All Questions
Tagged with homological-algebra chain-complexes
59 questions
2
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1
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391
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Endomomorphisms of Chain Complexes of vector spaces and determinants
Let $C_{\ast} : \cdots \to A_{2} \to A_{1} \to A_{0} \to 0$ be a chain complex of finite dimensional vector spaces over a field $K$.
And let $f_{\ast} : C_{\ast} \to C_{\ast}$ and $g_{\ast} : C_{\ast} ...
CommunityBot
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7
votes
2
answers
1k
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Resolutions of unbounded complexes and homotopy (co)limits.
I want to understand once and for all what the resolution of an unbounded complex is. I've been trying to read 'Homotopy limits in triangulated categories' by Marcel Bokstedt and Amnon Neeman and can'...
- 9,229
3
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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 ...
- 9,019
7
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2
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On the difference between a projective chain complex and a level-wise projective chain complex
Let R be an associative ring with a unit, and consider the standard projective model structure of non-negatively graded (left) R-module, $Ch_R$. A map $f:M\to N$ in $Ch_R$ is a weak equivalence if it ...
CommunityBot
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9
votes
1
answer
887
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Left adjoint of totalization?
There is a functor from bicomplexes to chain complexes sending a bicomplex to its associated total chain complex. Does this functor have a left adjoint, and if so, what is it?
- 45.7k
1
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0
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241
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Two-point desuspension for augmented chain complexes?
Let $X$ be a chain complex augmented over $\mathbf{Z}$ with augmentation $\varepsilon_X:X_0 \to \mathbf{Z}$. We define $[1](X)$ to be the $\mathbf{Z}$-augmented chain complex such that $[1](X)_0 = \...
- 19.6k
12
votes
1
answer
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Explicit description of the "simplicial tensor product" of chain complexes
Recall that there is an equivalence of categories (Dold-Kan) $$N:\mathrm{s}\mathbf{Ab}\simeq \operatorname{Ch}_{\geq 0}(\mathbf{Ab}):\Gamma$$ between simplicial abelian groups and (connective) chain ...
- 52.6k
1
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0
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80
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Extending Reedy dimension to augmented chain complexes of abelian groups
Recall that a normal continuously-graded finite interval is given by a pair $a=([a],f)$ consisting of:
1.) A finite totally ordered set $[a]=[a_0< \dots < a_n]$
2.) A grading function $f:U[a] \...
- 11
4
votes
0
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730
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Why chain complexes? [duplicate]
Possible Duplicate:
Motivating the category of chain complexes
Chain complexes (of, say, abelian groups) are fundamental in homological algebra and algebraic geometry. For example, using the ...
CommunityBot
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