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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I'm a beginner at derived categories... a derived categories morphism of objects is an isomorphism in the category...

In $kom(A)$ which is the cochain complexes of $A$... there are morphisms between the objects and they can also be 'isomorphisms in $A$'. What's that mean please & in which text can I read the ...
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thick$_{D^b(\text{mod} R)}\{X, R\}\cap $ thick$_{D^b(\text{mod} R)}\{ R\}\subseteq $thick$_{D^b(\text{mod} R)}\{X\}$ for every $X\in D^b($mod$R)$? [closed]

Let $R$ be a commutative Noetherian ring and mod $R$ be the abelian category of all finitely generated $R$-modules. Let $D^b($mod$R)$ denote the bounded derived category. Given $X_1,...,X_n\in D^b($...
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Coherent sheaves of an Abelian variety

I am currently studying the very basics of Abelian varieties over a field $k$. Besides this, I am trying to understand Fourier-Mukai transforms in Algebraic Geometry. My current aim is to study the ...
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Equality of maps in the derived category

I have two chain complexes $C$ and $D$ of $R$-modules over some ring $R$ and two chain maps $f,g: C \to D$. I know that both $f$ and $g$ induce the same map in homology, i.e. $f_* = g_*$. When can I ...
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Why is the category of perverse sheaves abelian? [duplicate]

I know the obvious formal answer: it's the heart of a t-structure. But why? The axioms of t-structure is so innocent and the result is so powerful. Is abelian subcategories of triangulated category ...
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3 votes
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Question about right derived functors and duality

UPDATE after some thinking (and without comments or answers) I want to ask the following: Say that I have a morphism $f:X\to Y$ of schemes and a induced morphism $f_*\colon \mathcal{S}(X)\to \mathcal{...
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A condition for the validity of an identity in the derived category of bounded complexes of modules.

Let $\mathcal{R}$ be a sheaf of (not necessarily commutative) rings on a topological space $X$ and let $M$ and $N$ be $\mathcal{R}-$modules such that $M$ is quasi-isomorphic to a finite complex of ...
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classical part functor in derived categories

Good morning to everyone, I am writing here because I need to understand some proofs about adjunctions in the settings of quasi-abelian categories. In this article http://www.numdam.org/issue/...
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4 votes
1 answer
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$ \operatorname{Hom}_{D(\mathcal{A})}(B, H^n(A^{\bullet}) ) \to \operatorname{Hom}_{D(\mathcal{A})}(B,A^{\bullet}[n] ) $ injective

Let $\mathcal{A}$ be an abelian category of finite homological dimension and $D^b(\mathcal{A})$ the associated derived category of bounded compexes. Let $A^{\bullet} \in D^b(\mathcal{A})$ and $n$ ...
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Canonical morphism in Huybrecht's Fourier-Mukai Transformations

Let $(\mathcal{D},T)$ be a $k$-linear triangulated category and $A, E \in \mathcal{D}$. Define as $A[i]:= T^i(A) $ the $i$-th twist under endomorphism $T: \mathcal{D} \to \mathcal{D}$. In the proof of ...
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3 votes
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How to learn math after your PhD is finished [closed]

Question: How does someone go about learning advanced topics in Math after they're done with their PhD? Specific example: You've done your undergrad and masters degrees in math and learned from ...
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7 votes
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The Mukai pairing

I am reading the book “Fourier-Mukai transforms in algebraic geometry” by Daniel Huybrechts. at the end of the page 132 and the beginning of the page 133, he introduced the Mukai pairing as follows: ...
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7 votes
1 answer
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Exterior tensor product of structure sheaves

I am reading the book "Fourier-Mukai transforms in algebraic geometry" by Daniel Huybrechts and to solve one of its questions, I came up to show that $$\mathcal{O}_{X_1}\boxtimes\mathcal{O}_{...
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Direct product totalization in the definition of hypercohomology

Let $X$ be a topological space, and $\mathcal{F}^\bullet$ be a cochain complex of sheaves on $X$. The hypercohomology $\mathbb{H}^i(X,\,\mathcal{F}^\bullet)$ is defined as $$\mathbb{H}^i(X,\,\mathcal{...
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3 votes
1 answer
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Derived Limits vs Limits in the Derived Category

This question is more or less a reference request. Assume $A$ is a (Grothendieck) abelian category, and we consider its derived category, and we know the inverse limit in $A$ is left exact, and we ...
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6 votes
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Exterior tensor product of complexes of coherent sheaves

I am reading the book “Fourier-Mukai transforms in algebraic geometry” by Daniel Huybrechts. In page 119, in Exercise 5.13, with the assumption $P_i\in D^b(X_i\times Y_i)$, $i=1,2$, he considers the ...
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5 votes
1 answer
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Simple objects with zero dimensional support

I am reading the book “Fourier-Mukai transforms in algebraic geometry” by Daniel Huybrechts. Before stating my question, let me state some definitions. Here, $X$ is a smooth projective variety. ...
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2 votes
1 answer
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Right derived functors

I am reading the book “Fourier-Mukai Transforms in Algebraic Geometry” by Daniel Huybrechts and I am trying to understand the proof of Corollary 2.68. This Corollary is as follows: Suppose $F:K^+(\...
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Questions about equivalence of Homotopy categories $D(R)$ and $\operatorname{Mod}_{HR}$

Let $R$ a ring and $D(R)$ it's derived category. Two questions: What are homotopy groups of $D(R)$? This terminology is used in this answer. Conjecture: by definition $D(R)$ is obtained from the ...
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2 votes
1 answer
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Definition of an ample sequence

I am reading the book “Fourier-Mukai transforms in algebraic geometry” by Daniel Huybrecht, on page 59, there is a definition for an “ample sequence” in an abelian category and another reference for ...
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Characterizing isomorphisms in the derived category

Let $D(A)$ be the derived category of an abelian category $A$ with the homotopy chain complex category $K(A)$. I want to characterize when morphisms in $K(A)$ become isomorphisms in $D(A)$. We can ...
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2 votes
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Derived category supported in a Serre subcategory of a locally noetherian category

This has now been cross-posted: https://mathoverflow.net/questions/404902/derived-category-supported-in-a-serre-subcategory-of-a-locally-noetherian-catego It is known that $Coh(\mathcal O_X)\subseteq ...
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Examples of applications of Beilinson-Bernstein localization?

I'm a sheaf theorist who knows a little representation theory but is not familiar with it. According to my memory, I was once told that one virtue of the Beilinson-Bernstein localization theorem is ...
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Derived Category idea from ncatlab

I have a question about following statement from ncatlab about derived categories: (https://ncatlab.org/nlab/show/derived+category#idea) Often in the literature, the term derived category refers to ...
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1 answer
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Discrete Spectra

What is a "discrete spectra" in context of homotopy theory/ derived category theory? It is for example mentioned here. Although it looks quite "googleable" I found nowhere a ...
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1 vote
1 answer
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Derived Tensor Product in Terms of Homotopy Groups

Let $R$ be a ring and $R-\operatorname{Mod}$ the category of $R$-modules. The tensor product functor $- \otimes_R -: (R-\operatorname{Mod}) \times (R-\operatorname{Mod}) \to R-\operatorname{Mod}-R$ ...
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1 vote
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Proof of derived tensor-hom adjunction

EDIT: this has now been cross-posted here. As far as I know, for $R,S,V,W$ rings and $M$ an $(R,W)$-bimodule, $N$ an $(R,S)$-bimodule and $L$ an $(S,V)$-bimodule, we have an isomorphism in D($V$-lmod) ...
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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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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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5 votes
1 answer
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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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3 votes
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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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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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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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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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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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1 vote
1 answer
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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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6 votes
0 answers
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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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2 votes
1 answer
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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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2 votes
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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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  • 1,162
1 vote
1 answer
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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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2 votes
1 answer
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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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4 votes
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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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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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3 votes
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$0$-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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1 answer
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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)=\...
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2 votes
1 answer
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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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2 votes
1 answer
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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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1 vote
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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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1 vote
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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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  • 1,066
1 vote
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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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