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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1answer
55 views

Bounded chain complexes and the bounded derived category

Let $\mathcal{A}$ be an abelian category and consider the following categories: $\mathbf{Ch} (\mathcal{A})$, the category of cochain complexes in $\mathcal{A}$. The full subcategories $\mathbf{Ch}^\...
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52 views

Can equivalent abelian categories have non-equivalent derived categories?

This is a point of stupid confusion for me. Let $s:\mathcal{A}\to\mathcal{B}$ be an equivalence of abelian categories. Does this functor induce a triangulated equivalence $\overline{s}: \mathbb{D}^b(\...
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43 views

Vanishing of Ext and Tor from Isomorphism of derived Hom with derived tensor product

Let $R$ be a Commutative Noetherian ring and let $\textbf{R} \text{Hom}_R(-,-)$ and $-\otimes_R^{\textbf{L}}-$ denote derived Hom and tensor product respectively. We recall that for any $R$-modules $M,...
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19 views

When is the derived functor of a representable functor representable?

Let $\mathcal{C}$ be an abelian category and $\mathcal{F}:\mathcal{C}\rightarrow \text{Set}$ a representable sheaf. Consider the $i$-th derived functor $R^i\mathcal{F}$ of $\mathcal{F}$. My question ...
2
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56 views

The Decomposition Theorem for a resolution of singularities

In learning about the celebrated Decomposition Theorem I have found that I am having trouble applying it in even the simplest situations. In particular, I'm considering an example where $f:X\to Y$ is ...
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28 views

Derived functor of additive functor

Let $\mathscr{C}$ be a category which admits enough injectives and let $\mathscr{I}$ be the full subcategory of injective objects. Let $F:\mathscr{C}\rightarrow\mathscr{C}'$ be an additive functor of ...
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83 views

Why is the nearby cycle of a family of elliptic curves the derived pushforward?

Consider a family of elliptic curves over the open disc $D$ in $\mathbb{C}$, which degenerate to the nodal elliptic curve over $0$, and let $f$ be the map to $D$. I'm interested in the sheaf on the ...
2
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0answers
90 views

Restriction-extension identities using six functors formalism

Recently fellow user Thorgott pointed out to me that flat restrictions of flat modules remain flat. That is, let $f : A \to B$ be a flat morphism of rings and $M$ be a flat $B$-module. Then ...
8
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1answer
66 views

Why is there no hyper-hypercohomology?

I am looking for a reference to answer the question in the title. Let me try to clarify a little what I mean: If a single sheaf $\mathscr F$ has a resolution $\mathscr G^\bullet$ by not necessarily ...
8
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1answer
106 views

Functors which are non-isomorphic but whose derived functors are isomorphic

I was wondering if there is a good/interesting examples of functors that are non-isomorphic as functors but whose derived functors are isomorphic? The example I have encountered so far, the derived ...
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31 views

When is it enough to consider roofs in the derived category?

In the derived category $D^b(\mathcal{A})$ of an abelian category $\mathcal{A}$, obtained by taking the Verdier quotient wrt. all quasi-isomorphism, a morphism is given by a roof (or span) $ X\...
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80 views

Identify cohomology of sheaves in mutations

Let $X$ be a special Gushel Mukai threefold, which is a double cover of a degree $5$ index $2$ Fano threefold which is a linear section of Grassmannian $Gr(2,5)$. Let $\mathcal{E}$ be the rank $2$ ...
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55 views

Vanishing of Tor group at every fiber implies flatness

I have some question on a proof in Huybrechts' "Fourier-Mukai Transfomation in Algebraic Geometry". Let $ f \colon S \to X$ be a morphism between Noetherian schemes, and we denote by $i_x \colon S_x ...
2
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50 views

$ \mathrm{ch}(F(A)) = \mathrm{ch}(F(B)) \implies \mathrm{ch}(A) = \mathrm{ch}(B) $ for autoequivalences?

Let $\mathcal{C} \subset D^b(X)$ be a subcategory of the derived category of coherent sheaves on a smooth projective variety $X$. Let $F : \mathcal{C} \to \mathcal{C}$ be an autoequivalence. Let $\...
0
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1answer
38 views

Universal dg-algebra of an $A_\infty$-algebra

In this document by Keller, proposition 2.1, it is stated that for every $A_\infty$-algebra $A$ there is a universal dg-algebra $U(A)$ w.r.t. the existence of an $A_\infty$-morphism $A\to U(A)$, and ...
2
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1answer
63 views

Yoneda product in the cohomology of a truncated polynomial algebra

Fix a ground field $k$, and let $A$ be the algebra generated over $k$ by symbols $X$ and $Y$ subject to the relations $X^2 = 0 = Y^2$ and $X Y = Y X$. This is augmented over $k$ by the map sending $X$ ...
2
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1answer
54 views

The derived category of the dual numbers?

Let $k$ be a field, and let $R = k[x]/x^2$ be the ring of dual numbers over $k$. Let $\mathcal D(R)$ be the derived category of $R$. I'm interested in getting a complete understanding of the structure ...
4
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1answer
74 views

How do Fourier-Mukai equivalences not contradict reconstruction theorems?

Apparently there is a thing called Fourier-Mukai equivalence in which the derived categories of coherent sheaves of two distinct schemes (e.g. an abelian variety and its dual) can be equivalent. On ...
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31 views

Derived functors via derived categories

I got stuck in the proof of III.6.8 of Gelfand, Manin: Methods of Homological Algebra which essentially asserts the existence of a right derived functor of a left exact functor in a category with ...
6
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49 views

Kan extensions and satellite functors

As is described for example in Cisinski's book on Higher Category theory, the left (right) derived functor of a functor between model categories $F: \mathcal{C} \rightarrow \mathcal{D} $ can be ...
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17 views

When $f$ induced $0$ homomorphism, then $f$ is homotopic $0$?

I am reading some materials about derived category. For one step in the proof of one theorem, it seems using the following consequence: Let $C_{\bullet}, D_{\bullet}$ be two chains of abelian ...
2
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34 views

The category of complexes modulo homotopy is triangulated

I am trying to understand why for an additive category $\mathcal{C}$ the category $K(\mathcal{C})$ of complexes over $\mathcal{C}$ modulo homotopy is triangulated: We take the shift functor as ...
2
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35 views

$D$ is triangulated. Then opposite functor $op:D\to D^{op}$ 's image is triangulated?

Let $D$ be a triangulated category where a triangle is by definition $X,Y,Z\in D$ and $T:D\to D$ automorphism s.t. $X\to Y\to Z\to T(X)$ is called a triangle. Consider opposite functor(contravariant)...
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38 views

Derived category of stack of vector bundles on (punctured) affine plane

Consider the open embedding of schemes $\mathbf A^2\setminus\{(0,0)\}\subset \mathbf A^2$ (say over $\mathbb C$). This induces a restriction functor on the stacks $$\mathrm{Vect}_n(\mathbf A^2)\to \...
3
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1answer
59 views

Pushforward as integral functor for Azumaya varieties

Consider $(X,\mathcal{A}_X)$ and $(Y,\mathcal{A}_Y)$ two Azumaya varieties over a field $k$. Recall that an Azumaya variety is the data of a variety and a sheaf of semisimple $\mathcal{O}_X$-algebra $\...
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1answer
48 views

Understanding a Line Bundle as a Spherical Twist

I am reading Segal's paper "All Autoequivalences are Spherical Twists", https://arxiv.org/abs/1603.06717, and I am trying to understand a detail in example 2.4. The situation is this: Let $X$...
2
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0answers
31 views

Cohomology functors in a triangulated category with t-structure.

Given a triangulated category $\mathcal{D}$ with a t-structure $(\mathcal{D}_{\leq 0}, \mathcal{D}_{\geq 0})$ the cohomology functor can be defined as \begin{equation} H^k := \tau_{\geq 0}\circ \tau_{...
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70 views

What fields of arithmetic/algebraic geometry benefit from infinity/derived techniques?

I'd like to get a better picture of how infinity/derived techniques become more important in algebraic/arithmetic geometry. I'd therefore like to know: What questions/subfields of algebraic/...
2
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1answer
53 views

Is the cone of the zero map $A \to B$ always $A[1] \oplus B$?

Let $\mathcal{D}$ be a triangulated category, with objects $A, B \in \mathcal{D}$. Is it true that $$A \xrightarrow{0} B \to A[1] \oplus B \to A[1] \tag{$*$}$$ is a distinguished triangle? If $\...
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17 views

Extension-closed subcategory $P(I)$ defined by stability condition $(Z, P)$.

Let $D$ be a triangulated category, and let $\sigma = (Z, P)$ be a Bridgeland stability condition on $D$. Let $I \subset \mathbb{R}$ be any interval (open, closed, or half-open). The category $P(I)$ ...
1
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1answer
30 views

A complex with prescribed cohomology

Let $\mathcal A$ be an Abelian category (I am happy to assume that $\mathcal A\cong \mathrm{Mod}(R)$ for some ring $R$ if that helps). Given the following long exact sequence in $\mathcal A$ $$ 0\to ...
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40 views

Definition of the Wedge Product of Chain Complex or Derived Object

Let $A$ be a commutative, unitary ring. The derived category of $A$-modules has a tensor product, namely the derived tensor product. To define the wedge product, one only need a tensor product and ...
1
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1answer
43 views

Does $H^i(E) = 0$ for $i \geq 0$ imply $\operatorname{Hom}_{\mathbf{D}(\mathcal{A})}(E, A) = 0$ for any $A \in \mathcal{A}$?

Let $\mathcal{A}$ be an abelian category, $\mathbf{D}(\mathcal{A})$ be the derived category, and consider an object $E \in \mathbf{D}(\mathcal{A})$ with $H^i(E) = 0$ for all $i \geq 0$. Is it true ...
2
votes
1answer
49 views

Does $\operatorname{Hom}_{\mathbf K (\mathcal{A})}(A, B) = 0$ imply $\operatorname{Hom}_{\mathbf D (\mathcal{A})}(A, B) = 0$?

Let $\mathcal{A}$ be an abelian category, let $\mathbf K(\mathcal{A})$ be the category of cochain complexes modulo homotopy, and let $\mathbf{D}(\mathcal{A})$ be the derived category. Let $A, B$ be ...
3
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1answer
66 views

IC complex on $\Bbb C = \Bbb C^* \sqcup \{0\}$

Let $X = \Bbb C$ stratified as $\Bbb C^* \sqcup \{0\}$. Let $\mathscr L$ be a local system on $\Bbb C^*$. How to describe $IC(U, \mathscr L)$ ? By describe I mean : compute its stalk at $0$, ...
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16 views

Cofibre of ring spectra

I was reading the following paper about how to generalise the classical derived algebra to the setting of spectra https://arxiv.org/pdf/1601.02473.pdf and I cannot understand a passage not explained ...
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0answers
33 views

Exact functors on the homotopy category

Let $F: \mathcal{A} \to \mathcal{B}$ be an additive functor between abelian categories. When is the induced functor $K(F): K(\mathcal{A}) \to K(\mathcal{B})$ between the homotopy categories exact (as ...
1
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1answer
62 views

Are short exact sequences in the heart of a bounded t-structure triangles?

This question came up while studying Bridgeland's Stability conditions on K3 surfaces, in the proof of Lemma 6.3. Suppose $\mathcal{D}$ is a triangulated category, and $(\mathcal{D}^{\leq 0}, \...
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0answers
56 views

Complexes of coherent sheaves as complexes of vector bundles

Let $X$ be a smooth projective variety. It is often said that the derived category $D^b (X)$ is "just" something like complexes of vector bundles, however I haven't been able to find an elementary ...
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1answer
51 views

Tensoring locally free sheaves preserves acyclicity

This is a statement in Huybrechts, Fourier Mukai Transform, pg78, where he defines the derived tensor. If $E^*$ is an acyclic complex bounded with all $E^i$ locally free. $F$ a coherent sheaf , ...
2
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0answers
22 views

Generation criterion for dg/stable linear category

I'm looking for the exact statement and the proof of the following statement: Let $\mathcal{C}$ be a small idempotent-complete pretriangulated dg category over $k$/stable $k$-linear category where $...
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0answers
30 views

DG algebra and its zeroth cohomology are derived equivalent

This is slightly related to this question: Can an algebra be morita equivalent to its dg-extension? . Suppose we have a DG-algebra $A$, such that $H^0(A)$ is Noetherian (both left and right) and $H^\...
1
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1answer
57 views

An exact triangle of (derived) sheaves.

For a sheaf $\mathcal{F}$, denote by $\mathcal{F}[n]$ ($n\in \mathbb{ Z}$) the complex of sheaves given by $$\mathcal{F}[n]_i=\begin{cases} \mathcal{F}&\text{if }i= n\\ 0&\text{otherwise}. \...
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0answers
24 views

Base change theorems, six functor formalism, and their generality

Let us suppose to have a Cartesian square of algebraic stacks (or schemes, for simplicity) of the form $$X\xrightarrow{g'} Y$$ $$f'\downarrow\ \ \ \ \downarrow{f}$$ $$Z\xrightarrow{g}W.$$ Several ...
2
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1answer
84 views

The cup product of etale cohomology

This is written in Fu Lei's "Etale cohomology theory", p362. Let $X$ be a scheme, $A$ a ring, $\mathscr{F,G}$ a sheaf of $A$-modules on $X$, $\mathscr{C^\bullet(F), C^\bullet(G)}$ be Godement ...
0
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1answer
33 views

Are There Examples of Triangulated Categories that $\bigcap_n\mathcal{D}^{\geq n}\neq\{0\} $?

In chapter 4 of Gelfand&Manin's book Methods in Homological Algebra they mentions the property that if a triangulated category $\mathsf{T} $ with the $t$-structure $(\mathcal{D}^{\leq 0} ,\mathcal{...
3
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0answers
121 views

Construct a coherent subsheaf of a quasi-coherent sheaf

I am trying to prove the following statement Given a Noetherian scheme $X$ and a surjective morphism of quasi-coherent sheaves on $X$: $$\mathcal{F}\xrightarrow{f} \mathcal{G} \to 0$$ where $\...
2
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1answer
139 views

In triangulated category, why do we use triangles, but not exact sequences?

Studying the theory of derived category, I came up with a little question. To define trianglated categories, first we choose a collection of sextuples, called the distinguished triangles. I know that,...
2
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0answers
49 views

Image of a line bundle on an elliptic curve under Fourier-Mukai transform

Let $E$ be an elliptic curve with a base point $p$. What is the image of the line bundle $\mathcal{O}_E(np)$ under the Fourier-Mukai transform with a kernel given by the Poincare bundle on $E\times E$?...
2
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1answer
87 views

A cochain complex with finitely generated cohomology is quasi-isomorphic to a cochain complex of finitely generated module

I was thinking over the following statement: Given a Noetherian ring $R$ and a bounded cochain complex $ C^{\bullet}$ with finitely generated cohomology, that is, $H^{i}(C^{\bullet})$ is finitely ...