All Questions
Tagged with kt.k-theory-and-homology derived-categories
15 questions
7
votes
0
answers
249
views
Phantoms and Geometry
Let $\mathcal{D}(X)$ be the bounded derived category of coherent sheaves on a smooth projective variety $X$. An autoequivalence $\Phi: \mathcal{D}(X) \to \mathcal{D}(X)$ is called phantom if it ...
- 71
3
votes
1
answer
265
views
Base change in Chriss-Ginzburg
Below is a fragment of the book by Chriss and Ginzburg. Proposition 5.3.15(b) is stated in $K$-theory. My question is, does the same conclusion (and proof?) of proposition 5.3.15(b) (i.e. base change) ...
- 2,964
5
votes
1
answer
380
views
Which complexes of coherent sheaves can be presented as countable homotopy limits of perfect complexes?
Let $X$ be a noetherian scheme (actually, I need the case where $X$ is proper over an affine scheme), $C$ is an object of the derived category $D_{coh}(X)$ of coherent sheaves on $X$. Under which ...
- 16.9k
2
votes
1
answer
267
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Semi-orthogonal decompositions over singular schemes
Where can I find any more or less explicit semi-orthogonal decompositions of derived categories of perfect complexes or of bounded derived categories for singular schemes that are proper over a ring R?...
- 16.9k
3
votes
0
answers
152
views
Bounded derived categories of which smooth projectives possess bounded t-structures whose hearts are categories of modules?
I am interested in $P$ that is smooth and proper over a field and such that the derived category of coherent sheaves $D^b(P)$ possesses a $t$-structure whose heart is the category of finitely ...
- 16.9k
4
votes
0
answers
325
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Which derived categories of coherent sheaves are equivalent (or "$t$-related") to derived categories of rings?
As far as I understand, it was Beilinson who proved that the bounded derived category of coherent sheaves $D^b(\mathbb{P}^n)$ is equivalent to the bounded derived category of a certain (non-...
- 16.9k
4
votes
1
answer
503
views
Mapping cone and derived tensor product
This question is in some sense a continuation to this question: Derived Nakayama for complete modules
For the setting: Let $A$ be a ring and let $I$ be some finitely generated ideal in $A$. Let $f\...
- 309
2
votes
2
answers
542
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Tensor product of mapping cones
Fix a ring $R$. If $A^*_i \to B^*_i \to C^*_i \to A^*_i[1]$ is a distinguished triangle of complexes of $R$-modules, for $i=1$ and $2$ (so $C_i^* = cone(f_i^*)$ where $f_i^*: A_i^*\to B_i^*$), is ...
- 2,964
3
votes
1
answer
285
views
Is there a notion of injective, projective, flat, dimension for a differential graded algebra?
Given a differential graded algebra $(A_\bullet,d)$, is there a well-defined notion of a K-injective, K-projective, K-flat dimension of a differential graded module, or even of the category of ...
- 1,716
1
vote
0
answers
70
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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 ...
- 16.9k
6
votes
1
answer
240
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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 ...
- 16.9k
5
votes
1
answer
417
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 ...
- 9,019
3
votes
1
answer
374
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 ...
- 9,019
51
votes
5
answers
5k
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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 ...
- 40.2k
4
votes
3
answers
540
views
When an exact embedding of abelian categories induces a full embedding of their derived categories?
Let $F:A\to A'$ be a (full) exact embedding of abelian categories. When $D(F):D(A)\to D(A')$ (or its bounded version) is a full embedding also?
I would be interested in any necessary or sufficient ...
- 16.9k