All Questions
25
questions
7
votes
0
answers
315
views
Bondal-Orlov' theorem for noncommutative projective schemes
My question is very simple.
Is Bondal-Orlov's theorem known for noncommutative projective schemes in the sense of Artin and Zhang?
The commutative version is the following :
Let $X, Y$ be smooth ...
- 869
5
votes
1
answer
178
views
A pushout diagram of derived categories coming from an open cover of schemes
Suppose $X=U\cup V$ is the standard open cover of $X=\mathbb{P}^1$ by two affine lines. The descent theorems say that the diagram (with all arrows restriction maps)
$\require{AMScd}$
\begin{CD}
D(X) @&...
- 53
6
votes
1
answer
465
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-...
- 1,837
3
votes
1
answer
170
views
Left adjoint for nested admissible categories
This question is motivated by the construction of the Kuznetsov component on a prime Fano threefold $X$ of index 1 (say genus $g \geq 6$, $g \neq 7, 9$):
$$
D^b(X) = \langle Ku(X), E, \mathcal{O}_X \...
- 275
4
votes
0
answers
162
views
External tensor product Calabi-Yau DG categories
Let $\mathcal{C}$ be a smooth proper DG-category such that the shift $[p]$ is a Serre functor for $D^{perf}(\mathcal{C})$ (we say that $D^{perf}(\mathcal{C})$ is $p$-Calabi-Yau). I am looking for a ...
- 7,250
7
votes
1
answer
612
views
Heart of a bounded $t$-structure on the derived category of coherent sheaves
Let $X$ be an elliptic curve and $D(X)$ the bounded derived category of $Coh(X)$, coherent sheaves on $X$. If $(D^{\leq 0}, D^{>0})$ is a bounded $t$-structure, then can we already say that the ...
- 71
4
votes
0
answers
161
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 ...
- 7,250
6
votes
1
answer
374
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} = \...
- 1,141
6
votes
1
answer
620
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 $\...
- 367
1
vote
1
answer
123
views
Tensoring with complex of finite flat dimension in derived category
Let $(R,m)$ be a Noetherian local ring, and $X$, $Y$ be complexes of finitely generated $R$ modules. Suppose $X$ is bounded above and $Y$ is bounded below. Let $S$ be an $R$-algebra of finite flat ...
- 323
4
votes
0
answers
113
views
Determining whether a morphism is the induced morphism?
Let $F\colon \mathcal A \to \mathcal B$ be a left exact functor between Grothendieck abelian categories. Given a morphism $f\colon A\to B$ in $\mathcal A$ and a morphism $g\colon RF(A)\to RF(B)$ in ...
- 3,039
5
votes
1
answer
478
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 ...
- 2,998
2
votes
1
answer
492
views
Question on lemma 3.5 of Beilinson's paper, Notes on Absolute Hodge Cohomology
I am reading Beilinson's paper, Notes on Absolute Hodge Cohomology (unfortunately I could find an link to this paper ), and I don't understand lemma 3.5.
For $A$ a Noetherian subring of $\mathbb{R}$ ...
- 2,971
3
votes
2
answers
504
views
Clarification on Hanamura's work on $t$ structure of triangulated category of mixed motives
In Hanamura's paper Mixed Motives and Algebraic Cycles III
http://intlpress.com/site/pub/files/_fulltext/journals/mrl/1999/0006/0001/MRL-1999-0006-0001-a005.pdf
He proved that if assume Grothendieck'...
- 2,971
7
votes
1
answer
699
views
Generalised Hodge Conjecture
Further to my question,
A Naive Question on Mixed Motives and Mixed Hodge Structures
that has received very good replies and suggestions, and I really appreciate it. I am going to ask a question on ...
- 2,971