ of char. $0$.
1) Is the abelian category $M(X)$ of $D$-modules on $X$, which are quasi-coherent as $O$-modules, a Grothendieck ...

**0**

votes

**1**answer

227 views

### Spectral sequences to compute Hom's in derived category

Does anybody have a good reference that lists spectral sequences that may be used to compute Hom sets in derived categories (of coherent sheaves, say)?

**3**

votes

**1**answer

139 views

### Characterizing the image of $D(A_f) \rightarrow D(A)$

Let $A$ be a ring and $f \in A$ an element. If $M$ is an $A$-module on which multiplication by $f$ is an isomorphism, then $M$ is in fact an $A_f$-module.
Now suppose that $C \in D(A)$ is a complex ...

**1**

vote

**3**answers

504 views

### Perverse sheaves and tensor product

If $X$ is a connected algebraic variety of finite type over $k$ (with $k$ a field of positive characteristic) of dimension $d$, and if $\mathcal{F}$ and $\mathcal{G}$ are perverse sheaves on $X$ so $(\...

**1**

vote

**0**answers

104 views

### Isomorphism passing to the derived category

Suppose to have an additive right exact functor $F: \mathcal A \rightarrow \mathcal B$ between Abelian categories and suppose that $F(A)=B$ for an object $A$ in $\mathcal A$. Denote with $D(\mathcal A)...

**2**

votes

**0**answers

179 views

### K theory and derived categories

Some months ago I studied Beilinson's paper about generators for the derived category of $\mathbb{P}^n$, "Coherent Sheaves on $\mathbb{P}^n$ and problems of linear algebra".
As next step, I moved to ...

**2**

votes

**0**answers

107 views

### Derived Categories provide a good Framework for Sheaf Cohomology?

I'm a bit new to this sheaf cohomology business. Can someone explain how derived categories provide a good setting for Sheaf Cohomology? I understand that sheaf coho arises as right derived functors, ...

**11**

votes

**2**answers

239 views

### Homotopy property of constructible sheaves on stratified spaces

Let $X$ be a stratified topological space (in my case $X$ is a compact space presented as a finite union of locally closed topological manifolds of finite dimension (strata) such that the closure of ...

**7**

votes

**1**answer

438 views

### what is the universal cover of GL(2,R)?

In the theory of Bridgeland stability conditions one has an action of the universal cover $G'$ of $G = GL^+(2,\mathbb R)$.
What is G'?
I know there is concrete description in terms of pairs (M,f)...

**3**

votes

**1**answer

98 views

### t-structure induced on the Verdier quotient ${\cal T}/\cal S$

Let $\mathfrak t$ be a $t$-structure on a triangulated category $\cal T$. Let $\cal S$ be a thick (or even non-thick) triangulated subcategory, and ${\cal T}/\cal S$ the Verdier quotient.
Is there a ...

**3**

votes

**0**answers

45 views

### Fracturing $t$-structures

$\def\tee{\mathfrak{t}}$ Let $\tee_1,\tee_2$ be two $t$-structures on a triangulated category $\cal T$; call them fracturing if the two fiber sequences $\tau^\le_1X\to X\to \tau^\ge_1X$ and $\tau^\...

**4**

votes

**1**answer

167 views

### On push-forward of the constant sheaf for fibrations

Let $f\colon E\to B$ be a fiber bundle with a connected fiber $F$, $f$ is proper. Let $\underline{\mathbb{C}}_E$ be the constant sheaf on $E$. Let $f_*(\underline{\mathbb{C}}_E)$ denote its direct ...

**24**

votes

**2**answers

565 views

### Complete the following sequence: point, triangle, octahedron, . . . in a dg-category

Let $\mathcal C$ be a pre-triangulated dg-category (or a stable $\infty$-category, if you wish).
An object $X$ in $\mathcal C$ gives a "point":
$$X$$
A morphism $X\xrightarrow f Y$ in $\mathcal C$ ...

**6**

votes

**2**answers

129 views

### Equivalence between a derived subcategory and a subcategory of the derived category

Let $\mathcal{A}$ be a subcategory of $\mathcal{C}$. Let $D(\mathcal{A})$ and $D(\mathcal{C})$ be the associated derived categories. We can define $D_\mathcal{A}(\mathcal{C}) = \{X \in \mathcal{C}\...

**0**

votes

**1**answer

110 views

### Projective resolutions of torsion modules [closed]

Let $l$ be a prime number, $n\in \mathbb{Z}$. Is it true that any finitely generated $\mathbb{Z}/l^n\mathbb{Z}$-module has a finite (left) resolution by free finitely generated $\mathbb{Z}/l^n\mathbb{...

**4**

votes

**1**answer

194 views

### When may “summand of” be dropped from the definition of perfect dg module?

Let $\mathcal{A}$ be a small dg category. In Section 1 of Lunts-Orlov http://arxiv.org/pdf/0908.4187v5.pdf, $Perf(\mathcal{A})$ is defined to be the full DG subcategory of $\mathcal{S}\mathcal{F}(\...

**3**

votes

**0**answers

209 views

### N-periodic derived categories

I have some seemingly basic questions about $N$-periodic derived categories to which I have not found answers in any of the usual places.
Let $R$ be a ring, and let $D(R)_{\mathbb Z/N\mathbb Z}$ ...

**6**

votes

**0**answers

71 views

### 2-functoriality of equivariant derived categories

I am wondering about the 2-functoriality in equivariant derived categories, and I hope that someone can clarify... (apologies if this is a stupid question)
For the more precise formulation, recall ...

**18**

votes

**4**answers

2k views

### determinant of a perfect complex

Say $K_\bullet$ is a bounded complex of vector bundles. I seem to want the determinant of $K_\bullet$ to be the alternating tensor product of the terms of the complex:
$\det(K) = \bigotimes_n \det(...

**4**

votes

**0**answers

93 views

### Short proof of the classification of representation-finite symmetric algebras up to stable equivalence

Assume $K$ is an algebraically closed field and $A$ a finite dimensional $K$-algebra. Assume additionally that $A$ is symmetric and representation-finite.
Then one has the following classification of ...

**6**

votes

**0**answers

191 views

### Is there an obstruction which classifies “quasi-isomorphism but not chain equivalence”?

Fix a ring $R$ and let $C_\bullet$, $D_\bullet$ be (possibly unbounded) chain complexes of $R$-modules. Assume that $f_\bullet:C_\bullet \to D_\bullet$ is a quasi-isomorphism: that is to say, $f$ is a ...

**3**

votes

**1**answer

190 views

### comparison of truncations

I am trying to understand the proof of Lemma 3.0.15 of this paper (Ben-Zvi, Nadler, Preygel - Integral transforms for coherent sheaves).
The context is of two triangulated categories $C,D$ with t-...

**2**

votes

**0**answers

122 views

### derived automorphisms of K3 surfaces of picard rank one

I am aware of work of Bayer-Bridgeland, which describes G=Aut(D(K3)) for a K3 of Picard rank one. Is it possible to use their result to give an explicit presentation of G, in terms of generators and ...

**1**

vote

**0**answers

122 views

### Verdier duality on excellent schemes

Let $f:X\rightarrow Y$ be a regular morphism between $k$-schemes which are noetherian and excellent with a funcion of dimension.
In the book by Illusie-Laszlo-Orgogozo, there is a theorem (4.4.1 in ...

**3**

votes

**2**answers

191 views

### Definition of the differential of the Cone of a morphism of complexes [closed]

Let $(F^\bullet,d_F)$ and $(G^\bullet,d_G)$ be two complexes in an abelian category $\mathbf{A}$.
The complex cone $Cone(\varphi)^\bullet$ of a morphism of complexes $\varphi:F^\bullet \to G^\bullet$ ...

**7**

votes

**0**answers

83 views

### Non-Standard Derived Equivalences of Non-Flat Algebras

I read that for algebras $R$ and $S$ (over a commutative ring), assuming that $R$ or $S$ is flat, the existence of a derived equivalence $\mathcal{D}(R) \to \mathcal{D}(S)$ implies the existence of an ...

**2**

votes

**1**answer

132 views

### How do you rigidify a Bousfield localization?

I'm learning about Bousfield localizations. For a triangulated category satisfying some axioms, a Bousfield localizations can be described as an idempotent functor $L:D \to D$.
I thought there is a ...

**2**

votes

**0**answers

99 views

### Factorization of a map through a square

Assume to have an abelian category $\mathcal{A}$, and consider its derived category $\mathcal{D^b(A)}$). Let $F:\mathcal{D^b(A)}\rightarrow\mathcal{D^b(A)} $ be a functor between triangulated ...

**8**

votes

**2**answers

282 views

### derived categories as presentable DG-categories

Let $A$ be a ring. Is it true that the DG category of unbounded complexes of $A$-modules, localized by quasi-isomorphisms, is cocomplete and compactly generated? What would be a reference for that and ...

**5**

votes

**0**answers

150 views

### Extension of sheaf of Azumaya algebras and derived equivalence

Suppose there is a smooth variety $X$ and a sheaf of algebra $\mathcal{B}$. Let $Z\subseteq X$ be a closed subvariety, whose codimension is large (say $\geq 2$). If the restriction of $\mathcal{B}$ to ...

**4**

votes

**1**answer

181 views

### Does derived equivalence of the fibres imply derived equivalence of the total spaces?

Let $f:X\to B$ and $g:Y\to B$ be smooth morphisms of complex projective varieties. Assume that for every closed point $b\in B$, the fibres $X_b=X\times \kappa(b)$ and $Y_b$ are derived equivalent. ...

**1**

vote

**1**answer

113 views

### Derived equivalence of families of dual abelian varieties

Let $B$ be a smooth projective complex variety and $\pi:X\to B$ a smooth projective map whose fibres $X_b$ are abelian varieties. Let $\psi:Y\to B$ be the naturally associated bundle such that the ...

**4**

votes

**0**answers

169 views

### Derived equivalent varieties with differing integral Mukai-Hodge structures?

For a smooth projective complex variety $X$ of dimension $n$, let $H^i(X)$ denote its integral Hodge structure of weight $i$. Define $\tilde{ H^0}(X) = \bigoplus H^{2i}(X)\otimes \Bbb Z(i)$ and $\...

**8**

votes

**0**answers

163 views

### List of known Fourier Mukai partners?

I'm familiar with some examples of pairs of derived equivalent varieties, for example an abelian variety and its dual, a K3 surface and certain moduli schemes on it, or the Pfaffian-Grassmannian ...

**1**

vote

**0**answers

192 views

### Quantities associated to deformed sheaves

I am trying to figure out what happens to "quantities" associated to a sheaf when one deforms it. I am actually interested in deforming a bounded complex of coherent sheaves but I want to make the ...

**2**

votes

**1**answer

308 views

### Will a tilting sheaf over Z which is a generator over Q be a generator modulo every prime?

Assume you have a smooth quasi-projective scheme $X$ (you can actually assume $X$ is projective over an affine scheme of finite type) defined over $\mathbb Z$ (or if you prefer, a discrete valuation ...

**2**

votes

**1**answer

204 views

### An alternative definition of pseudo-coherent complex

Let $(X,\mathcal{O}_X)$ be a scheme or a general ringed space. First recall that a complex of $\mathcal{O}_X)$-modules $\mathcal{E}^{\bullet}$ is called strictly perfect if $\mathcal{E}^{\bullet}$ is ...

**20**

votes

**1**answer

528 views

### Why would one “attempt” to define points of a motive as $\operatorname{Ext}^1(\mathbb{Q}(0),M)$?

I'm a novice when it comes to motives. (I've read multiple introductory texts.)
I'm attempting to read Galois Theory and Diophantine geometry by Minhyong Kim. In it, he says that "One might attempt, ...

**13**

votes

**1**answer

580 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.
...

**5**

votes

**1**answer

290 views

### An equivalence of derived categories by Happel-Reiten-Smalø

I have a problem in understanding the proof of a theorem by Happel-Reiten-Smalø. The original reference is this article
http://arxiv.org/abs/0911.4473
.
I write down the text of the theorem and a ...

**1**

vote

**2**answers

309 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 ...

**1**

vote

**1**answer

148 views

### Can one drop the locally free assumption in projection formula on a projective bundle?

Let $X$ be a noetherian scheme over $\mathbb{C}$, and let $E$ be a locally free sheaf of finite rank over $X$. Then we have the projective bundle $f: \mathbb{P}(E)\rightarrow X$.
Now $f$ is a flat ...

**2**

votes

**0**answers

72 views

### space of stability condition for an elliptic curve

Let $E$ be an elliptic curve. I want to understand why $\mathrm{Stab}(E)/\mathrm{Aut}(D^b(E))$ is a $\mathbb{C}^\times$-bundle over the moduli space $\mathbb{H}/\mathrm{SL}(2,\mathbb{Z})$ of elliptic ...

**5**

votes

**1**answer

97 views

### Injective model structure on sheaves of bounded complexes of $A$-modules

The following might be very well known for people who works with model categories, but I do not find the answer.
Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain ...