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Results tagged with homological-algebra
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user 310
(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.
50
votes
Accepted
How do I know the derived category is NOT abelian?
The following nicely does the trick I think...
Lemma Every monomorphism in a triangulated category splits.
Proof: Let $T$ be a triangulated category and suppose that $f\colon x\to y$ is a monomorphis …
- 8,681
22
votes
Accepted
triangulated vs. dg/A-infinity
I don't really think that triangulated categories are abominable, but they certainly have their problems which are a result of having forgotten the higher homotopies. For instance, non-functoriality o …
- 8,681
13
votes
Some intuition behind the five lemma?
One can think of the five lemma in terms of the two four lemmas. I think this makes it clearer... for instance drop the $A_1$ and $B_1$ from your diagram. If the maps from $A_2$ and $A_4$ to $B_2$ and …
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11
votes
Accepted
Is the chain homotopy category, K(Ab), an Abelian category? By Ab, I mean the category of A...
You might want to have a look at my answer to this question.
- 8,681
10
votes
Accepted
Classifying triangulated structures on a graded category
Generally speaking a unique lifting does not exist and I believe it is open as to what the possible liftings can be.
As an example of the non-uniqueness consider a slight variant of the particular ca …
9
votes
Accepted
Matrix factorization categories beyond the isolated singularity case
The answer to (1) is yes for any local abstract hypersurface $S$ whose singular locus is closed (which is barely a hypothesis, and free in the case of interest). Let us write $\mathrm{Sing} \;S$ for t …
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9
votes
Accepted
Freyd-Mitchell for triangulated categories?
There are some things like what you ask for but as Tyler points out one needs restrictions on the categories one can consider.
Any algebraic triangulated category which is well generated is equivalen …
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9
votes
distinguished triangles and cohomology
I just wanted to point out that this failure is quite standard rather than pathological. As a starting point it can go wrong more generally than Tyler points out. For instance there exist triangles wh …
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5
votes
Accepted
When do chain complexes decompose as a direct sum?
One result that guarantees such a decomposition comes from looking at the homological support of such complexes (assuming that $R$ is commutative so we have a tensor product). The homological support …
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5
votes
How to think about model categories?
One can also view model structures as a solution to the problem of when is a localization of a category locally small. In other words when one wants to invert a collection of morphisms the morphisms i …
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4
votes
Accepted
Tate Cohomology via stable categories
To address Hanno's question about checking that composition gives a graded-commutative ring structure on $End^{*}(\mathbb{Z}) = \oplus_i [\mathbb{Z}, \Omega^{-i} \mathbb{Z}]$ suppose first that
$a \st …
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4
votes
Accepted
Sources for exact triangles in triangulated categories.
This question has kind of been bothering me since I started thinking about it - I am far from an expert on KK-theory but I thought I'd throw something out there and maybe someone else will see it and …
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3
votes
Derived category with total cohomology finite dimensional: is there a better name for it?
I'm not sure if this category has a particular name - usually until someone cares enough to give one of these a name or nice notation they just have long unwieldy names. I can suggest some notation th …
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1
vote
When does direct image with proper support have a right adjoint?
So we know that such an adjunction exists for a closed immersion or for an open immersion where we get $(f_*, f^!)$ the pushforward and subsheaf with supports and $(f_!, f^*)$ the extension by zero an …
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