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Results tagged with homological-algebra
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user 2191
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
11
votes
1
answer
263
views
What are the projective dimensions of big fraction fields?
Let $A$ be an integral domain, $B$ is its fraction field. Can the projective dimension of the $A$-module $B$ be greater than $1$? This surely cannot happen if the spectrum of $A$ is countable (since t …
- 16.9k
4
votes
2
answers
349
views
Are hearts of all $t$-structures on smashing triangulated categories closed with respect to ...
Let $T$ be a triangulated category closed with respect to (small) coproducts, and $t$ be (an arbitrary!) a $t$-structure on $T$. I have noted that the heart $\underline{Ht}$ of $t$ is closed with wit …
- 16.9k
6
votes
1
answer
240
views
For which exact couples do associated spectral sequences degenerate at $E_1$?
It is well known that a bigraded exact couple of objects of an abelian category yields a spectral sequence (cf. https://ncatlab.org/nlab/show/exact+couple#SpectralSequencesFromExactCouples). My questi …
- 16.9k
5
votes
0
answers
518
views
Spectral sequences coming from filtrations (Postnikov systems) in triangulated categories: r...
Let $c^i: M^{\ge i+1}\to M^{\ge i}$ for $i\in \mathbb{Z}$ be an sequence of morphisms in a triangulated category $C$ and assume that $M^{\ge i}$ are equipped with compatible morphisms into an object $ …
- 16.9k
2
votes
Accepted
Is the dual of a compact generator also a compact generator of the derived category of a var...
Let me try to sketch an argument (though I am not quite sure in details).
A theorem of Neeman implies that a compact object $M$ in a compactly generated triangulated category $T$ is a generator if an …
- 16.9k
5
votes
2
answers
984
views
On various relations between "additional axioms" for AB4 and Grothendieck abelian categories
Let $A$ be an abelian category that has a generator and satisfies the AB4 axiom. I would like to understand (better) the relations between various additional "restrictions" on $A$.
So here is my list …
- 16.9k
5
votes
1
answer
365
views
Which triangulated categories are subcategories of compact objects "somewhere"?
Let $T$ be a small triangulated category. Under which conditions there exists a triangulated category $B$ closed with respect to (small) coproducts such that $T$ fully embedds into the subcategory of …
- 16.9k
5
votes
0
answers
321
views
Do differential objects form triangulated categories?
Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to A\stackrel{a}{\t …
- 16.9k
3
votes
0
answers
188
views
Can one complete a morphism of commutative triangles to a "commutative cube" in a triangulat...
This question is a continuation of Can one extend a morphism of commutative triangles to a morphism of octahedral diagrams?.
I am deeply grateful for the contributions there; they roughly say that gi …
- 16.9k
1
vote
0
answers
70
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 …
- 16.9k
5
votes
0
answers
225
views
Can triangulated categories be "approximated by countable subcategories" (that are triangula...
For a given (finite) set of (objects and) morphisms $f_i$ in a triangulated category $C$ I am interested in a (non-full!) triangulated subcategory $C'\subset C$ of "small size" that would contain them …
- 16.9k
4
votes
0
answers
162
views
When there exists some "cone" of a morphism of (ind-representable) cohomological functors?
I am interested in cohomological functors from a certain small triangulated category $C$ to abelian groups.
The question is: given a tranformation $F\to G$ of two functors of this sort, is it possibl …
- 16.9k
1
vote
A conservative, non faithful functor between triangulated categories
You can start with the derived category of graded polarizable Hodge structures or with the category of Hodge modules (over a complex variety $X$). These categories possess natural weight structures (a …
- 16.9k
6
votes
1
answer
237
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 exampl …
- 16.9k
5
votes
1
answer
833
views
Extension-closed subcategories of triangulated categories as "almost exact" categories
Did anybody study those subcategories of triangulated categories that are closed with respect to "extensions" (in the sense of distinguished triangles; in particular, any such $B$ is additive)? If we …
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