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Results tagged with homological-algebra
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user 1784
(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.
38
votes
Accepted
Example Wanted: When Does Čech Cohomology Fail to be the same as Derived Functor Cohomology?
Q1: A very simple example is given in Grothendieck's Tohoku paper "Sur quelques points d'algebre homologique", sec. 3.8. Edit: The space is the plane, and the sheaf is constructed by using a union of …
- 4,492
21
votes
Accepted
When are there enough projective sheaves on a space X?
About Jon Woolf's answer, it seems to me that the condition that "$x$ is a closed point" was implicitly used: the extension by zero $Z_A$ is only defined for a locally closed subset $A$ (see e.g. Tenn …
- 149k
6
votes
When are there enough projective sheaves on a space X?
Searching for various examples and counterexamples for sheaves, it is sometimes helpful to look at partially ordered sets with the poset topology: a set $U$ is open if and only if $x \in U$ and $x < y …
- 8,004
9
votes
Ways of formulating homological algebra without diagram chasing
But isn't doing homological algebra without diagram chasing akin to doing geometry without circles, or doing algebra without multiplication? It's just the nature of the subject, a big part of it (and …
- 13.1k
7
votes
Accepted
Can injective modules over R give non-injective sheaves over Spec R?
Let me put this here for the sake of clarity. As was noted by Emerton in a comment above, this answer to a related Math Overflow question shows that the answer is no, for an injective $R$-module $I$, …
CommunityBot
- 1
11
votes
Existence of projective resolutions in abelian categories
Way too optimistic: in many abelian categories there are not enough projectives (and in the dual category there are not enough injectives).
The most standard example is sheaves of abelian groups on …
CommunityBot
- 1
46
votes
Accepted
Why are spectral sequences so ubiquitous?
Let's say you have a resolution $0\to A\to J^0\to J^1\to\dots$ (of a module, a sheaf, etc.) If $J^n$ are acyclic (meaning, have trivial higher cohomology, resp. derived functors $R^nF$), you can use …
- 13.1k
17
votes
Accepted
Differences between reflexives and projectives modules
Well, the answer is well known of course. For a finitely generated module over a commutative normal Noetherian domain TFAE
M is reflexive
M is torsion-free and equals the intersection of its locali …
- 13.1k
2
votes
Accepted
How to construct pair of adjoint functors from category A to category A_D(category of diagrams)
If D is small and A has enough projectives and has infinite sums then $A^D$ has enough projectives. For the proof, see Weibel, "An introduction to homological algebra", 2.3.13 on p.43. It contains the …
- 13.1k