Questions tagged [derived-categories]

For questions about the derived categories of various abelian categories and questions regarding the derived category construction itself.

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84 views

Is the derived category of local systems equivalent to the derived category of sheaves of vector spaces with local system cohomology?

Let $k$ be a field and $X$ a topological space. Write $\mathrm{Sh}(X)$ for the category of sheaves of vector spaces on $X$, and $\mathrm{Loc}(X)$ for the subcategory of local systems of finite ...
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2answers
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The relation between t-structures and derived category

Let $\mathcal{D}$ be a triangulated category and a $t$-structure $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ on $\mathcal{D}$. The heart of the $t$-structure, $\mathcal{A}=\mathcal{D}^{\leq 0} \cap ...
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Reflexive vs. pseudo-coherent abelian groups

Recall that a module M over some ring R is pseudo-coherent if it admits a resolution whose terms are finitely generated projective R-modules. Such a module is reflexive when regarded as an object in ...
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Derived symmetric powers and determinants

Given a vector bundle $V$ (on a scheme $X$, say), I can form $Sym(V[1])$, the symmetric algebra (in the derived/graded sense) on the shift of $V$; in other words this is the Koszul complex of the zero ...
2
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Does direct image via proper map preserve coherence of unbounded complexes?

As for the title, I'm considering a proper map $f : X \rightarrow Y$ of Noetherian schemes and I'm trying to understand whether the direct image $Rf_{\ast} : D_{qc}(X) \rightarrow D_{qc}(Y)$ sends the ...
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Reference for equivariant derived Künneth formula

I'm looking for a reference for the following statement in as much generality as possible, assuming it is correct. Let's $X$ and $Y$ be "spaces" with a $G$-action. We can take the $G$-product defined ...
3
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156 views

Universal property for derived category of coherent sheaves

Let $X$ be a scheme, and let $D^{*}(X)$ be the unbounded (resp. unbounded, resp. bounded below/above, etc) derived category of coherent sheaves on $X$. The work of Robalo establishes a universal ...
4
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85 views

Understanding a step in proof of sheaf version Verdier duality

Warning: This question is likely low-level for MathOverflow. My apology that there is almost surely something basic I miss. So all proofs I can find factors through a particular statement, which goes ...
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Why is a DG-enhancement of the derived bounded category an enhancement?

I asked this question on math.stackexchange with no luck, so I thought I would try here. In order to make mirror symmetry more compatible with homological machinery, I understand it is common to give ...
7
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1answer
225 views

Skew differential graded algebra

A sigma, or skew, derivation is a natural generalisation of the notion of derivation depending on an algebra automorphism $\sigma$ which when equal to $id = \sigma$ reduces to the usual notion of a ...
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121 views

Yoneda Extensions in derived categories

If given an abelian category $\mathcal{A}$, we can consider the bounded derived category $D^b(\mathcal{A})$. For two objects $A,B \in \mathcal{A}$, we know that there is a natural identification ...
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1answer
313 views

idea and intuition behind triangulated category [closed]

I have some trouble in understanding the significance of some axiom of triangulated category. if someone could explain me each axiom with some intuition,and explain me the intuition behind the ...
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1answer
95 views

Explicit locally free resolution of a perfect complex $E\oplus F\to (E\oplus F)\otimes \mathcal{O}_X(D)\to (E\otimes \mathcal{O}_X(D))|_D$

Let $X$ be a smooth projective variety over $\mathbb{C}$. Let $E,F\to X$ be 2 holomorphic vector bundles and $D\hookrightarrow X$ be a smooth divisor. Denote by $\mathcal{O}_X(D)$ the line bundle ...
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Does a homologically bounded dg A-module admit a “locally finite” semi-free resolution

Let $A$ be a bounded dg-algebra whose underlying algebra is Noetherian and such that $H^*(A)$ is Noetherian. Let $M$ be a cohomologically bounded dg-module over $A$, whose cohomology groups are ...
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83 views

Dualizing complex description in Stacks project

The question is closely related to this one this one (more precisely the reference the comment by AGl earner) and is aimed to understand the proof of Lemma 20.2 from notes from Stacks notes from ...
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1answer
110 views

Morphisms on fibre products

Let $X$ and $Y$ be two quasi compact, separated schemes over $k$, and consider the fibre product $X \times Y$. If we call $p_1$ and $p_2$ the two projections, and we take perfect complexes $F_1, F_2 \...
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Determinantal identities for perfect complexes

Let $S$ be a noetherian scheme. Let $V,W$ be vector bundles on $S$. There is a canonical isomorphism of line bundles $$ {\rm det}(V\otimes W)\cong{\rm det}(V)^{\otimes{\rm rk}(W)}\otimes{\rm det}(W)^{\...
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2answers
308 views

Homology of perfect complexes

I apologize in advance if this question is basic. If $P_{\bullet}$ is a perfect complex over say a ring $R$ such that $H_{i}(P_{\bullet})=0 $ if $i\neq n$ $H_{i}(P_{\bullet})=E$ if $i=n$ is $E$ ...
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0answers
95 views

Correct reference for a proposition in a paper of Kapranov-Vasserot

In the paper "Kleinian singularities, derived categories and Hall algebras" Math. Ann. 316 (2000) of Kapranov-Vasserot, the authors write in page 569 that the complex $\mathcal{L}'$ (defined in p.568) ...
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2answers
192 views

Faithfully flat modules over a group algebra

Suppose we have the following data: 1) A group ring $\mathbb{Z}[G]$, where $G$ is a torsion free group. 2) $M_{\bullet}$ a bounded (above and below) chain complex of $\mathbb{Z}[G]$-modules such ...
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Extension-closed subcategory $P(I)$ defined by stability condition $(Z, P)$ and an interval $I \subset \mathbb{R}$

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 $P(I)$ ...
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Stalks of perverse cohomology sheaves?

For a complex of sheaves $\cal{F}^{\bullet}$ on a variety $X$, a useful fact is that the stalks of the cohomology sheaves of $\mathcal{F}^{\bullet}$ agree with the cohomology groups of the complex of ...
3
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0answers
69 views

Fourier Mukai kernel which gives an equivalence only in one direction

If $X$ and $Y$ are two schemes and $F \in Perf(X \times Y)$, then we can define a functor from $Perf(X)$ to $Perf(Y)$ as the Fourier Mukai transform $\Phi^{X \rightarrow Y} = q_{\ast}(F \otimes p^{\...
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193 views

Derived category of a fiber product

Let $X = Y \times_Z W$, where $X,Y,Z,W$ are Noetherian schemes, and consider the pullback diagram associated to $X, Y, Z, W$. We have a diagram $$ \require{AMScd} \begin{CD} D(Z) @>>> D(Y)\\ @...
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1answer
223 views

When is $\mathcal{D}(\mathcal{F}):\mathcal{D}(\mathcal{A})\to \mathcal{D}(\mathcal{B})$ fully faithful?

Let $\mathcal{A}$ and $\mathcal{B}$ be two abelian categories and let $\mathcal{F}:\mathcal{A}\to \mathcal{B}$ be an additive functor. Assume that $\mathcal{F}$ is exact and let $\mathcal{D}(\mathcal{...
4
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0answers
81 views

Morphism in a Verdier quotient

Let $\mathcal{T}$ be a triangulated category and take $\mathcal{S}$ a triangulated subcategory. Consider the Verdier quotient $\mathcal{T} \left/ \mathcal{S} \right.$, morphisms in this category are ...
2
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0answers
175 views

State of the art on strong full exceptional collections

I am trying to understand on which kinds of varieties we can find strong full exceptional collections of sheaves. (Classical examples are of course projective spaces (Beilinson) and Grassmann ...
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3answers
1k views

So what exactly are perverse sheaves anyway?

Is there a way to define perverse sheaves categorically/geometrically? Definitions like the following from lectures by Sophie Morel: The category of perverse sheaves on $X$ is $\mathrm{Perv}(X,F):=...
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56 views

Characterization of degeneracy of spectral sequence of a fiber bundle at the second term

Let $f\colon E\to B$ be a fiber bundle of compact manifolds with fiber $F$. Assume that the push-forward $Rf_*(\underline{\mathbb{F}})$ in the derived category of the constant sheaf with coefficients ...
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1answer
105 views

Relative version of the cohomology product

Let $f\colon X\to Y$ be a continuous map of 'nice' topological spaces (e.g. $f$ is a smooth map of smooth manifolds; $f$ might be assumed to be proper although I am not sure it is relevant). Let $\...
3
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1answer
199 views

A question on the proof of $D^b(coh(X))\simeq D^b_{coh}(Qcoh(X))$

Proposition 3.5 of "Fourier-Mukai Transforms in Algebraic Geometry" by Huybrechts claims that the is an equivalence of categories $$ D^b(coh(X))\overset{\sim}{\to} D^b_{coh}(Qcoh(X)) $$ where $D^b(coh(...
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0answers
150 views

Do we have $D^b_{coh}(X)\simeq D^b(coh(X))$ for a compact complex manifold $X$?

Let $X$ be a compact complex manifold and $\mathcal{O}_X$ be the structure sheaf of holomorphic functions. We call a sheaf of $\mathcal{O}_X$-module $\mathcal{F}$ coherent if it satisfies the ...
6
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1answer
162 views

Indecomposable objects in bounded derived category of $\mathbb C[x]/x^2$-mod

We know for any principal ideal domain, objects in the bounded derived category are all formal hence we can classify those objects with finitely generated cohomology using structure theorem for ...
2
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0answers
167 views

What's wrong with higher dimensional nearby cycles?

Suppose we have a complex algebraic variety $X$ with a map $f: X \to \mathbb{C}$ with $Y=f^{-1}(0)$. Let $\overset{\sim}{\mathbb{C}}$ be the universal cover of $\mathbb{C}-\{0\}$ and consider the ...
4
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0answers
210 views

Properties of microlocalization

Let $i: M\hookrightarrow X$ be the inclusion of a closed submanifold in a smooth manifold $X$. I denote by $T_MX$ the normal bundle to $M$ in $X$, by $T^{\ast}_MX$ its dual bundle, and by $D^b(X)$ the ...
5
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1answer
158 views

About “strict” short exact sequences in quasi-abelian subcategory of a derived category

I'm reading Bridgeland's Stability conditions on K3 surfaces. In Lemma 4.4 there appears a full quasi-abelian subcategory $\mathscr{A} \subset \mathscr{D}$ of a triangulated category $\mathscr{D} = \...
1
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1answer
120 views

Computation of extension groups in the category of pairs $(M,f)$

Let $A$ be a unitary commutative ring, and let $B$ be an $A$-algebra. We consider the category whose objects are pairs $\textbf{M}=(M,f)$ where $M$ is an $A$-module and where $f$ is a $B$-linear ...
6
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1answer
130 views

Relative Ext of Avramov-Martsinkovsky as a derived Hom

Avramov-Martsinkovsky (http://mathserver.neu.edu/~martsinkovsky/Relative.pdf) have defined an exotic version of Ext between two modules over (for simplicity) Gorenstein rings. The basic idea of their ...
7
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1answer
437 views

Derived categories and classical theorems in homological algebra

So far I have studied fundamental part of derived category theory, for example, the existence of derived functors, the "composition of derived functors", and so on. Now I came up with some questions ...
12
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1answer
589 views

A concrete example of the deficiency of triangulated categories?

There seems to be a general sentiment that triangulated categories are not the "correct" notion to use because mapping cones of morphisms are unique, but only up to non-unique isomorphism. Does ...
6
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1answer
319 views

Derived Category of the derived critical locus, is it the category of Matrix Factorizations?

Let $W \in \mathbb{C}[x_1, \dots, x_n]=R$ be a polynomial with an isolated critical point at the origin. A Matrix Factorizations for $W$ consists a $\mathbb{Z}/2\mathbb{Z}$-graded finite free $R$-...
6
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1answer
443 views

Grothendieck-Verdier duality without the noetherian condition

The Grothendieck-Verdier duality: $$ Rf_*\big(R\mathcal{H}\textit{om}_X^\bullet(\mathcal{E}^\bullet,f^!\mathcal{F}^\bullet)\big) \cong R\mathcal{H}\textit{om}^\bullet_Y(Rf_*\mathcal{E}^\bullet,\...
5
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0answers
261 views

How to understand the Fourier-Sato transform and microlocalization functors?

Given a smooth real vector bundle $\pi: E \to M$ I can look at the (bounded from below) derived category of sheaves on $E$. Since $E$ admits a very natural action of $\mathbb{R}^{\geq 0}$ by scaling, ...
11
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2answers
281 views

Obstruction to splitting an object in derived category into a sum of two-term complexes

Let $\mathcal{A}$ be an abelian category, and $D$ its bounded derived category. An object $M \in D$ may be described as a list of cohomology objects $H^i = H^i(M)$ together with some complicated ...
2
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0answers
159 views

Comparing definitions of cotangent complex

Consider the following two ways of defining the cotangent complex of a ring map $R \rightarrow A$ (Let $P^{\bullet} \rightarrow A$ be a polynomial resolution): As the complex $\Omega^1_{P^{\bullet}/R}...
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0answers
184 views

Gluing for derived category of coherent sheaves

Let $X$ be a scheme and assume $X=U \cup V$ for two affine schemes $U_0$ and $U_1$. If $\mathcal F'$ and $\mathcal F''$ are some (coherent) sheaves on $U$ and $V$ respectively such that $\mathcal F'|_{...
6
votes
1answer
502 views

Derived base change in étale cohomology

Given a commutative square of ringed topoi $$\begin{array}{ccc}X'\!\! & \overset{f'}\to & Y'\!\! \\ \!\!\!\!\!{\small g'}\downarrow & & \downarrow{\small g}\!\!\!\! \\ X & \...
2
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0answers
160 views

Derived category of coherent sheaves with a codimension $\geq$ 1 support

Let $X$ be some smooth algebraic variety. I would like to understand the relation between the following two categories: $D^b_{cd,1}\text{Coh}(X) \subset D^b\text{Coh}(X)$: the full subcategory of the ...
2
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0answers
65 views

Gerstanharber bracket and derived Hom

Let $A$ be a honest algebra or more generally, a DG algebra. It is known that the Hochschild cochain complex is quasi-isomorphic to the derived Hom complex, i.e. one has $$\mathrm{HH}^{\bullet}(A,\,A)...
6
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1answer
444 views

Category of $\mathcal{D}$-modules on a singular variety

Take $X\to V$ a closed embedding, where $X$ is not necessarily smooth, $V$ is affine and smooth. Define the category $\mathcal{C}$ of $\mathcal{D}$ modules on $X$ to be the full subcategory of $\...

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