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Results tagged with homological-algebra
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(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.
8
votes
4
answers
733
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Tensored Over Abelian Groups?
Suppose I have an category additive category C (i.e. the hom sets are enriched in abelian groups and there are finite direct sums). Suppose further that C has cokernels. Then I can make C tensored ove …
- 27.5k
5
votes
2
answers
662
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Terminology: Is there a name for a category with biproducts?
Many people are familiar with the notion of an additive category. This is a category with the following properties:
(1) It contains a zero object (an object which is both initial and terminal).
Thi …
- 27.5k
12
votes
2
answers
1k
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Failure of Fin. Presented and Fin. Generated Modules to be Abelian Categories?
Let R be a ring. I'm trying to understand when the categories of finitely presented R-modules and finitely generated R-modules can fail to be abelian categories.
Poking around on the internet has le …
- 27.5k
5
votes
1
answer
269
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Classifying Algebra Extensions over a fixed extension?
There are lots of "Ext groups" in homological algebra which measure extensions of various things. I'm sure there must be a homological algebra machine for computing the following, and I'm hoping that …
- 27.5k
34
votes
2
answers
5k
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Example Wanted: When Does Čech Cohomology Fail to be the same as Derived Functor Cohomology?
I want to know exactly how derived functor cohomology and Cech cohomology can fail to be the same.
I started worrying about this from Dinakar Muthiah's answer to an MO question, and Brian Conrad's com …
- 27.5k
25
votes
4
answers
3k
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A Peculiar Model Structure on Simplicial Sets?
I'm wondering if there is a Quillen model structure on the category of simplicial sets which generalizes the usual model structure, but where every simplicial set is fibrant? I want to use this to do …
- 27.5k
27
votes
13
answers
4k
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Homological algebra for commutative monoids?
Homological algebra for abelian groups is a standard tool in many fields of mathematics. How much carries over to the setting of commutative monoids (with unit)? It seems like there is a notion of sho …
- 27.5k