Questions tagged [derived-categories]
For questions about the derived categories of various abelian categories and questions regarding the derived category construction itself.
776
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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 ...
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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 ...
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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 ...
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1
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Non-uniqueness of $C$ with $f_!(C) = f_*(1_{\mathcal{C}})$
$\newcommand{\Cc}{\mathcal{C}}$
$\newcommand{\Dd}{\mathcal{D}}$
$\newcommand{\Z}{\mathbb{Z}}$
$\newcommand{\Q}{\mathbb{Q}}$
$\newcommand{\tensor}{\otimes}$
$\newcommand{\colim}{\rm colim}$
$\...
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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 ...
- 1,685
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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? ...
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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 $\...
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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\...
- 203
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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 \...
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(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 $D^b(P^1 \times P^2)$.
...
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(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). ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
- 1,685
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1
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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 ...
4
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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}_{...
- 43
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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 ...
- 1,021
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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-...
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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-...
- 16.5k
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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 ...
- 1,986
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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 ...
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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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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 ...
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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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$L^r_M = i_* \circ \hat{L}^{r-1}_M \circ i^*$ 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.
...
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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 ...
- 1,952
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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 ...
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Naïve 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 naïve pullback functor $f^\circ:=\mathcal D_{X\to Y}\otimes_{f^{-1}\...
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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. ...
- 1,952
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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 ...
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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 $1$($\...
- 1,952
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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, ...
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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 $X$ ...
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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)...
- 1,952
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When is an algebra derived indecomposable?
Call a finite dimensional (acyclic) quiver $K$-algebra A derived indecomposable in case $A$ is not derived equivalent to an algebra of the form $B \otimes_K C$.
For example when the number of simples ...
- 26.1k
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Does formation of the derived $\infty$-category preserve pushouts?
Let $B\leftarrow A\to C$ be a diagram of commutative rings, and let $\mathcal{D}(A)$ be the derived $\infty$-category of $A$-modules (as in Lurie's "Higher Algebra"). Then is there an ...
- 1,129
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Cone of a morphism of complexes that are concentrated in degree $0$ and $1$
Let $R$ be a ring and $f:A\to A'$ and $g:B\to B'$ be morphisms of $R$-modules. Let $h:C_{\bullet}\to C_{\bullet}'$ be a morphism of $R$-module complexes fitting in a morphism of distinguished ...
- 1,479
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Smallness condition for augmented algebras
I'm not sure this question is research level question. Sorry in advance.
Hypothesis
$k$ is a commutative ring.
$A$ is an augmented $k$-algebra.
$A^e$ is defined as the $k$-algebra $A\otimes_{k}A^{op}$...
- 511
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2
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On the link between homology and homotopy
In the last semester I learned homological algebra and higher category theory/homotopy theory.
But I am kind of confused when I try to really understand the link between the two subjects (this is ...
- 546
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Cone of morphism in families
I am working in derived category $D^b(X)$ of coherent sheaves on a smooth projective varitey.
Let $E,F$ be two sheaves on $X$, with $\mathrm{R}Hom(E,F)=k\oplus k[-1]$, I consider the following ...
- 1,952
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$H^{p,q}(X)$ versus $H^{q}(X, \bigwedge^p TX)$
Let $X$ be a Kahler manifold. To $X$ one can associate the cohomology groups $H^{p,q}(X)$, and $H^{(0,q)}(X, \bigwedge^p TX)$ with $TX$ being the holomorphic tangent bundle of $X$.
Is there a general ...
- 319
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Variety over $\mathbb{F}_p$ that does not embed into flat scheme over $\mathbb{Z}/p^2\mathbb{Z}$
Let $X\to\mathrm{Spec}\:\mathbb{F}_p$ be a smooth proper morphism. Is there a closed immersion $X\to Y$ where $Y$ is flat of finite type over $\mathbb{Z}/p^2\mathbb{Z}$?
As mentioned in the comments ...
user158636
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Computing the derived $I$ invariants $D(G\operatorname{Rep})\to D((G/I)\operatorname{Rep})$
In Scholze's paper The Langlands–Kottwitz approach for the modular curve (published version) about $\operatorname{GL}_2$ when he wants to find the relation between the semisimple trace and the usual ...
- 1,043
4
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0
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159
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detecting a semi-free module from its bar-resolution
Let $A$ be a DG-algebra over a field (say $k$). A DG-module $M$ over $A$ is said to be semi-free if it admits an exhaustive filtration $0 = M_0 \subset M_1 \subset \ldots \subset M_p = M$ such that ...
- 7,200
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0
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107
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Compute Cech cohomology with two open sets
Let $(X,\mathcal{O}_X)$ be a topological ringed space and $(U,V)$ be an open covering of $X$ (i.e. $U$ and $V$ are two open subsets of $X$ such that $U\cup V=X$). Let $\mathcal{F}$ be a sheaf of $\...
- 1,479
4
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0
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126
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Morphism of distinguished triangles where one of the arrows is a quasi-isomorphism
Let $R$ be any ring and let $A\to B\to C\to [1]$ and $A'\to B'\to C'\to [1]$ be distinguished triangles of complexes of $R$-modules. Let $f:A\to A'$, $g:B\to B'$ and $h:C\to C'$ be morphism of ...
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