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Results tagged with homological-algebra
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user 43054
(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.
14
votes
Accepted
is the tensor product of projective modules again projective?
Since $P_1$ is projective there exists $Q_1$ $A_1$-module and an isomorphism
$$ P_1\oplus Q_1 = A_1^{\oplus I_1}$$
for some index set $I_2$. Analogously there exists $Q_2$ $A_2$-module and an index se …
- 16.5k
5
votes
About the cone being unique up to non-unique isomorphism
Another theory (equivalent, I believe, to derivators) which has the aim of compensating for this shortcoming of triangulated categories is the theory of ∞-categories.
These are, essentially, categori …
- 16.5k
3
votes
Injective resolution for right derived functor
I think you have a little confusion between right and left derived functors. Let me repeat the definition using the language of Kan extensions. In the following for any abelian category $A$ we will wr …
- 16.5k
38
votes
Accepted
Replacing triangulated categories with something better
My opinion, and that of many other people although not of everyone, is that the "correct" notion is that of stable ∞-category.
Now, this is not a category in the strictest sense, rather a generalizat …
- 16.5k
55
votes
Accepted
Why stable $\infty$-categories?
I already answered some version of this question in this answer, but let me try to expand a bit on the concrete advantages in mathematical practice. For understanding the following you need to take on …
- 16.5k
5
votes
Accepted
Which triangulated categories are subcategories of compact objects "somewhere"?
I do not know of an answer for a general triangulated category (non-topological triangulated categories are very unusual), but as soon as you ask for some more structure the thesis follows very quickl …
- 16.5k
21
votes
Accepted
Why does K-theory need schemes to be Noetherian?
You don't need the Noetherianness hypothesis to talk about K-theory. But the definition you propose in your question is not suited for the most general case. From a notion of K-theory we want at least …
- 16.5k
9
votes
Accepted
Contractible chain complex from non-contractible space
These are known as acyclic spaces (note that since $\tilde C_*(X)$ is a bounded below complex of projectives, it being contractible is equivalent to its homology being trivial).
There's an extensive …
- 16.5k
6
votes
Accepted
mod p (odd) cohomology of dihedral groups
EDIT: Thanks a lot to Mike Miller for pointing out in the comments significant simplifications to the proof I wrote
First suppose that $p$ does not divide $n$. Then we have
$$H^*(D_{2n};\mathbb{F}_p) …
- 16.5k
14
votes
Accepted
Is the derived category of perfect complexes idempotent complete?
The derived category of perfect complexes is idempotent complete, because it is the sub category of compact objects in the derived category of quasi coherent sheaves (which is idempotent complete by t …
- 16.5k
11
votes
Künneth formulas/theorem for bordism groups and cobordisms?
The Künneth formula for ordinary homology as you present it works only when $R$ is a PID (or more generally of cohomological dimension 1).
For a general well-behaved homology theory[1] (this includes …
- 16.5k