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Results tagged with homological-algebra
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user 1631
(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.
6
votes
1
answer
491
views
A curious construction of a chain complex and its homology
... curious to me, that is.
Suppose two module filtrations $$ \cdots < A_3 < A_2 < A_1 < \cdots $$ and $$ \cdots < B_3 < B_2 < B_1 < \cdots $$ are comparable in the sense that for all $j$, $ B_{j+1} …
- 1,987
4
votes
2
answers
964
views
About higher Ext in R-Mod
So, in $R-Mod$, we have the rather short sequence
$\mathrm{Ext}^0(A,B)\cong Hom_R(A,B) $
$\mathrm{Ext}^1(A,B)\cong \mathrm{ShortExact}(A,B)\mod \equiv $, equivalence classes of "good" factorization …
- 1,987
4
votes
0
answers
301
views
Connecting group ring, abelianization
For reasons arising in algebraic topology, I'm wanting to better understand the relations between two functors from groups to abelian groups, $\mathbb{Z}[\cdot]$ and $\operatorname{ab}$; group ring an …
- 1,987
4
votes
Tracking spectral sequence differentials
Without meaning to be snarky, I think there's some confusion here about what spectral sequences are for. In particular, the sense in which a SpSeq (or even a long exact sequence!) serves as a computa …
- 1,987
2
votes
Accepted
Homology or cohomology?
(CW because it's more an over-long comment than a real answer.)
I think there are too many competing normalizations to make a good choice. In lieu of sensible default, call one of them homology, and …
2
votes
Heuristic explanation of why we lose projectives in sheaves.
Sorry if this is silly; but might it have something to do with needing in the sheaf category to consider the sheafified presheaf cokernel in order to talk about projections? that is, I (think I) can i …
- 1,987