# Questions tagged [derived-categories]

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

589
questions

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votes

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121 views

### External tensor product Calabi-Yau DG categories

Let $\mathcal{C}$ be a smooth proper DG-category such that the shift $[p]$ is a Serre functor for $D^{perf}(\mathcal{C})$ (we say that $D^{perf}(\mathcal{C})$ is $p$-Calabi-Yau). I am looking for a ...

**3**

votes

**1**answer

326 views

### Pseudocoherent analogue of compact + nuclear = dualizable?

$\DeclareMathOperator\RHom{RHom}\DeclareMathOperator\Map{Map}\DeclareMathOperator\id{id}\DeclareMathOperator\colim{colim}$Let $(\mathcal A,\mathcal M)$ be a (normalized) analytic ring defined in ...

**2**

votes

**1**answer

309 views

### A Fourier-Mukai equivalence between non trivial component of cubic threefold and degree 14 prime Fano threefold

Consider a cubic threefold $Y$ and its associated degree $14$ prime Fano threefold $X$, we have the equivalences of non-trivial components of $D^b(Y)$ and $D^b(X)$, i.e, $\mathcal{A}_X\cong\mathcal{B}...

**14**

votes

**2**answers

521 views

### Recovering an abelian category from the Ext of its simple objects

Let $C$ be an abelian category, assume for simplicity that $C$ is enriched over $Vect_k$ (vector spaces over $k$) for some fixed field $k$.
Suppose also that $C$ is both Artinian and Noetherian, so ...

**2**

votes

**1**answer

343 views

### Does $R\hat{f}_*\mathcal{F}=\hat{f}_*\mathcal{F}$ hold for affine adic morphisms?

Let $f:X\to Y$ be an affine morphism of locally Noetherian schemes. By this, we know that $Rf_*\mathcal{F}=f_*\mathcal{F}$ for any quasi-coherent sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules (the ...

**7**

votes

**0**answers

179 views

### Direct summands of a pushforward in the derived category of coherent sheaves

For a Noetherian scheme $X$, let $D^b(X)$ denote the bounded derived category of coherent sheaves on $X$.
Let $X$ be a Noetherian scheme, $i:Y \hookrightarrow X$ a closed subscheme and $F$ an object ...

**5**

votes

**1**answer

525 views

### Derived category of abelian sheaves on a site equivalent to sheaves on the derived category of abelian groups

Reading Scholze's notes on Condensed Mathematics it is mentioned that when considered as $\infty$-categories,
$$ D(\operatorname{Cond(Ab)}) \cong \operatorname{Cond}(D(\operatorname{Ab}))$$
and that ...

**8**

votes

**0**answers

99 views

### Equivariant coherent sheaf category for unipotent group actions

Suppose $U$ is a complex algebraic unipotent group. Let $X$ be a projective variety with a $U$-action. For simplicity, we may assume that there are only finite many $U$ orbits on $X$. The primary ...

**8**

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130 views

### What's the relationship between spherical twist functors and tilting?

I've been reading about connections between Coxeter groups and preprojective algebras, and I keep running into two operations on the derived categories of preprojective algebras which seem very ...

**5**

votes

**1**answer

1k views

### Proposition 5.13 (ii) in Scholze's Perfectoid Spaces

In Proposition 5.13 (ii) in Scholze's Perfectoid Spaces, we have $R \to S$ a morphism of $\Bbb F_p$-algebras and the assumption that the relative Frobenius $\Phi_{S/R}$ induces an isomorphism $R_{(\...

**8**

votes

**1**answer

227 views

### Checking exactness of a triangle on stalks

Suppose I have a triangle
$$A \to B \to C \to A[1]$$
in $D(Ab(X))$, the derived category of abelian sheaves on some topological space $X$. For each $x \in X$, there is an exact functor $D(Ab(X)) \to D(...

**1**

vote

**1**answer

70 views

### Rigid, maximal rigid and cluster-tilting objects

Let $\mathcal{D}$ be a $k$-linear, Hom-finite triangulated category with a Serre functor $\mathbb{S}$. An important class of objects in $\mathcal{D}$ are the cluster-tilting objects, which have many ...

**6**

votes

**1**answer

426 views

### Functorial kernel in derived category

By the work of Verdier, we know that cones in a triangulated category $\mathcal{T}$ are functorial if and only if $\mathcal{T}$ is semisimple abelian. However, in these notes, it is said that
In the ...

**3**

votes

**0**answers

98 views

### Cellularization functor and cohomological dimension

I'm little bit confused by the following problem. And I was hoping for some help.
Here is the set up: I have an associative ring $R$. Let $M$ and $N$ two $R$-modules such that
$F_{n}\rightarrow F_{n-1}...

**3**

votes

**1**answer

343 views

### Duality between $D^b(\mathbb{Z})$ and $D(\mathrm{Solid})^\omega$

My question is about Corollary 6.1(ii) in Lectures on Condensed Mathematics by Scholze (page 41). Here is the claim:
The derived category $D(\mathrm{Solid})$ is compactly generated, and the full ...

**4**

votes

**0**answers

107 views

### Non-uniqueness of $C$ with $f_!(C) = f_*(1_{\mathcal{C}})$

$\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\tensor}{\otimes}$
$\newcommand{\colim}{\rm colim}$
$\...

**5**

votes

**1**answer

169 views

### What is the smallest group for which Broué's abelian defect group conjecture has not yet been verified?

Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $k:=\overline{\mathbb{F}_p}$.
Let $b$ be a $p$-block of $kG$ with abelian defect group $D$. Let $H:=N_G(D)$. Let $c$ be the Brauer ...

**4**

votes

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80 views

### Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts have enough injectives?

For which smooth projective $P$ over a field there exists a bounded $t$-structure $t$ on the bounded derived category of coherent sheaves $D^b(P)$ such the heart $Ht$ of $t$ has enough injectives? ...

**1**

vote

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44 views

### Bound on Hochschild dimension of a dg-algebra

Consider a dg-algebra $A$, is there any way I can estimate the Hochschild dimension, or global dimension of $A$?
More precisely the algebra that I am considering is the Endomorphism dg-algebra $\...

**2**

votes

**0**answers

61 views

### Computing $m_3$ of an $\mathrm{Ext}$-algebra

I currently am studying $A_{\infty}$-obstructions and to compute them I need to compute at least the $A_3$-data of an $\mathrm{Ext}$-algebra.
More precisely, I have a functor $F:\mathcal{D}\left(X\...

**3**

votes

**1**answer

96 views

### Derived functor and bi-module

If A and B are finite dimensional k-algebras, k is a field. $_{A}G\in A-mod$ is a Gorenstein projective module, then we have $RHom_{A}(G,A)\simeq Hom(G,A)$ since $Ext_{A}^{i}(G,A)=0$ for any $i\in \...

**1**

vote

**1**answer

212 views

### (Bridgeland stability conditions)How to get the heart of a bounded t-structure on $D^b(P^1 \times P^2)$?

I have already known how to get the heart of a bounded t-structure on $D^b(P^n)$ by Macri`s paper,
https://arxiv.org/abs/math/0411613.
However I cannot purpose analogously on $D^b(P^1 \times P^2)$.
...

**2**

votes

**1**answer

229 views

### (Bridgeland stability conditions) How can I get the heart of a bounded t-structure on $D^b(P^3)$?

In the article, Bayer, Arend; Macrì, Emanuele; Toda, Yukinobu, Bridgeland stability conditions on threefolds. I Bogomolov-Gieseker type inequalities, J. Algebr. Geom. 23, No. 1, 117-163 (2014). ...

**3**

votes

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130 views

### Derived category of an abelian monoidal category

For any abelian category $\mathcal{A}$, we can consider its derived category $\mathcal{D(A)}$, which is naturally triangulated. If $\mathcal{A}$ is endowed with a monoidal structure (bilinear with ...

**7**

votes

**1**answer

242 views

### Heart of a bounded $t$-structure on the derived category of coherent sheaves

Let $X$ be an elliptic curve and $D(X)$ the bounded derived category of $Coh(X)$, coherent sheaves on $X$. If $(D^{\leq 0}, D^{>0})$ is a bounded $t$-structure, then can we already say that the ...

**10**

votes

**1**answer

450 views

### K-equivalence ⇒ isomorphism of Chow motives?

An old conjecture of Bondal–Orlov–Kawamata predicts that K-equivalent varieties are D-equivalent, see Kawamata's paper D-equivalence and K-equivalence for definitions. In particular this applies to ...

**2**

votes

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100 views

### The derived $\infty$-category of sheaves on a site is closed symmetric monoidal

Let $X$ be a quasicompact semiseparated scheme. I am trying to recover the (closed) symmetric monoidal structure on $\mathcal{D}(\mathrm{QCoh}(X))$, the derived $\infty$-category of quasicoherent ...

**3**

votes

**1**answer

132 views

### Derived Category of strictly simplicial algebraic space vs. systems of objects in the derived categories

Let $X_{\bullet}^+$ be a strictly simplicial algebraic space and for a morphism $\delta:[m]\to[n]$ in $\Delta^+$, let $\delta:X_n\to X_m$ also denote the associated map (by abuse of notation). Then ...

**6**

votes

**1**answer

266 views

### What is equivariant chains on a representation sphere?

For a finite group $G$ and a finite-dimensional real representation $V$ of $G$, denote by $S^V$ the one-point compactification of $V$, with basepoint at infinity.
What is the reduced chain complex $...

**5**

votes

**1**answer

122 views

### Rickard's strengthening of Broué's abelian defect group conjecture and the lifting of some equivalences up to splendid derived equivalences

Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $K:=\overline{\mathbb{F}_p}$.
Let $b$ be a $p$-block of $G$ with abelian defect group $D$. Let $H:=N_G(D)$. Let $c$ be the Brauer ...

**2**

votes

**1**answer

236 views

### Conflicting definitions of RHom

I am trying to understand the bifunctor $R\operatorname{Hom} : D(\mathcal{A}) ^{op} \times D(\mathcal{A}) \to D(\operatorname{Ab})$ (I am also interested in the total right derived functor of the ...

**3**

votes

**1**answer

155 views

### 2-limits of triangulated categories

Let $\mathcal{D}_{i}$ be a family of triangulated categories, labelled by a countable poset $I$ with a lowest element. Suppose further that for $i\leq j$, we have exact functors $F_{i,j}: \mathcal{D}_{...

**6**

votes

**1**answer

121 views

### Serre functor on the category $Perf(A)$, $A$ - k-algebra

Consider a finite-dimensional $k$-algebra $A$ of finite global dimension. Then it is known that the Serre functor on $D^b(mod-A)$ exists and is given by the Nakayama functor. The proof goes something ...

**9**

votes

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141 views

### 3x3 lemma in triangulated categories

I am currently reading Le Stum's Rigid Cohomology and have encountered the following passage (proof of Proposition 5.2.16):
The deduction made here seems to be purely "triangulated category-...

**4**

votes

**0**answers

162 views

### Which derived categories of coherent sheaves are equivalent (or “$t$-related”) to derived categories of rings?

As far as I understand, it was Beilinson who proved that the bounded derived category of coherent sheaves $D^b(\mathbb{P}^n)$ is equivalent to the bounded derived category of a certain (non-...

**3**

votes

**0**answers

74 views

### Is the secondary Euler characteristic a categorical trace?

Context: The ordinary Euler characteristic of a complex (satisfying appropriate finiteness conditions so that all cohomology groups are finite-dimensional over some field ''k'', say, and only finitely ...

**10**

votes

**1**answer

930 views

### Computations in condensed mathematics, page 32-34

I started reading the Lectures on Condensed Mathematics. I am looking at the material at page 32-34. I have three fundamental computation questions:
At the last line of pg 32 - it seems to imply that ...

**4**

votes

**1**answer

182 views

### Questions about $\text{Perf}(A)$ of dg algebra $A$

[ALEXEY ELAGIN AND VALERY A. LUNTS, p.4.] Recall that triangulated category $\text{Perf}(A)$ is defined as the full thick triangulated subcategory of $D(A)$ generated by the dg $A$-module $A$.
[...

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vote

**0**answers

110 views

### Perverse sheaves and maximal genus Gopakumar-Vafa invariants

Let $f: X \to Y$ be a proper morphism between complex varieties (the varieties as well as the map may be non-smooth) and let $\phi \in \text{Perv}(X)$ be a perverse sheaf on $X$. Given this data, it ...

**21**

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536 views

### What is the status of a result of Kontsevich and Rosenberg?

In their influential paper Noncommutative Smooth Spaces (https://arxiv.org/abs/math/9812158), Kontsevich and Rosenberg define the notion of a noncommutative projective space. In Section 3.3 they ...

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vote

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62 views

### $L^r_M = i_* \circ \hat{L}^{r-1}_M \circ i^*$ by the projection formula and the Poincare duality

This is a question arising when I am reading
M. A. A. de Cataldo, L. Migliorini - The Hard Lefschetz Theorem and the topology of semismall maps, Ann. sci. École Norm. Sup., Serie 4 35 (2002) 759-772.
...

**2**

votes

**1**answer

157 views

### Cohomology of normal bundle and tangent bundle on Gushel-Mukai threefold

Let $X$ be a smooth general ordinary Gushel-Mukai threefold. There is an embedding $X\rightarrow\mathrm{Gr}(2,5):=G$. Consider the normal bundle $\mathcal{N}_{X|G}$, how to compute cohomology of this ...

**11**

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254 views

### Is there a bestiary of “derived 2-vector spaces”?

The appendix "A bestiary of 2–vector spaces" of Bartlett, Douglas, Schommer-Pries, Vicary, "Modular categories as representations of the 3-dimensional bordism 2-category" analyzes ...

**3**

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56 views

### Naive pushforward of D-modules and Gauss--Manin connection

Suppose that $f\colon X\to Y$ is a morphism of smooth quasi-projective varieties over a field of characteristic $0$. We then have a naive pullback functor $f^\circ:=\mathcal D_{X\to Y}\otimes_{f^{-1}\...

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54 views

### Is there a direct way to show Fano surface of lines and conics on the pairs of Fano threefolds isomorphic?

I am considering the following setting:
Let $(Y_d, X_{4d+2})$ be the pair of degree $d$ and index 2 Fano threefold $Y_d$ and degree $4d+2$ index 1 Fano threefold and both of them are Picard number 1. ...

**5**

votes

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144 views

### Smoothness of a variety implies homological smoothness of DbCoh

I have been told that $D^bCoh(X)$ is homologically smooth if $X$ is a smooth variety, and I am trying to construct a proof. My background is not in algebra, so I apologize for elementary questions.
It ...

**11**

votes

**1**answer

411 views

### Embedding of a derived category into another derived category

I am considering the following two cases:
Assume that there is an embedding: $D^b(\mathcal{A})\xrightarrow{\Phi} D^b(\mathbb{P}^2)$and the homological dimension of $\mathcal{A}$ is equal to $1$($\...

**5**

votes

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215 views

### When is the cotangent complex perfect?

Let $X\rightarrow S$ be a proper flat morphism of schemes.
When is the cotangent complex $L_{X/S}$ perfect ?
It is well known, that for local complete intersections the cotangent complex is perfect, ...

**2**

votes

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141 views

### 2 K3s and cubic fourfolds containing a plane

Two K3 surfaces show up when talking about cubic fourfolds containing a plane. Let $P\subset X\subset \mathbb{P}^5$ be the plane inside the cubic. Since $P$ is cut out by 3 linear equations then $X$ ...

**3**

votes

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231 views

### Are two versions of Kuznetsov components equivalent?

Let $X$ be a degree $10$, genus $6$ index $1$ and Picard number $1$ smooth Fano threefold. There are two versions of Semi-orthogonal decompositions. The First version is $$D^b(X)=\langle\mathrm{Ku}(X)...