**5**

votes

**0**answers

78 views

### flatness and derived completion

Let $A$ be a local ring of maximal ideal $\mathfrak{m}$. Let $\hat{A}$ be its completion.
If $A$ is noetherian , then we know that $A\rightarrow\hat{A}$ is faithfully flat.
If $A$ is not noetherian, ...

**4**

votes

**1**answer

194 views

### Massey products and $A_{\infty}$ structures

I know the general theorem of Kadeishvili which says that, for a DGA $C$, when $H^{i}(C)$, $i\geq 0$, is free, $H(C)$ can be made into an $A_{\infty}$ algebra. If my understanding is correct, the ...

**8**

votes

**1**answer

169 views

### Can Enriques Surfaces have non-trivial TWISTED Fourier-Mukai partners?

It is a well-known fact that for an Enriques surface $Y$, if $D^b(Y)\cong D^b(X)$ for some smooth projective variety $X$, then $X\cong Y$. In other words, Enriques surfaces have no non-trivial ...

**1**

vote

**0**answers

214 views

### Cubic fourfold and K3 surface: geometric constructions of Hodge isometry

Hodge structure on K3 surface (the middle line of Hodge diamond is 1 20 1) is similar to the Hodge structure of cubic fourfold (the middle line of Hodge diamond of primitive cohomology is 0 1 20 1 0). ...

**4**

votes

**0**answers

171 views

### Can we obtain a derived category from an additive category? Like a category of Banach modules?

Let $A$ be a Banach algebra, let $A$-mod be the category of left Banach modules (as defined in Helemskii's "Banach and locally convex algebras"), $A$-mod is an additive category, but not abelian ...

**6**

votes

**1**answer

203 views

### An example of an object in $D^b_{\text{coh}}(\mathbb{P}^2)$ which is not formal

We know that for a curve $X$, any object $\mathcal{E}^{\bullet}$ in the derived category $D^b_{\text{coh}}(X)$ is formal, i.e. $\mathcal{E}^{\bullet}$ is quasi-isomporphic to the direct sum of its ...

**0**

votes

**0**answers

109 views

### Isomorphism in a derived category of chain complexes with rational coefficients

Let $C$ be the category of quasi-projective schemes over a base field $k$ (maybe I will need some assumptions on $k$). Let $Ab(C_{\tau})$ be the category of Abelian sheaves on a site $C_{\tau}$, where ...

**0**

votes

**0**answers

59 views

### Extending natural transformations of triangulated functors on $D^b(\mathbb P^1)$

Let $\mathcal T = D^b(\mathbb P^1)$, the bounded derived category of $\mathbb P^1$ over a field $k$. It is well known that $\mathcal T$ admits a strong and full exceptional sequence $\{\mathcal O, ...

**4**

votes

**0**answers

133 views

### Characterization of closed immersions at the level of perfect complexes

Let $f : X \to Y$ be a morphism of quasi-compact quasi-separated schemes.
Is there a necessary and sufficient condition on the inverse image functor $\DeclareMathOperator{Perf}{\mathrm{Perf}} f^* : ...

**1**

vote

**2**answers

203 views

### A question on triangulated category of matrix factorizations

Let $X$ be an algebraic variety and $W:X\rightarrow\mathbb{A}^1$ a regular map, then the triangulated category of matrix factorizations $D^b(X,W)$ is defined to be
...

**1**

vote

**0**answers

95 views

### Exact Functors from Perverse Sheaves

Pretty sure this is a simple question, but there's something I'm missing here.
For context, I'm reading 'An Elementary Construction of Perverse Sheaves' by MacPherson and Vilonen. A key aspect of the ...

**21**

votes

**1**answer

599 views

### Which properties of a variety are detected by its derived category of coherent sheaves?

Context: I'm giving an informal seminar/reading group collection of talks on derived categories, following on from earlier talks giving the abstract definition. I am starting to talk about ...

**0**

votes

**0**answers

165 views

### Triangulated category of singularities of quotient

Let $X$ be a variety, the triangulated category of singularities $D_{sg}^b(X)$ is obtained by taking the quotient of $D^b(X)$ by the category of perfect complexes. Suppose there is a group $G$ acting ...

**1**

vote

**0**answers

108 views

### Ext and cup products and subvarieties

I am trying to understand Remark 11.3 in Huybrechts's amazing book on derived categories (FM transforms in AG).
He starts with smooth projective varieties $j\colon Y \subset X$ and aims to describe ...

**3**

votes

**1**answer

253 views

### unbounded derived category of a $\infty$-topos

In HTT(Higher Topos Theory) Remark7.3.1.19, it it sketched that the proper base change theorem for $\infty$-topos implies the usual proper base change theorem in (unbounded) derived category. However, ...

**9**

votes

**1**answer

116 views

### Fourier-Mukai functors being identity on objects

Let $X$ be a projective variety over $\mathbb{C}$, denote by $D^b(X)$ the bounded derived category of coherent sheaves on $X$. Suppose we have a Fourier-Mukai functor $\Phi_{X\rightarrow ...

**0**

votes

**0**answers

42 views

### Derived tensor product and restriction

Let $A$ be a commutative ring, and let $B$ be a commutative $A$-algebra.
We have a restriction map $(-)_A : D(B) \to D(A)$ which takes an object of the derived category of $B$-modules $M$, and ...

**2**

votes

**0**answers

81 views

### Is there any numerical obstruction for all perfect complexes on a scheme being strictly perfect?

Let $X$ be a scheme and $E^{\cdot}$ be a cochain complex of sheaves of $\mathcal{O}_X$-modules.
We call $E^{\cdot}$ a strictly perfect complex if $E^{\cdot}$ is a bounded (in both direction) complex ...

**2**

votes

**0**answers

203 views

### Why care about Fourier-Mukai partners?

Two (smooth, projective, complex?) varieties are called Fourier-Mukai partners if they have equivalent derived categories of coherent sheaves. On the other hand, my general impression is that cool ...

**0**

votes

**1**answer

233 views

### A functorial isomorphism in derived category

This question is a direct continuation of Question 1 in this post: Two basic questions on derived categories
Let $f\colon \mathcal{A}\to\mathcal{B}$ be a left exact functor between two abelian ...

**3**

votes

**2**answers

532 views

### Two basic questions on derived categories

Let $\mathcal{A}, \mathcal{B}$ be two abelian categories with sufficiently many injective objects (in my case these are categories of sheaves of vector spaces on a manifold).
Let $f_*\colon ...

**2**

votes

**0**answers

117 views

### Do all full exceptional sequences of a triangulated category have the same length?

Let $\mathcal{D}$ be a $k$-linear triangulated category and $\langle E_n,E_{n-1},\ldots, E_0\rangle$ be a sequence of objects of $\mathcal{D}$. We call $\langle E_n,E_{n-1},\ldots, E_0\rangle$ an ...

**1**

vote

**0**answers

124 views

### Thickness of the category of perfect complexes with finite length homology

Let $R$ be a commutative Noetherian local ring and let $D(R)$ be the derived category of $R$-modules. Recall that a chain complex $C_\bullet$ of modules over $R$ is called perfect if it is isomorphic ...

**1**

vote

**0**answers

111 views

### Why $D^b(S)\cong D^b_{\text{Car}}(\text{cosq}(X\rightarrow S))$?

Let $p: X\rightarrow S$ be a map between topological spaces and we can construct the simplicial space $\text{cosq}(X\rightarrow S)$ where $X_0=X$, $X_1=X\times_S X$ and
$$
X_n=\underbrace{ X\times_S ...

**6**

votes

**1**answer

266 views

### Idea and intuition behind Penrose transform

I would like to know what a Penrose transform is, or more precisely, what is it intended to be - I'm interested in ideas, intuition and some examples of application.
My knowledge of differential ...

**5**

votes

**0**answers

147 views

### Compact objects and ind-objects in triangulated categories

question : let $A$ be triangulated category compactly generated by subcategory $A^c$ of compact objects. Consider category of ind-objects $Ind(A^c)$. Is there relation between $A$ and $Ind(A^c)$? ...

**4**

votes

**0**answers

172 views

### Does GKZ's reflexivity theorem imply the Plucker formula?

Let $S\subset\mathbb{P}^n$, Gelfand-Kapranov-Zelevinsky defined its dual variety $S^\vee\subset\mathbb{P}^{n^\ast}$. In this paper (http://arxiv.org/pdf/math/0111179v1.pdf), the author obtained the ...

**2**

votes

**0**answers

112 views

### Unbounded derived category that is not left-complete

Let me first recall some definition: Let $A$ be a Grothendieck Abelian category. Then, then category $\mathrm{Ch}(A)$ (I am using homological indexing) admits a combinatorial model structure (see for ...

**3**

votes

**1**answer

119 views

### A canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}_{r+2}$ (references)

According to several articles I could find, a canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}_{r+2}$, where $r \geq 2$.
I don't know how to obtain this ...

**5**

votes

**2**answers

254 views

### Is the homotopy category of a ring also the derived category of another ring?

Let $R$ be an associative ring. Let $K(R)$ be the category of chain complexes of $R$-modules and chain homotopy classes of maps between them, and let $D(R)$ be its localization with respect to ...

**3**

votes

**1**answer

327 views

### Reference for comparison of heart cohomology with standard cohomology

I'm looking for a reference for the following fact (which I believe to be true and should be easy for people who understand how spectral sequences arise from filtrations).
Let A,B be two hearts of ...

**1**

vote

**0**answers

152 views

### Derived category of product of complex manifolds

Let $X$ and $Y$ be a compact complex manifold. Is it possible to describe the derived category $D^b(X\times Y)$ of coherent sheaves in terms of $D^b(X)$ and $D^b(Y)$? I am particularly interested in ...

**7**

votes

**2**answers

399 views

### Equivariant motivic sheaves

Thanks to the work of Cisinski-Deglise: http://arxiv.org/abs/0912.2110, we now have a triangulated category of `motivic sheaves' available that admits the standard yoga of the six functors.
Is there ...

**23**

votes

**4**answers

1k views

### What (if anything) unifies stable homotopy theory and Grothendieck's six functors formalism?

I know of two very general frameworks for describing generalizations of what a "cohomology theory" should be: Grothendieck's "six functors", and the theory of spectra.
In the former, one assigns to ...

**0**

votes

**0**answers

115 views

### Why “Fourier”-Mukai? [duplicate]

The Fourier-Mukai functor is one of the most important tools to work with in the derived category. While it is clear why the name of S.Mukai appears there,
why does Joseph Fourier appear in the name ...

**1**

vote

**1**answer

226 views

### Morphism between Fourier-Mukai functors implies the morphism between kernels?

Suppose $X,Y$ are smooth varieties over $\mathbb{C}$, and let $K_i \in D^b(X \times Y), i=1,2$ be objects in the derived category of bounded complex of coherent sheaves on $X \times Y$. Then there are ...

**2**

votes

**1**answer

61 views

### dg-flat complexes and their characters

Let $\otimes$ denotes the usual tensor products of complexes and symbols live in the category of chain complexes of $R$-modules. Let $X$ be a dg-flat complex (i.e. $X_n$ is flat for each n and ...

**3**

votes

**2**answers

315 views

### Derived category of a hypersurface

Let $X$ be a smooth projective variety over $\mathbb{C}$, and $H \subset X$ be a smooth hypersurface.
Many properties of an ambient variety $X$ could somehow inherit to the hypersurface $H$, I was ...

**3**

votes

**1**answer

185 views

### flat descent for perverse sheaves

Let $E \in D^{b}_{c}(X,\overline{\mathbb{Q}}_{l})$ where $X$ is a $k$ scheme of finite type for a field $k$.
Let $Y\rightarrow X$ a finite flat surjective morphism such that $f^{*}E$ is perverse and ...

**5**

votes

**1**answer

195 views

### Equivariant Formality

Let $G$ be a finite group and $\mathcal{A}$ be a $dg$-algebra. Assume $G$ acts on $\mathcal{A}$, i.e. there exists a homomorphism $G\to {\rm Aut}_{dg}(\mathcal{A})$.
Assume further there exists a ...

**4**

votes

**0**answers

118 views

### Formal DG-algebras

Sorry for this question but I really have difficulties with model categories.
Usually a $dg$-algebra $A$ is called formal, if there exists a $dg$-algebra $B$ and quasi-isomorphisms $$A\leftarrow B\to ...

**0**

votes

**2**answers

223 views

### Injective resolution for right derived functor

This question is base on my previous question, and I repeat it here:
Suppose $X$ is a projective variety and $D^{+}(X)$ is the derived cateogry of bounded below complexes of sheaf of ...

**4**

votes

**2**answers

592 views

### Why is the derived tensor product only defined for bounded above derived categories?

In "Residues and Duality" by Hartshorne, the derived tensor $\otimes$ only defined for the bounded above categories (see Chapter I, section 4), that is one has
$$\otimes: D^{-}(X) \times D^{-}(X) \to ...

**9**

votes

**1**answer

475 views

### Are $D^b_{coh}(X)$ and $D^b(Coh(X))$ derived equivalent?

Let $X$ be a variety. Let $D^b(Coh(X))$ be the derived category of bounded complexes of coherent sheaves on $X$, and $D^b_{coh}(X)$ be the derived category of bounded complexes of sheaves of ...

**5**

votes

**2**answers

294 views

### Equivariant derived category and invariant divisor

I'm looking for a reference of the following (folklore?) result.
Let $X$ be a smooth projective variety equipped with a $G=\mathbb{Z}/2\mathbb{Z}$ action (we consider the simplest case, everything ...

**0**

votes

**0**answers

259 views

### generators for derived category

Let $G$ be a algabraic group $G$ over a field $k$. We denote by $D^b(\mathrm{Repr}(G))$ the derived category of finite dimensional representations. Under what kind of assmumptions one has a generating ...

**4**

votes

**0**answers

163 views

### Grothendieck group of intersection of quadrics

Conjecturally, for a complete intersection $X$ the group $K_0(X)$ is finitely generated iff $H^{p,q}(X)=0$ for $p≠q$. This holds if and only if $X$ is a quadric, a cubic surface, or an ...

**0**

votes

**0**answers

171 views

### Hodge structure of abelian surfaces

In my case, I have an abelian surface $A$ of (2,8)-polarization, and I have some finite group (may not be abelian group) $G$ acting on $A$ without fixed point. I want to understand when there is a ...

**3**

votes

**0**answers

431 views

### unique enhancement for derived categories

I have two questions concerning the existence and uniqueness of enhancements in the following cases:
i.) Let $A$ be a finite dimensional $k$ algebra of finite global dimension. Does the triangulated ...

**7**

votes

**0**answers

263 views

### Reconstructing the Chow ring from the derived category

Let $X$ be a smooth projective variety and write $\mathbf{D}(X)$ for its triangulated category of perfect complexes of quasi-coherent sheaves.
Recall that $\mathbf{D}(X)$ determines the Grothendieck ...