All Questions
69 questions
13
votes
1
answer
1k
views
When do six operations work?
This question comes (heavily edited) from my notes, thus slightly unusual structure.
We know that algebraic maps have very strict structure, and in many settings the operations ...
- 32.5k
6
votes
2
answers
442
views
Existence of functorial (K-)flat resolutions?
I am wondering about the following: Suppose $X$ is a reasonable scheme or stack with the resolution property. (So, all quasi-coherent sheaves admit a surjection from a flat sheaf.) Then I believe ...
- 15.6k
2
votes
0
answers
133
views
Formulation of cap product in group-equivariant sheaf cohomology + applications?
Originally asked on Math SE but it was suggested I move it here.
Suppose one has a distinguished cocycle in the group-equivariant sheaf cohomology $\Phi \in H^n(X, G, \mathcal{F})$ for a "nice&...
- 101
2
votes
0
answers
154
views
Non-triviality of a morphism
Let $X$ be a smooth Gushel–Mukai fourfold and $Y$ a smooth hyperplane section, which is a Gushel–Mukai threefold. I consider semi-orthogonal decomposition of $X$ and $Y$:
$$D^b(X)=\langle\mathcal{O}_X(...
- 12.9k
2
votes
0
answers
136
views
dg-Künneth formula for qcqs schemes
Let $X$ and $Y$ be qcqs schemes over a field $k$ (or I am happy to assume any nice condition up to smooth (quasi-)projective varieties if that makes the folloiwng question true). Let us define $Perf(X)...
- 256
-1
votes
1
answer
177
views
When morphism of complexes is homotopic to 0?
Let $f\colon A^\bullet\to I^\bullet$ be a morphism of bounded below complexes in an abelian category. Assume all $I^i$ are injective objects. Assume also that $f$ induces the zero map on cohomology.
...
- 38k
1
vote
0
answers
111
views
Kunneth formula for hypercohomology
Let $A_{\bullet}$ and $B_{\bullet}$ be two bounded complexes of sheaves over a variety $X$. Is there a Kunneth-like formula for the hypercohomology of the tensor product $A_{\bullet}\otimes B_{\bullet}...
- 494
2
votes
0
answers
119
views
dg-natural transformation between FM functors and Hom between kernels
The question is related to Morphism between Fourier-Mukai functors implies the morphism between kernels?
Consider a complex smooth projective variety $X$ and the bounded derived category $D^b(X)$, it ...
CommunityBot
- 1
3
votes
1
answer
147
views
What do we know about a sheaf $M$ if we know its derived fibers $\mathsf{L}x^* M$, for $x\in X(k)$?
Let $X$ be a scheme over a field $k$. (Feel free to assume that $X$ is an algebraic variety, if needed.) Also, let $M^\bullet$ be a complex in the derived category of quasi-coherent sheaves $\mathsf{D}...
- 39.3k
6
votes
1
answer
526
views
How does the Balmer spectrum fail to describe the algebraic geometry of categories of non-compact objects?
In trying to understand the higher algebraic geometry of the stable homotopy category, one thing I've come across repeatedly is the claim that one should only consider the Balmer spectrum of a tt-...
- 1,969
5
votes
2
answers
3k
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 II, section 4, p.93), that is one has
$$\otimes: D^{-}(X) \...
- 615
2
votes
1
answer
239
views
How to complete $i^*i_*\mathcal{F}\to \mathcal{F}$ into an exact triangle for a smooth divisor $i: X\hookrightarrow Y$?
Let $Y$ be a smooth algebraic variety and $i: X\hookrightarrow Y$ be a smooth divisor. We consider the derived functors $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$ and $i_*: D^b_{coh}(X)\to D^b_{coh}(Y)$. By ...
- 39.3k
2
votes
0
answers
244
views
Do we have a left adjoint of $i^*$ for a closed immersion $i$?
Let $i: X\hookrightarrow Y$ be a closed immersion of varieties. We have the derived pullback functor $i^*: D^b_{coh}(Y)\to D^b_{coh}(X)$.
My questions is: can we construct a left adjoint of $i^*$ in ...
- 9,019
11
votes
1
answer
2k
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 ...
- 9,229
1
vote
1
answer
303
views
A question about a truncated object
I was hoping someone could help me with the understanding of a particular truncated object. Here are some background:
For any object $A$ in an abelian category $\mathcal{A}$, we can view $A$ as an ...
- 40.2k
1
vote
0
answers
72
views
Bound on Hochschild dimension of a dg-algebra
Consider a dg-algebra $A$, is there any way I can estimate the Hochschild dimension, or global dimension of $A$?
More precisely the algebra that I am considering is the Endomorphism dg-algebra $\...
- 213
12
votes
1
answer
577
views
Embedding of a derived category into another derived category
I am considering the following two cases:
Assume that there is an embedding: $D^b(\mathcal{A})\xrightarrow{\Phi} D^b(\mathbb{P}^2)$and the homological dimension of $\mathcal{A}$ is equal to $1$($\...
- 1,161
6
votes
0
answers
201
views
Smoothness of a variety implies homological smoothness of DbCoh
I have been told that $D^bCoh(X)$ is homologically smooth if $X$ is a smooth variety, and I am trying to construct a proof. My background is not in algebra, so I apologize for elementary questions.
It ...
3
votes
1
answer
125
views
Smallness condition for augmented algebras
I'm not sure this question is research level question. Sorry in advance.
Hypothesis
$k$ is a commutative ring.
$A$ is an augmented $k$-algebra.
$A^e$ is defined as the $k$-algebra $A\otimes_{k}A^{op}$...
- 701
4
votes
0
answers
168
views
detecting a semi-free module from its bar-resolution
Let $A$ be a DG-algebra over a field (say $k$). A DG-module $M$ over $A$ is said to be semi-free if it admits an exhaustive filtration $0 = M_0 \subset M_1 \subset \ldots \subset M_p = M$ such that ...
- 7,300
5
votes
0
answers
214
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 ...
- 9,870
1
vote
1
answer
168
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 ...
- 39.3k
7
votes
2
answers
612
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 $E$ ...
- 1,759
1
vote
2
answers
287
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 ...
- 12.3k
3
votes
0
answers
424
views
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 ...
- 63.9k
2
votes
0
answers
315
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 ...
- 988
4
votes
0
answers
205
views
Sheaf-type property for Derived Categories?
Suppose $X$ is a finite dimensional complex space (I'm happy to restrict to $X$ being a scheme of finite type over $\mathbb C$ as well). I'm wondering if the following sheaf-like properties hold for ...
- 356
3
votes
1
answer
185
views
How to check that exceptional sequence of vector bundles on Fano variety is helix foundation
Let $X$ be smooth Fano variety with $\operatorname{Pic}(X) = \mathbb{Z}$ of dimension $m$ with canonical class $K$, and $E_0,...,E_n$ is exceptional sequence of $(n+1)$ vector bundles in $D^b(Coh(X))$....
- 39.3k
2
votes
1
answer
276
views
Can one always find a bigger global resolution
Let $X$ be a scheme. Let $E$ be a perfect complex of coherent sheaves on $X$ and suppose it admits two global resolutions $ F$ and $F'$. By global resolution I mean that both $F$ and $F'$ are quasi-...
- 42.9k
15
votes
4
answers
2k
views
What is the relation between the Lie bracket on $TX$ as commutator and that coming from the Atiyah class?
Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction to the global existence ...
- 13.6k
4
votes
0
answers
258
views
Generators of unbounded derived categories of (quasi-)coherent sheaves
An object $T$ in a triangulated category $\mathcal{D}$ is called a generator if $T^\perp=0$, which means that for any nonzero $X$ in $\mathcal{D}$, there are $i\in\mathbb{Z}$ and a nonzero morphism $T[...
- 263
5
votes
1
answer
510
views
General existence theorem for cup products
I'm curious if it is possible to formulate cup products and prove that they exist in a general way which would subsume a lot of examples: e.g. group cohomology, sheaf cohomology for sheaves on ...
10
votes
1
answer
342
views
Vanishing natural transformation exact triangle
This question is a follow-up to this question I asked some time ago. Let $X$ be a smooth projective variety of dimension $n$ over $\mathbb{C}$. Let $\omega \in H^{n}(X,K_X)$, $\omega \neq 0$. Let
$$A ...
- 7,300
2
votes
1
answer
342
views
A question on some lemmas in Orlov's "Triangulated Categories of Singularities and D-Branes in Landau-Ginzburg Models" (Exts vanishing)
I'll write the two lemmas I have questions about, and then ask my questions. For reference, I'm using the following definition of Gorenstein:
$\mathbf{Definition\ 1.15}$ A local noetherian ring $A$ ...
- 39.3k
4
votes
0
answers
235
views
Serre duality graded singularity category
Let $R$ be a local Gorenstein ring of Krull dimension $d$ with an isolated singularity. Defined $D_{sing}(R)$ as the Verdier quotient $D^b(R)/Perf(R)$). Then, a famous result of Auslander says that ...
- 7,300
9
votes
0
answers
506
views
Categorification of definitions in the context of the derived category of quasi-coherent sheaves
Let $SpecA=X$ be an affine noetherian scheme. Let $QCoh(X)$ denote the derived (stable $\infty$-)category of quasi-coherent sheaves on $X$. There are the following special full subcategories spanned ...
- 7,789
3
votes
2
answers
1k
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 ...
CommunityBot
- 1
6
votes
1
answer
929
views
Different definitions of derived functors
In principle one uses the notion of derived category, and the other doesn't.
Suppose $F: \mathcal A \to \mathcal B$ is a left exact (additive) functor between abelian categories, and suppose the ...
- 9,481
2
votes
1
answer
1k
views
Simple example of a perfect complex not isomorphic to a strictly perfect complex?
I'm looking for the simplest possible example (one that's easy to remember) for the situation described in the title. More precisely I'm looking for the following example:
A (probably has to be ...
- 4,111
7
votes
0
answers
268
views
Identifying and reconstructing the derived category from its auto-equivalences
Background: Given a smooth irreducible algebraic variety $Y$ with $\omega_Y$ or $\omega_Y^{-1}$ ample. Then Bondal-Orlov theorem states that if there exists any other smooth algebraic variety $Y'$ ...
- 1,981
7
votes
1
answer
566
views
Smoothness of a projective variety via the derived category
Let $X$ be a smooth projective integral variety over an algebraically closed field $k$. Let $Y$ be a (not necessarily smooth) projective integral variety over $k$.
Assume that $D^b(X) \cong D^b(Y)$.
...
- 2,808
5
votes
1
answer
314
views
Is the dual of a compact generator also a compact generator of the derived category of a variety?
Let $X$ be a variety (or more generally a quasi-compact, separated scheme) and $D(X)$ be the derived category of complexes of $\mathcal{O}_X$-modules with quasi-coherent cohomologies. Let $\mathcal{E}$...
- 16.9k
3
votes
1
answer
479
views
K-injective (also known as hoinjective) complexes of sheaves of modules
Let $(X,\mathcal O_X)$ be a ringed space (if necessary, assume that it is a scheme with suitable hypotheses). Given two complexes of sheaves $\mathcal F$ and $\mathcal G$ of $\mathcal O_X$-modules, ...
- 2,079
2
votes
0
answers
178
views
Modern dictionary for "old" homological terms
I'm trying to build a little dictionary between old Homological algebra for local rings and the slightly more modern approach via derived functors.
Let $X = SpecA$ be a spectrum of a local ring $(A,...
- 7,789
4
votes
2
answers
587
views
Question about the proof of Prop I.7.4 in Hartshorne's Residues and Duality
Let $F : \mathcal{A}\rightarrow\mathcal{B}$ be an additive functor of abelian categories, such that $F$ has cohomological dimension $\le n$. Suppose $\mathcal{A}$ has enough injectives. Let $P\subset\...
- 7,527
7
votes
1
answer
2k
views
What is the negative cyclic homology of a smooth projective variety?
Let X be a smooth and projective variety. Hochschild homology and cohomology have a very simple definition in terms of Ext groups of the diagonal of X. The Hochschild-Kostant-Rosenberg (HKR) theorem ...
- 8,972
0
votes
0
answers
116
views
Is the pull back of a compact generator under field extension again a compact generator?
Let $\mathcal{T}$ be a triangulated category which has arbitraty direct sums. An object $E\in \mathcal{T}$ is called compact if the functor Hom$(E,-)$ commutes with arbitrary direct sums. A compact ...
- 9,019
6
votes
0
answers
656
views
The derived version of the Grothendieck spectral sequence
Consider the (very well known) Grothendieck spectral sequence for composition of functors $\mathcal F: \mathcal A \to \mathcal B$ and $\mathcal G: \mathcal B \to \mathcal C$ between abelian categories ...
- 167
1
vote
1
answer
492
views
Spectral sequences to compute Hom's in derived category
Does anybody have a good reference that lists spectral sequences that may be used to compute Hom sets in derived categories (of coherent sheaves, say)?
- 196
0
votes
1
answer
266
views
Projective resolutions of torsion modules [closed]
Let $l$ be a prime number, $n\in \mathbb{Z}$. Is it true that any finitely generated $\mathbb{Z}/l^n\mathbb{Z}$-module has a finite (left) resolution by free finitely generated $\mathbb{Z}/l^n\mathbb{...
- 31.2k