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Results tagged with homological-algebra
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user 488857
(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.
3
votes
2
answers
373
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Is there an elementary reason that this colocalisation map of complexes is a quasi-isomorphism?
A fact about triangulated categories is that (exact) localisation functors and so-called colocalisation functors come in pairs, making an exact localisation triangle. I've tried to come up with less t …
- 532
10
votes
Accepted
Factoring through projective modules is an equivalence relation
This equivalence relation is just a quotient of abelian groups. It will be cleaner to show that the $\mathrm{PHom}_R(M, N)$ is a subgroup of $\mathrm{Hom}_R(M,N).$ Indeed, if $f:M\to P_1 \to N$, $g: …
- 532
4
votes
Extensions of $G$-modules parametrized by $H^1$
No. If $G$ is the trivial group and $q =p$ is a prime, $\operatorname{Ext}^1_G(V, W) = \operatorname{Ext}^1_{\mathbb{Z}}(\mathbb{Z}/ p, \mathbb{Z} / p) \cong \mathbb{Z}/p$ but $H^1(G, V^\vee\otimes_\m …
- 532
4
votes
The Krull dimension of the tensor product of rings
You are looking at infinite tensor products so we really shouldn't expect the infinite product to have finite dimension in most cases. Your second example was simply an infinite tensor product of $\ma …
- 532