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13 votes
0 answers
680 views

Singular chains generated by manifolds with corners --- does it really work?

Usually we define singular homology via the complex of singular chains: $$C_\ast(X)=\bigoplus_{n\geq 0}\mathbb Z\langle\sigma:\Delta^n\to X\rangle$$ where the right hand side denotes the free abelian ...
John Pardon's user avatar
  • 18.7k
5 votes
2 answers
683 views

Model categories and chain complexes

I'm fairly new to thinking about homological algebra and chain complexes in their own right, i.e outside of isolated examples such as for constructing simplicial homology, or for computing $Ext$ ...
asd's user avatar
  • 168
5 votes
1 answer
2k views

Cohomology of derived tensor product of complexes and Künneth spectral sequence

Let $R$ be any commutative ring, let $V^\bullet$ and $W^\bullet$ be (co)chain complexes of $R$-modules, indexed cohomologically. We can also assume that they have both cohomology in nonpositive ...
Francesco Genovese's user avatar
3 votes
0 answers
68 views

Hermitian structure for complexes of vector bundles

Does it exist a different notion of Hermitian metric for complexes of vector bundles, besides the obvious data of a metric for each vector bundle? Same question for connections. In particular is there ...
BinAcker's user avatar
  • 789
3 votes
0 answers
244 views

Reference Request: The Categorification of $\mathbb{Z}$ as cochain complexes of vector spaces

Just as the fact that a categorification of $\mathbb{N}$ is the category of finite dimensional vector spaces, a categorification of $\mathbb{Z}$ (in my mind) is the category of bounded cochain ...
Zhaoting Wei's user avatar
  • 9,019
2 votes
0 answers
64 views

contracting homotopies of Koszul resolution of $\mathbb{C}[x_1, \ldots, x_n]$ and $\mathbb{C}_{q}[x_1, \ldots, x_n]$

Let $A : = \mathbb{C}[x_1, \ldots, x_n],$ $A_q : =\mathbb{C}_q[x_1, \ldots, x_n] = \mathbb{C} \langle x_1, \ldots, x_n \rangle / (x_ix_j = q x_jx_i)$. By Koszul resolution I mean $$\ldots \to A \...
Invincible's user avatar