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Results tagged with homological-algebra
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user 18263
(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
How to visualize the Microsupport of a Sheaf?
I'd been hoping for months that someone would come along and answer this question: every time I encounter the definition of microsupport, my brain responds with a flash of anger followed by a protract …
- 15.5k
7
votes
0
answers
653
views
Is there an obstruction which classifies "quasi-isomorphism but not chain equivalence"?
Fix a ring $R$ and let $C_\bullet$, $D_\bullet$ be (possibly unbounded) chain complexes of $R$-modules. Assume that $f_\bullet:C_\bullet \to D_\bullet$ is a quasi-isomorphism: that is to say, $f$ is a …
- 15.5k
16
votes
1
answer
359
views
Moduli space of boundary maps with prescribed chain and homology groups?
Let $R$ be a reasonable ring (maybe I mean a PID, or $\mathbb{Z}$, and when sufficiently desperate, a field). Now consider fixed sequences $C_n$ and $H_n$ of $R$-modules, which are tame in every possi …
- 15.5k
3
votes
1
answer
147
views
Classifying space for homology endomorphisms supported on a graph?
Let $X$ be a reasonable topological space (say one that has the homotopy type of a finite CW complex) and consider a subset $\Gamma$ of $X \times X$ so that the projection $p:\Gamma \to X$ onto the fi …
- 15.5k
5
votes
Accepted
Algebraic Morse theory
It's always nice to see people working on discrete Morse theory.
Answer 1
It is an "if and only if". Meaning: the partial order $\prec$ is defined by $\alpha \prec \gamma$ if and only if $\gamma$ pr …
- 15.5k
7
votes
2
answers
2k
views
isomorphic spectral sequences => quasi-isomorphic filtered chain complexes?
Let $(C,\partial)$ and $(C',\partial')$ be chain complexes of $R$-modules where $R$ is a (commutative) ring. Let $F$ and $F'$ be finite filtrations of $C$ and $C'$ respectively, i.e., $$\varnothing = …
- 15.5k