# Abstract Nonsense

## 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.