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Results tagged with homological-algebra
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user 2039
(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.
5
votes
0
answers
672
views
Pullbacks of Abelian Categories and their Ext-Groups
Let $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ be abelian categories and $F: \mathcal{A} \to \mathcal{C}$ and $G: \mathcal{B} \to \mathcal{C}$ be functors between them. It is then possible to defi …
- 12.1k
4
votes
0
answers
473
views
Tor over graded rings
Let $R$ be a graded ring (concentrated in nonnegative dimensions and maybe bounded from above). For every positive natural number $n$, denote by $R\to\tau_{\leq n}R$ the $n$-truncation and by $\tau_{\ …
- 12.1k
8
votes
2
answers
525
views
Non-vanishing $\mathrm{lim}^1$-term for the cohomology of a CW-complex
Let $h$ be an additive cohomology theory. If we want to compute $h^*(X)$ for an infinite CW-complex $X$, a standard method is to use the Milnor sequence
$$ 0 \to \mathrm{lim}^1_k h^{n-1}(X^{(k)}) \to …
- 12.1k
13
votes
Accepted
generalized universal coefficient sequence
There is more than one possible generalization. The most common is the universal coefficient spectral sequence. Given a (homotopy) commutative ring spectrum $E$ and a spectrum $X$, there is under cert …
- 12.1k