Abstract Nonsense

Crushing one theorem at a time

Chain Homotopy


Point of Post: In this post we discuss the notion of chain homotopic chain complexes and prove that they give isomorphic homology objects.

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Motivation

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We know that if we are given two chain complexes \mathbf{C} and \mathbf{D} and a chain map f=\{f_n\} this induces an arrow H_n(f):H_n(\mathbf{C})\to H_n(\mathbf{D}). A fairly obvious question then is when two different chain maps f,g:\mathbf{C}\to\mathbf{D} induce the same map on homology objects. This leads us naturally to the notion of a chain homotopy between which roughly states that f and g differ by something that trivially gets sent to zero.

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Chain Homotopy

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Let \mathbf{C} and \mathbf{D} be two chain complexes. A sequence \{f_n\} of maps with C_n\xrightarrow{f_n}D_{n+p} is called a map of degree p from \mathbf{C} to \mathbf{D}For example, a chain map is an example of a degree zero map \mathbf{C}\to\mathbf{D}.

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We say that two chain maps f,g:\mathbf{C}\to\mathbf{D} are homotopic (which we shall denote f\simeq g) if there exists a map s:\mathbf{C}\to\mathbf{D} of degree one such that f_n-g_n=d_{n+1}\circ s_n+s_{n-1}\circ\partial_n.

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The important theorem about homotopic maps is the following:

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Theorem: Let f,g:\mathbf{C}\to\mathbf{D} be two chain maps. Then, if f\simeq g then H_n(f)=H_n(g) for all n\in\mathbb{Z}.

Proof: It suffices to prove this in R\text{-}\mathbf{mod}. Note then that an element of H_n(\mathbf{C}) is an element of \ker\partial_{n-1}/\text{im }\partial_n and so looks like x+\text{im }\partial_n. We see then that

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(H_n(f)-H_n(g))(x+\text{im }\partial_n)=s_{n-1}(\partial_n(x))+d_n(s_{n}(x))+\text{im }d_n=0

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since \partial_n(x)=0 (since x\in\ker\partial_n) and d_n(s_n(x))\in\text{im }d_n. \blacksquare

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References:

[1] Weibel, Charles A. An Introduction to Homological Algebra. Cambridge [England: Cambridge UP, 1994. Print.

[2] Schapira, Pierre. “Categories and Homological Algebra.” Web. <http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf&gt;.

[3] Rotman, Joseph. An Introduction to Homological Algebra. Dordrecht: Springer, 2008. Print.

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April 29, 2012 - Posted by | Algebra, Homological Algebra | , ,

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