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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43 views

Definition of derived Hom when both inputs are chain complexes

Let $\mathcal{A}$ be an abelian category and let $A \in obj(\mathcal{A})$, and let $D(\mathcal{A})$ denote the derived category of $\mathcal{A}$. Given a left-exact functor $F: \mathcal{A} \to \...
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28 views

Thick triangulated subcategory in non-abelian case

A full triangulated subcategory $\mathcal{S}$ of a triangulated category $\mathcal{T}$ is called thick (or épaisse) iff it is closed under extensions (https://ncatlab.org/nlab/show/thick+subcategory). ...
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2answers
212 views

Derived Hom and tensor of chain complexes homologically concentrated in degree zero

Let $R$ be a commutative ring, and let $X,Y\in \mathcal D_0(R)$ (i.e. $X,Y$ are represented by chain complexes with only non-zero homology at the $0$-th spot). Then, is it true that $$\text{H}_n \...
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30 views

Derived Tensor Product of Bimodules

Let $A,B,C$ be sheaves of (possibly noncommutative) rings, $M$ a $(B,A)$-bimodule and $N$ a $(A,C)$-bimodule. The object I am trying to understand is $M\otimes_{A}^{L}N$. I can choose a resolution $S^{...
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26 views

Right adjoint of pushforward in the category of sheaves vs category of coherent sheaves.

Does upper shriek pullback in the category of coherent sheaves coincide with when we take it in the category of sheaves of abelian groups? Let $f:Z\hookrightarrow X$ be a closed immersion, let $f_{gp}^...
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24 views

Duality of upper shriek and pullback.

Given a morphism of schemes $f:X\rightarrow Y$, we have two functors $f^*$ and $f^!$ from the derived category of sheaves of $k$-vector space on $Y$ to $X$. It is well known at least in the context of ...
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27 views

Six Operations: a natural transformation $i^! \to i^*$

Let $i : Z \to X$ be the inclusion of a closed subset of sufficiently nice spaces to support the six operations. I'm reading a paper where the author says there is a natural transformation of functors ...
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41 views

Derived inverse image and support of coherent sheaves

The following is more or less Exercise 3.30 in Huybrechts "Fourier-Mukai-Transforms in Algebraic Geometry": Let $X$ be a projective scheme over a field $k$, and $\mathcal{F}^{\bullet} \in \...
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1answer
47 views

Length of locally free resolution complex

Let $X$ be a projective scheme and let $C^{\bullet} = \left\{\dots\rightarrow C^i\rightarrow C^{i+1}\rightarrow\dots\right\}$ be a perfect complex in $D^b(X)$. As $X$ carries an ample family of line ...
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70 views

If the homotopy category $\mathsf{K}(\mathsf{A})$ is abelian, then $\mathsf{A}$ is semi-simple.

Let $\mathsf{A}$ be an abelian category. I understand that if the homotopy category $\mathsf{K}(\mathsf{A})$ is abelian, then $\mathsf{K}(\mathsf{A})$ is semi-simple. (There's a proof of this in ...
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1answer
67 views

Spectral Sequences from Derived Categories

I'm trying to understand how derived categories "replace" spectral sequences. More specifically the derived category statement of Grothendieck Spectral Sequence vs the normal version. I been ...
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39 views

Injectivity of derived Hom functor base change map

According to Tag 0E1V of Stacks, given a ring map $R \rightarrow R'$ and two $R$-modules $K,M$, there is a base change map $$ R\mathop{\mathrm{Hom}}\nolimits _ R(K, M) \otimes _ R^\mathbf {L} R' \...
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1answer
59 views

Clarifying a possible error in wikipedia regarding generation of t-structures

I am very much a beginner to this subject but I can't make sense of something in the Wikipedia page for t-structures. I am thinking it is an error but I not qualified enough to trust that I'm not just ...
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60 views

Interchanging derived Hom and the shift functor

Let $R$ be a Commutative Noetherian ring and $D(R), D^{-}(R), D^+(R)$ denote respectively the derived category of chain complexes (bounded below and bounded above resp.). Let $\sum^n$ denote the $n$-...
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1answer
26 views

Exact triangle in derived categories

Suppose there is an exact triangle $A \to B \to C \to A[1]$ in $D^b(\mathcal A)$, where $A,B,C$ are concentrated in degree $0$. Does it follows that there is a short exact sequence $0 \to A \to B \to ...
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1answer
95 views

Is every derived category (isomorphic to a category) locally small?

Given an abelian category $A$, we can form its derived category $D(A)$ as follows: Start with the category of (co)chain complexes in $A$. Construct the homotopy category $K(A)$ by keeping the objects,...
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35 views

Relationship between different definitions of localising subcategory of triangulated category

Let $\mathcal{T}$ be a triangulated category. If $\mathcal{T}'$ is a triangulated subcategory, modern literature calls it a localising subcategory if $\mathcal{T}'$ is closed under taking $\mathcal{T}$...
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41 views

$0$-th cohomology of derived tensor product.

I want to show the following lemma: Lemma 4.8. Let $M$ and $N$ be two complexes of $A$-module whose homology is concentrated in negative degrees. Then there is an isomorphism $$ H^0(M \otimes^{\...
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1answer
33 views

Does an equivalence of derived categories necessarily preserve the structure sheaf?

Let $(X,\mathcal{O}_X)$ and $(Y,\mathcal{O}_Y)$ be schemes such that their derived categories of modules are equivalent via $F:D(\mathcal{O}_X)\to D(\mathcal{O}_Y)$. Is it true that $F(\mathcal{O}_X)=\...
2
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1answer
47 views

What does the derived functor send roofs of one derived category to?

For reference, I am working out of Pramod Achar's notes here, and my confusion centres around definition 7 on page 15. For $F: \mathcal A \to \mathcal B$ a left exact functor, and $\mathcal R$ an ...
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1answer
55 views

Meaning of certain cohomology classes and morphisms in Prop 3.3 of Deligne-Illusie's paper on mod $p^2$ liftings and decompos. of the de Rham complex

I am still struggling with Deligne and Illusie's paper (https://eudml.org/doc/143480). They say on page 261, in the course of the proof of theorem 3.3: The class $e(K)$ (which is associated to the ...
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58 views

Values of a sheaf on the etale covers of an affine space.

Let's assume $\mathcal{F}$ is a sheaf of abelian groups on the big etale site of $k$-schemes. As an example, we know that values of $\mathcal{F}$ on any Zariski open subset of $\mathbb{A}_{k'}^n$ (for ...
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59 views

Čech cohomology of sites: How does a “composite” morphism induce an element of $\operatorname{Ext}^{2}$?

This is yet another question concerning Deligne and Illusie's paper on $W_{2}(k)$ liftings and the degeneration of the Hodge-de Rham spectral sequence (https://eudml.org/doc/143480). In the proof of ...
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23 views

Cone of the map $\tau_{\leq 0} A^{\bullet} \rightarrow A^{\bullet}$.

The article Proposition 3.27. Let $\mathcal{A}$ be an abelian category and let $A^{\bullet} \stackrel{f}{\rightarrow} B^{\bullet} \stackrel{g}{\rightarrow} C^{\bullet}$ be a complex in $\mathcal{C}(\...
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40 views

Cohomology(ies) of simplicial sheaves

Let $X$ be a topological space and denote by $Ab(X)$ the category of abelian sheaves on $X$. My question is on the category of simplicial abelian sheaves $[\Delta^{op},Ab(X)]$. A natural way to define ...
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39 views

Existence of a lift

Let $V_1$, $V_2$, $V_3$, $V_4$ be four holomorphic vector bundles over a smooth manifold. Let $f_1\in Hom(V_0,V_1)$, $f_2\in Ext^1(V_1,V_2)$ and $f_3\in Hom(V_2,V_3)$, such that $f_2\circ f_1=0$ and ...
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41 views

Serre theorem and Ext

Let $X$ be a smooth projective variety over a field, $\mathcal{L}$ an ample invertible sheaf over $X$ and $\mathcal{F}$ a coherent sheaf. One of the theorems of Serre says that there exists an integer ...
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1answer
37 views

Is a perfect complexes locally quasi-isomorphic to a bounded complex of free modules of finite type?

I'm a little confused by the definition of perfect complex on a scheme $X$, namely a complex $E^{\cdot}$ of $\mathcal{O}_X$-modules such that for every $x\in X$ there exists $U\ni x$ open such that $E^...
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1answer
103 views

Canonical bundle and skyscraper sheaf

Let $Y$ be a smooth projective variety, $\omega_Y$ its canonical bundle and $k(y)$ the skyscraper sheaf. In a proof in Huybrechts' book, he uses the "restriction" map $r_{y_1,y_2} \colon \...
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47 views

Definition of derived category

Let $\mathcal{A}$ be an abelian category. I know that the derived category of $\mathcal{A}$, $\mathrm{D}(\mathcal{A})$, is obtained from the homotopy category of complexes $\mathbf{K}(\mathcal{A})$ ...
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1answer
39 views

Homological dimension of a curve

I'm studying Huybrechts' "Fourier-Mukai transforms in algebraic geometry". In the $3^{rd}$ chapter, when he shows that any object in $D^b(C)$, with $C$ a smooth projective curve, is $\oplus \...
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34 views

Hom, H$^n$ and tensor product

If $X$ is a scheme, $\mathcal{F}$ is a quasi-coherent sheaf and $L$ is an ample line bundle, why it holds $H^0(X, \mathcal{F} \otimes L^n) = Hom(L^{-n}, \mathcal{F})$?
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1answer
62 views

Semisimple perverse sheaf?

Let $\mathcal F$ be a complex of constructible sheaves on a stratified algebraic variety $X$ of dimension $d$. I read (*) that if $\mathcal F$ is perverse and $\mathscr H^j(\mathcal F) = 0$ for $j \...
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Question about tensor product of cohomologies

The following result is claimed to be easy in the paper by Foxby, Bounded complexes of flat modules, 2.1: Let $R$ be a commutative ring and $X,Y$ complexes in $D(R)$ which are cohomologically bounded ...
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1answer
68 views

Derived tensor product , independence of resolution

Right after Lemma 20.26.13 We have the following paragraph of how to derive the tensor product. It claims that the end result is independent of choice of K-flat resolution Suppose we take $L^\bullet ...
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1answer
51 views

A definition of defect groups

In Rouquier's 2006 ICM talk (see here), he gives the following definition of the defect group of a block $b$ (definition 2.1.2) A subgroup $D$ of $G$ is a defect group of the block $b$ if it is ...
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0answers
60 views

Mistake in derived category computation

This might be a stupid mistake but I do not see where it is. Let $f:X \to Y$ be a morphism between smooth projective varieties (over $\mathbb C$), and let $F\in D^b(X)$, $G\in D^b(Y)$. Some version of ...
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52 views

Explicit description of morphisms in derived category

I'm working with the definition of derived category $D(\mathcal{A})$ of an abelian category $\mathcal{A}$ as the category whose objects are chain complexes and morphisms $X\to Y$ are classes of roofs ...
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13 views

Is a pairing on $D^\mathrm{b}(\mathcal{C})$ non-degenerate iff it is on iamges projectives?

$\newcommand\C{\mathcal{C}} \newcommand\D{D^\mathrm{b}(\C)} \newcommand\id{\mathrm{id}} \newcommand\End{\operatorname{End}} \newcommand\Hom{\operatorname{Hom}} $I am trying to understand this paper ...
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1answer
51 views

Superfluous assumptions on Vietoris-Begle mapping theorem in Iversen's Cohomology of Sheaves

I am trying to prove the Vietoris-Begle mapping theorem as indicated in Iversen's Cohomology of Sheaves on page 203. The statement is the following: Let $f: X\rightarrow Y$ be a proper map between ...
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0answers
61 views

An abelian category $\mathcal{C}$ is a full subcategory of the bounded derived category $D^{b}(\mathcal{C})$

I want to prove that an abelian category $\mathcal{C}$ is a full subcategory of the bounded derived category $D^{b}(\mathcal{C})$. As the objects of $D^{b}(\mathcal{C})$ consist only of bounded ...
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79 views

A path to reaching the frontiers of Quantum Field Theories.

I am very excited about all the ideas surrounding quantum field theories (with or without the use of infinity categories). When I look around I see the following interesting topics: Extended TQFTs ...
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45 views

Is there a functor $\mathit{Ch}(D^+(\mathcal{C})) \to D^+(\mathcal{C})$ of derived categories?

Assume I have an abelian category $\mathcal C$ with enough projectives. Say the indecomposable projectives $P_i$ are indexed by $i\in I$ for some index set $I$. Let $F\colon \mathcal C \to \mathcal C$ ...
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1answer
65 views

Nonexistence of spherical objects

This is example 8.10(vi) (on page 170-171) in Huybreachts' Fourier-Mukai transform in algebraic geometry. Let $X$ be a variety of dimension at least three with nontrivial $H^2(X,O_X)$. I want to show $...
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0answers
140 views

Inverse Limit of Complexs and Homology

Let $\mathcal{A}$ be an abelian category and denote by $D(\mathcal{A})$ its derived category. Let $K_n\in D(\mathcal{A})$ be an inverse system of complexes in $\mathcal{A}$ viewed as elements of the ...
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90 views

How to show two complexes in derived categories are isomorphic?

I am new to derived categories, and I am trying to prove the following statement: Let $\mathcal A$ be an abelian category, and $A,B$ be objects in it. Let $C^\bullet \in D(\mathcal A)$ such that $$H^{...
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1answer
112 views

Are RHom and Ext the same thing?

The right-derived functor of the Hom functor is called the Ext functor or the RHom functor. The choice between calling it Ext or RHom doesn't seem to depend on anything which I can see. Are Ext and ...
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1answer
83 views

Down-to-earth introduction to DG categories

What are down-to-earth (i.e without $\infty$-categories for example) introduction to DG-categories ? I am mostly interested by applications to algebraic geometry. I know basic of category theory, ...
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1answer
27 views

Concentrated complexes determined by cohomology groups?

Let $C$ be a chain complex such that $H^0(C)=E$ and $H^i(C)=0$ for all $i\neq 0$. Is it true that $C$ is quasi-isomorphic to $C':\cdots \to 0 \to E \to 0 \to \cdots$ ? Apparently, $C$ and $C'$ have ...
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49 views

Bondal-Orlov Theorem

I'm trying to read a proof of the reconstruction theorem by Bondal and Orlov asserting that an equivalence of derived categories of coherent sheaves $F:D^b(X)\cong D^b(Y)$ gives isomorphism between ...