Questions tagged [derived-categories]
Use this tag for questions about a particular construction of homological algebra of an abelian category A that refines and in a certain sense simplifies the theory of derived functors defined on A.
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Geometric interpretation of a generalized Euler sequence
Although similar arguments work for $\mathbb P^n$ for any $n$, let us deal with $X=\mathbb P^1_k$ for simplicity. Recall there is a so-called Euler sequence on $X$ (twisted by $\mathcal O_X(1)$):
$$ 0 ...
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Webeil's Intro to Homological Algebra: does theorem 10.6.3 implicitly reindex cochain complexes to chain complexes?
Let $R$ be a ring, let $\mathbf{D^-(R-mod)}$ denote the derived category of bounded above cochain complexes of $R$-modules, and consider the total tensor product functor
$$
\otimes_R^\mathbf{L}: \...
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mapping cone and derived functor
Let $F:\mathcal{C}\rightarrow \mathcal{D}$ be a left exact functor between abelian categories. If $f:A\to B$ is a morphism in $D^+(\mathcal{C})$, do we have
$$\operatorname{Cone}(\operatorname{R}F(A)\...
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Characterizing pushforwards of sheaves under Galois covers
Let $\pi: Y \to X$ be a Galois cyclic cover with automorphism group $G$ generated via $g$, that arises from a line bundle $L$ on $X$ that is $n$ torsion.
I will give lots of context, but my question ...
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What is extension closure in triangulated categories?
The term extension closure appears in some papers constructing t-structures on triangulated categories, for example in Section 1.2 of
Bayer, Arend; Macrì, Emanuele; Toda, Yukinobu, Bridgeland ...
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Formulation of cap product in group-equivariant sheaf cohomology + applications?
Note: moved to Math Overflow here.
Suppose one has a distinguished cocycle in the group-equivariant sheaf cohomology $\Phi \in H^n(X, G, \mathcal{F})$ for a "nice" -- at least locally ...
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Closed embedding is not faithful on derived category
I was surprised to hear that if $j: Y \to X$ is a closed embedding, then $j_*: D^b(Y) \to D^b(X)$ is in general not faithful (note you shouldn't expect fullness, since $\operatorname{Ext}^1(O_p, O_p) \...
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Derived $\mathrm{Hom}$-functors commute with finite direct products?
Let $\mathcal{R}_X$ be a sheaf of (commutative) rings on a topological space $X$. Let $\mathcal{M}_X, \mathcal{N}_X, \mathcal{P}_X$ be $\mathcal{R}_X$-modules.
Do the derived $\mathrm{Hom}$-functors ...
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Nearby cycle complex of a scheme over a DVR with finite number of singularities
Let $R$ be a DVR and $S = \mathrm{Spec}(R)$ with closed point $s$ and generic point $\eta$. Let $X\to S$ be a scheme over $S$, and denote by $X_s$ and $X_{\eta}$ the special and generic fibers. We ...
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Constructing "maximal" quotient that is a local system on $\mathbb{C}^×$
Consider $\mathbb{C}^×$ with its analytic topology and a sheaf $M$ of $\mathbb{C}$-vector spaces on it. I want to find a way to find a subsheaf $N\subset M$ such that
$N$ has no subsheaves that are ...
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Is there some $Ext$ group that classifies extensions in the **Derived Category**?
Let $\mathcal{A}$ be an abelian category, we have $D(\mathcal{A})$
For $A,C \in \mathcal{A}$, we know that $Ext^1(A,C)$ classifies extensions $0 \to C \to B \to A \to 0$
I want an analog for $Der(\...
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Is a smooth projective variety that is derived equivalent to an abelian variety necessarily an abelian variety?
We say smooth projective varities are derived equivalent if their bounded derived categories of coherent sheaves are equivalent. Thanks to Orlov's work, we know a lot of facts about derived equivalent ...
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If Spectra are analogous to chain complexes, why do we have both $KG(\mathbb{Z}), \mathbb{S}$
I've heard that $Sp$ is analogous to the derived category $Der(Ch\mathbb{Z})$ (I will thus refer to those two categories as the left and right side respectively below).
Namely, every spectra is a ...
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Fourier-Mukai partners that are birational at every point
Let $X$ and $Y$ be smooth projective irreducible varieties that are Fourier-Mukai partners, i.e., have exact equivalent derived categories $D^b(X) \simeq D^b(Y)$. It is well-known that if a derived ...
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reference request support of derived tensor product
I currently am working on trying to compute some Hochschild cohomology of some scheme. However, I should be able to do it as soon as I know/have a reference for the following natural statement:
Let $X$...
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Exercise 2.27 in Huybrecht's "Fourier-Mukai transforms"
I have some questions about Exercise 2.27 from
Huybrecht's script "Fourier-Mukai transforms in algebraic geometry":
Exercise 2.27: Suppose $0 \to A \xrightarrow{f} B \to C
\to 0$ is a
...
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Shortest path from undergrad to the (co)tangent complex?
After reading the first two answers to this question, I've become interested in understanding the concept of (co)tangent complex as a way to get some intuition about homotopical algebra, being ...
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Can the following criterion of torsion-free objects be generalized to derived category?
Let $\mathcal{A}$ be a "good" abelian category (The category I really care about is $R$-Mod for some noncommutative noetherian algebra $R$ over some field $k$, but I think the question still ...
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Semi-orthogonal decomposition of derived category of Calabi-Yau manifolds?
Let $X$ is a smooth projective variety over some field with trivial canonical bundle $\omega_X\cong\mathscr{O}_X$. In the book Fourier-Mukai transforms of AG by D. Huybrechts, he claim that there are ...
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Derived Completion
I am reading Lecture III of Bhatt, Bhargav. Geometric aspects of prismatic cohomology. Fall 2018.
Let $A$ be a commutative ring, $f\in A$ and $M$ an $A$-module. The module can be seen as a complex $M\...
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Vanishing of $\text{Ext}^i_{X}(E, F)$ vs. $\text{Ext}^i_{\mathcal O_x}(E_x, F_x)$ for $E,F \in \mathcal D^b(\text{Coh } X)$
Let $X$ be a Noetherian scheme and $E,F \in \mathcal D^b(\text{Coh } X)$. Let $x\in X$ be a closed point. Then, is there any connection between $\text{Ext}^i_{\mathcal O_x}(E_x, F_x)$ and $\text{Ext}^...
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Product, coproduct, and convolution of chain complexes
I am currently reading Gelfand and Manin's Methods of Homological Algebra, and at some point (IV. 10. Exercise 2 I guess? The numbering in the book is quite inconvenient to get around, so I'm ...
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Compare two definition of $Rf_!$ (derived pushforward with proper support)
I'm talking about etale sheaves.
For a morphism $f:X\to Y$ of schemes, there are two definition of $Rf_!:D(X)\to D(Y)$.
The usual derived functor:
$\forall\mathcal{F}\in Sh(X)$, let $f_!\mathcal{F}$ ...
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left bounded chain complex $Z$ such that $Z \otimes_R^{\mathbf L} (R_P/PR_P)$ is uniformly right bounded as $P$ varies over prime ideals of $R$
Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $Z$ be a left bounded chain complex of finitely generated $R$-modules. If there exists an integer $n$ such that for every prime ideal $P$ of $R$, ...
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Derived tensor product of finitely generated module of finite projective dimension and bounded above chain complex of finitely generated free modules
Let $(R,\mathfrak m)$ be a reduced Noetherian local ring. Let $0\neq G$ be a finitely generated $R$-module of finite projective dimension. Let $M$ be a bounded above chain complex of finitely ...
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When is Fourier-Mukai transform an equivalence?
Let $X, Y$ be varieties over a field $k$ and $F \in D_{qc} (X \times Y).$ It defines the Fourier-Mukai map $\Phi_F: D_{qc} (X) \to D_{qc} (Y)$ by $\Phi_F (Q) = pr_{2, *} (pr_{1}^{*} (Q) \otimes F) $ (...
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Morphisms in derived category
I'm beginning to study abelian and derived categories, and I'm not understand morphisms in derived categories. I understand that morphisms are roofs under an equivalence relationship, but I don´t know ...
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Why is the upper row in (III.21) a distinguished triangle?
I am reading Methods of Homological Algebra from Gelfand/ Manin.
In chapter three (Derived categories and derived functors) in the section 4 (Derived categories as the localization of the Homotopic ...
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Trianguled bifunctor $\mathcal{D}^{-}(A)^{op} \times \mathcal{D}^{+}(A) \to \mathcal{D}^{+}(A)$
Setting: Let $\mathcal{C(A)}$ be the category of complexes, $\mathcal{K(A)}$ the homotopy category, $\mathcal{D(A)}$ the derived category with respective essential images with $\mathcal{K^*(A)},\...
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equivalence between $K^b(\mathcal{P}) $ and $ D^{b}(\mathcal{P})$
In my homological algebra notes I found the proof that the homotopy category $K^{-}(\mathcal{P})$ is equivalent to the derived category $D^{-}(\mathcal{A})$ when $\mathcal{A}$ has enough projectives.
...
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On the right cancellation in localization of category. Is it necessary?
I recently faced the concept of derived category, introduced as localization of homotopy category. I tried to verify of the axioms and I stucked in the following: In the definition of multiplicative ...
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Grothendieck group of the heart of a bounded $t$-structure.
My ultimate goal is to prove that given a heart $\mathcal{C}$ of a bounded $t$-structure $(\mathcal{T}^{\leq 0}, \mathcal{T}^{\geq 0})$ in a triangulated category $\mathcal{T}$, that the Grothendieck ...
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Are derived functors of sheaf-hom out of structure sheaf computing cohomology sheaves of complexes?
$\newcommand{\OX}{\mathcal{O}_X}\newcommand{\Hom}{\mathcal{H}\text{om}}\newcommand{\C}{\mathcal}\newcommand{\OXM}{\OX\text{-mod}}$
Are the following three things true?
$\hom_{\OXM}(\OX,\C{F})\cong F(...
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Can projective objects in triangulated categories be detected via Hom vanishing?
Projective objects can be defined in any category https://en.m.wikipedia.org/wiki/Projective_object. Now let $(T,\Sigma)$ be a triangulated category. Then, is it true that an object $X$ in $T$ is a ...
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Passage from $\mathrm{Hom}_{D(\operatorname{mod} R)}(\Sigma^{-n-1} M, N)$ to Yoneda $\text{Ext}^1_R(\Omega^n M, N)$ for $R$-modules $M, N$
Let $R$ be a commutative Noetherian ring, let $\operatorname{mod} R$ be the abelian category of finitely generated $R$-modules, and let $D(\operatorname{mod} R)$ be its derived category. Let $M, N \in ...
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Cochain is isomorphic to the direct sum of its cohomologies
The problem is Lemma 13.27.9 of The Stacks Project and the statement is as follows:
Let $\mathcal{A}$ be an abelian category. Let $K$ be an object of $D^b(\mathcal{A})$ such that $\mathrm{Ext}^p_\...
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The morphism induced in the derived category and in the homotopy category
During a lecture my professor made an observation, she said that it can happen the following fact but she never explained how/why: let $\mathcal{A}$ a category; there are examples of morphisms $f\...
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Again, the Nakayama functor
The question has already been asked here, but without an answer, so I want to ask it again.
Let $A$ be a fd. $k$-algebra, $\mathcal{I}$ and $\mathcal{P}$ be the categories of fd. injective and ...
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Invariants of objects in $D(\mathcal{A})$ for non-hereditary category $\mathcal{A}$
$\newcommand\A{\mathcal{A}}$Let $\A$ be an additive category, and $D(\A)$ be its derived
category (i.e. the category of chain complexes of $\A$ localized
at quasi-isomorphisms). It is easy to show ...
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Expliciting description of the connecting homomorphism between Yoneda Ext groups
I have learned from Stacks 06XP that there is a natural interpretation of higher $\mathrm{Ext}$ groups as Yoneda extensions. Where a $i$-th Yoneda extension of $B$ by $A$ is an equivalence class of ...
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Why can Scholze define a tensor product on $D(\operatorname{CondAb})$?
In the paragraph above Warning 2.8 of https://www.math.uni-bonn.de/people/scholze/Condensed.pdf , Scholze remarks that you can use the fact that $\operatorname{CondAb}$ has enough projectives to ...
user960774
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Reference for homological algebra and derived categories/functors.
Like the title says, I need to get familiar with derived categories and functors.
I am going to study some stuff related to stacks and infinity-stacks, and I need to learn this abstract side of ...
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Why, exactly, does the derived category localisation factor through the "homotopy category" of cochain complexes?
$\newcommand{\A}{\mathcal{A}}\newcommand{\C}{\mathsf{C}}\newcommand{\D}{\mathfrak{D}}\newcommand{\ho}{\mathsf{Ho}}\newcommand{\K}{\mathcal{K}}\newcommand{\kom}{\mathrm{Kom}}\newcommand{\H}{\mathsf{H}}$...
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How proof The shape of the two hearts in $\mathcal{D}^b(A_n)$ is the same if the graded undergrphas of the Ext quiver of heart of are equal
According to Yu Qiu 's Ext-quivers of hearts of A-type and the orientation of associahedron,
We know hearts in $\mathcal{D}^b(A_n)$ one by one corresponds to the Ext quivers and are precisely the ...
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Derived Category of Arrow Categories
Let $\mathscr{A}$ be an abelian category. Then, taking an arrow category does not commute with taking derived categories and this is the main point of a derivator - taking an arrow category first ...
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Syzygy in projective resolution and thick closure
Let $R$ be a Commutative Noetherian ring, and let $\text{mod } R$ denote the abelian category of finitely generated $R$-module. Consider the bounded derived category $D^b(\text{mod } R) $ which is a ...
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Construction of truncations in a $t$-structure
I'm reading about $t$-structures in the book D-Modules, Perverse Sheaves, and Representation Theory by Hotta, Takeuchi, and Taniski. Given a $t$-structure $(\mathcal{D}^{\leq 0}, \mathcal{D}^{\geq 0})$...
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Is there a direct proof that $\mathcal O(n) \in \text{thick}_{D^b(\mathbb P^1)}\{\mathcal O, \mathcal O(1)\} $ for all $n\in \mathbb Z$
Let $k$ be an algebraically closed field, and let $\mathbb P^1$ denote $\mathbb P^1_k$. Let $D^b(\mathbb P^1)$ be the bounded Derived Category of Coherent Sheaves on $\mathbb P^1$. Let $\text{thick}_{...
- 1,368
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Triangulated category of bounded constructible complexes of $\mathbb{Q}_l$-sheaves
Let $k$ be a field, $l$ be a prime number invertible in $k$, $X/k$ be an algebraic variety, then the triangulated category $D_c^b(X)$ of bounded constructible complexes of étale $\mathbb{Q}_l$-sheaves ...
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An exercise on derived category in Weibel's book: commuting $\mathbf{R}\mathrm{Hom}_R$ with $\otimes^{\mathbf{L}}_R$ in a "weird" way
My question is on the following exercise in Weibel's book: Exercise 10.8.3.
Exercise 10.8.3: Let $R$ be a commutative ring and $C$ a bounded complex of finite $\mathrm{Tor}$-dimension over $R$. Show ...
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