**1**

vote

**1**answer

109 views

### Can one drop the locally free assumption in projection formula on a projective bundle?

Let $X$ be a noetherian scheme over $\mathbb{C}$, and let $E$ be a locally free sheaf of finite rank over $X$. Then we have the projective bundle $f: \mathbb{P}(E)\rightarrow X$.
Now $f$ is a flat ...

**2**

votes

**0**answers

52 views

### space of stability condition for an elliptic curve

Let $E$ be an elliptic curve. I want to understand why $\mathrm{Stab}(E)/\mathrm{Aut}(D^b(E))$ is a $\mathbb{C}^\times$-bundle over the moduli space $\mathbb{H}/\mathrm{SL}(2,\mathbb{Z})$ of elliptic ...

**17**

votes

**1**answer

402 views

### Why would one “attempt” to define points of a motive as $\operatorname{Ext}^1(\mathbb{Q}(0),M)$?

I'm a novice when it comes to motives. (I've read multiple introductory texts.)
I'm attempting to read Galois Theory and Diophantine geometry by Minhyong Kim. In it, he says that "One might attempt, ...

**4**

votes

**1**answer

222 views

### An equivalence of derived categories by Happel-Reiten-Smalø

I have a problem in understanding the proof of a theorem by Happel-Reiten-Smalø. The original reference is this article
http://arxiv.org/abs/0911.4473
.
I write down the text of the theorem and a ...

**1**

vote

**1**answer

239 views

### Will a tilting sheaf over Z which is a generator over Q be a generator modulo every prime?

Assume you have a smooth quasi-projective scheme $X$ (you can actually assume $X$ is projective over an affine scheme of finite type) defined over $\mathbb Z$ (or if you prefer, a discrete valuation ...

**0**

votes

**0**answers

168 views

### Quantities associated to deformed sheaves

I am trying to figure out what happens to "quantities" associated to a sheaf when one deforms it. I am actually interested in deforming a bounded complex of coherent sheaves but I want to make the ...

**3**

votes

**1**answer

68 views

### Injective model structure on sheaves of bounded complexes of $A$-modules

The following might be very well known for people who works with model categories, but I do not find the answer.
Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain ...

**4**

votes

**1**answer

126 views

### Interchanging the tensor product with infinite product

Let $R$ be a $k$-algebra (not necessary commutative) and let $\mathbf{D}(R)$ be its derived category (right modules). I'm interested in the class of objects $V$ of $\mathbf{D}(R^{op})$ having the ...

**4**

votes

**2**answers

182 views

### Vanishing natural transformation and strong generator

Let $X$ be a smooth projective variety (over the field of complex numbers). Let $T$ be strong generator of $D^b(X)$ : this means that every object in $D^b(X)$ can be obtained in a given finite number ...

**4**

votes

**0**answers

147 views

### object in D^bCoh(P^2) with prescribed RHom's

Let $\mathbb{C}P^2$ denote the projective plane.
From reading the section of http://homepages.math.uic.edu/~coskun/gokova.pdf
which surveys Gieseker stable sheaves, I have understood that there are ...

**2**

votes

**1**answer

135 views

### Derived pullback of the coarse moduli morphism

Let $f: \mathcal{X}\to X$ be a morphism from a smooth DM-stack $\mathcal{X}$ to its coarse moduli space $X$. Assume that $X$ is also smooth. Is it true that $Lf^*$ is fully faithful and induces an ...

**0**

votes

**1**answer

120 views

### perfect modules over polynomial algebra

This may be obvious. My question is short:
$R$ is the polynomial algebra $\mathbb{k}[X_{1},\dots , X_{n}]$. Is the $R$-module $\mathbb{k}$ perfect in the sense that $\mathbb{k}$ is a compact object ...

**1**

vote

**0**answers

60 views

### On (universal) additive functors making a given complex contractible: examples?

Let $M=(M^i)$ be a (cohomological) complex of objects of some additive category $A$ (I am mostly interested in "short" complexes; yet one may also consider an unbounded $M$). I am interested in those ...

**2**

votes

**0**answers

99 views

### Full exceptional collections in derived category of coherent sheaves on non-compact varieties

Let $X$ be a smooth algebraic variety over $\mathbb{C}$, and $D^b(X)$ its bounded derived category of coherent sheaves. Then a full exceptional collection will lead to significant simplifications in ...

**1**

vote

**2**answers

88 views

### Showing that the stable module category of a ring $R$ restricted to maximal Cohen-Macaulay objects is trivial if $\text{gldim } R < \infty$

(In the following, a (not necessarily commutative) ring $R$ is Gorenstein if it has finite injective dimension as a module over itself on either side, and a finitely generated (right) $R$-module is ...

**6**

votes

**1**answer

196 views

### When the restriction of a derived functor to a subcategory is the derived functor of the restriction

Let $\mathcal{D},\mathcal{E}$ be abelian categories and $\mathcal{C}$ be a Serre subcategory of $\mathcal{D}$.
Let $D(\mathcal{C}), \, D(\mathcal{D})$ denote the derived categories of ...

**1**

vote

**0**answers

49 views

### Do flat resolutions guarantee the existence of Tor (without enough projectives)?

Let $\mathcal A$ be an abelian category with a symmetric monoidal structure $\otimes$. Suppose that $\mathcal A$ does not have enough projectives, but every object has a flat resolution Then, is the ...

**5**

votes

**1**answer

130 views

### Left orthogonals to compact objects in triangulated categories: existence and “control”?

Let $C$ be a compactly generated triangulated category. Can it contain a non-zero object $M$ such that there are no non-zero morphisms FROM $M$ into compact objects? I would be grateful for any ...

**3**

votes

**1**answer

248 views

### Equivariant Derived Category

Can someone give me a reference for the following or an idea on why it is true? (This is taken from remark 1.5 on page 5 of http://arxiv.org/abs/0810.0794.)
Suppose we have an algebraic group $G$ ...

**0**

votes

**1**answer

178 views

### Grothendieck-Verdier duality for affine morphisms

Suppose $X,Y$ are varieties over $\mathbb{C}$, $Y$ is smooth and $X$ is Gorenstein ($X$ is not smooth in my case). Let $f: X \to Y$ be an affine morphism, and each fibre of $f$ has the same dimension ...

**2**

votes

**1**answer

137 views

### Ext groups in the equivariant derived category

I apologize in advance that this question is probably too basic for MO, but I reckoned I would not get an answer on Math.Stackexchange.
I am starting to learn about perverse sheaves, the ...

**10**

votes

**2**answers

1k views

### Derived functors vs universal delta functors

I would like to understand the relationship between the derived category definition of a right derived functor Rf (which involves an initial natural transformation n: Qf → (Rf)Q, where Q is the ...

**4**

votes

**0**answers

127 views

### Reference for Grothendieck's duality and Cousin, Dualizing and Residual complexes

I am a graduate student currently reading Hartshorne's Residues and Duality. In order to reach the construction of the right adjoint $f^!$ of $Rf_*$ for some special types of maps of locally ...

**3**

votes

**1**answer

381 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 ...

**7**

votes

**4**answers

728 views

### Is there a (satisfying) proof that cellular cohomology is isomorphic to simplicial cohomology that doesn't use relative cohomlogy?

That singular and de Rham cohomologies of a smooth manifold are isomorphic has two proofs that I know of. The classical one uses Stokes' theorem to give the isomorphism explicitly. The second proof ...

**1**

vote

**1**answer

155 views

### A delicate question about derived functors

Let $A\subseteq B \subseteq C$ be three triangulated categories, such that $A$ is a full triangulated sub-category of $B$, and $B$ is a full triangulated sub-category of $C = K(R)$.
Let $F: C \to D$ ...

**10**

votes

**1**answer

304 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 ...

**2**

votes

**0**answers

83 views

### Do copairings provide dualities in derived categories?

Here is an elementary fact about vector spaces. Let $V,W$ be vector spaces over a field $\mathbb K$ and let $c : \mathbb K \to V \otimes W$ be an element of the tensor product. Then $c$ determines ...

**1**

vote

**0**answers

79 views

### Is there any explicit result on the triangulated category of singularities of a curve?

This question is related to this MO question.
Let $X$ be a projective curve over a field $\mathbb{C}$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category ...

**2**

votes

**1**answer

144 views

### Could we extend the exact sequence $K^0(X)\to K_0(X)\to K_0(D_{sg}(X))\to 0$ to the left?

Let $X$ be a variety over a field $k$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category of perfect complex $Perf(X)$. It is clear that $Perf(X)$ is a ...

**4**

votes

**0**answers

207 views

### On a theorem of Hopkins-Neeman-Thomason on generators of thick subcategories of perfect complexes

Notations and background. Let $R$ be a commutative noetherian local ring and let $D(R)$ denote the derived category of the category of R-modules. A strictly perfect complex on $R$ is a bounded complex ...

**4**

votes

**0**answers

162 views

### 2-periodic derived equivalence

Let $A$ and $B$ be finite-dimensional algebras with finite global dimension over some field (in fact I am thinking of rational incidence algebras of finite posets).
Suppose we know that $A$ and $B$ ...

**2**

votes

**1**answer

198 views

### Is $K^0(X)\to K_0(X)$ monomorphic for a noetherian scheme $X$?

This question is related to the MO questions What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes? and Does a fully faithful functor between triangulated categories induce ...

**0**

votes

**0**answers

112 views

### Derived categories of modules categories

Does anyone know if there is a note or a paper about the derived category of the category $\sigma[M]$ where $M$ is a left module over a ring?, and some uses of this.

**4**

votes

**1**answer

258 views

### What is the applications of the dg-enhancements of derived categories of sheaves

Let $X$ be a scheme and let $D^b_{\text{coh}}(X)$ be the derived category of complexes of sheaves with bounded, coherent cohomologies.
We know that the category $D^b_{\text{coh}}(X)$ has some ...

**1**

vote

**0**answers

143 views

### higher direct images of O(E)

I hope this is well known, I just could not work it out myself.
Say I have a variety X (smooth and projective over C is my usual setup) with a smooth subvariety Z. Let f: BL_Z(X) --> X be the blowup ...

**0**

votes

**0**answers

129 views

### Is it a correct description of the bounded above derived category of coherent sheaves?

Let $X$ be a (Noetherian) scheme. Let $D^{-}_{\text{coh}}(X)$ be the derived category of complexes of $\mathcal{O}_X$-modules with bounded above and coherent cohomologies.
Do we have the following ...

**3**

votes

**1**answer

194 views

### Is the derived category of perfect complexes idempotent complete?

Let $\mathcal{C}$ be a category. We call a morphism $\alpha: X\rightarrow X$ an idempotent if $\alpha^2=\alpha$ in $\mathcal{C}$. We call $\mathcal{C}$ is $\textit{idempotent complete}$ if any ...

**4**

votes

**0**answers

176 views

### When do we have $D_{\text{perf}}(\text{Qcoh}(X))\simeq D_{\text{perf}}(X)$?

Let $(X,\mathcal{O}_X)$ be a scheme (or more generally a ringed space). We know that in general the derived category of complexes of quasi-coherent modules $D(\text{Qcoh}(X))$ is not equivalent to the ...

**2**

votes

**1**answer

122 views

### An alternative definition of pseudo-coherent complex

Let $(X,\mathcal{O}_X)$ be a scheme or a general ringed space. First recall that a complex of $\mathcal{O}_X)$-modules $\mathcal{E}^{\bullet}$ is called strictly perfect if $\mathcal{E}^{\bullet}$ is ...

**3**

votes

**1**answer

103 views

### Is the image of a idempotent morphism in $\mathcal{K}(\mathcal{A})$ defined in the naive way?

Let $\mathcal{A}$ be an abelian category and $\mathcal{K}(\mathcal{A})$ be the homotopy category of chain complexes in $\mathcal{A}$. It is well-known that $\mathcal{K}(\mathcal{A})$ is idempotent ...

**5**

votes

**1**answer

162 views

### Relations between functors in a recollement

Consider a recollement situation like the following
by the very definition of the various functors it follows that $i^* j_*=0$, and $j^! i_* = 0 = j^* i_!$. Also, $j^! i_! = 0 = j^* i_*$ by ...

**6**

votes

**0**answers

117 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, ...

**7**

votes

**2**answers

461 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 ...

**4**

votes

**1**answer

260 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 ...

**1**

vote

**0**answers

289 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). ...

**8**

votes

**1**answer

224 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 ...

**3**

votes

**1**answer

336 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 ...

**4**

votes

**0**answers

198 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 ...

**0**

votes

**0**answers

125 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 ...