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Results tagged with homological-algebra
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user 155668
(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.
4
votes
1
answer
465
views
Homology with local systems
Let $X$ be a connected topological space with abelian fundamental group. Let $\mathcal{L}$ be a $\mathbb{Z}$-valued local system on $X$.
Suppose that I know the full homology $H_*(X;\mathbb{Z})$. Are …
- 177
6
votes
0
answers
432
views
Yoneda product on Ext
Let $M$ be a closed manifold and consider the constant sheaf $\mathbb{R}_M$. I have heard that the Yoneda product on $\operatorname{Ext}(\mathbb{R}_M, \mathbb{R}_M)= H^*(M; \mathbb{R})$ coincides with …
- 177
4
votes
1
answer
418
views
Contractible chain complex from non-contractible space
Recall that a chain complex $(C_*,d)$ of abelian groups is contractible if it is homotopic to the zero map. Or equivalently: there exists a degree 1 map $F: C_* \to C_*$ such that $\operatorname{Id}= …
- 177
3
votes
1
answer
272
views
Hochschild homology of acyclic complex
Let $A$ be a differential graded algebra over a commutative ring $R$. Suppose that $H_*(A)=0$, i.e. $A$ is acyclic.
Question: Does this imply that the Hochschild homology $HH_*(A)$ also vanishes ide …
- 177