Abstract Nonsense

Crushing one theorem at a time

Exact Sequences and Homology (Pt. III)

Point of Post: This is a continuation of this post.

Homology Objects

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As we have stated before, there is absolutely no reason to believe that a given chain complex is going to be exact and so we’d like to have a measure of how far a chain complex is from being exact. So, how exactly should we go about doing this? Let’s start off with a slightly easier problem by figuring out we can tell if a chain \mathbf{C}  is exact at a given term, say at C_n.

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Well, by the mere fact that C_{n+1}\xrightarrow{\partial_{n+1}}C_n\xrightarrow{\partial_n}C_{n-1} is exact gives an exact sequence

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0\to \text{im }\partial_{n+1}\to \ker \partial_n\quad\mathbf{(1)}

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The question is, when is this an isomorphism? The key to when this is an isomorphism lies within a very interesting property of abelian categories. Let us call a category balanced if every monic and epi arrow is an isomorphism. For example, R-\text{-}\mathbf{Mod} is balanced for every ring R since if an R-map M\to N is both a mono and an epi (i.e. injective and surjective) then there exists an inverse R-map N\to M, and thus the map M\to N is an isomorphism. Well, as one can probably guess from the buildup, every abelian category is balanced. While there are more elementary ways to see this, it follows quite immediately from the metaprinciple afforded to us by the embedding theorem.

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Thus, armed with the fact that abelian categories are balanced, we can safely say that the natural map in \mathbf{(1)} is going to be an isomorphism if and only if it is an epimorphism. But, we know from the basic theory that we have the exact sequencece

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0\to\text{im }\partial_{n+1}\to\ker \partial_n\to \text{coker}(\text{im }\partial_{n+1}\to\ker\partial_n)\to 0

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and, in fact, we know that \text{im }\partial_{n+1}\to\ker\partial_n will be an epimorphism if and only if we have the following equality: \text{coker}(\text{im }\partial_{n+1}\to\ker\partial_n)=0.

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For notational convenience let us denote \text{coker}(\text{im }\partial_{n+1}\to\ker\partial_n) as H_n(\mathbf{C}) and call it the n^{\text{th}} homology object of \mathbf{C}. Thus, we see that \mathbf{C} is exact at C_n if and only if H_n(\mathbf{C}) is zero, and thus \mathbf{C} is exact if H_n(\mathbf{C}) is zero for all n\in\mathbb{Z}.

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These homology objects are precisely the measures of how far a given chain complex is from being exact.

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If our category \mathscr{A} admits abritrary coproducts (as is often assumed), such as in R\text{-}\mathbf{Mod} for some ring R, we can define the total homology object H(\mathbf{C}) of \mathbf{C} to be \displaystyle \bigoplus_{n\in\mathbb{Z}}H_n(\mathbf{C}). One can then see that \mathbf{C} is exact if and only if H(\mathbf{C}) is zero.

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For modules there is a simple description of H_n(\mathbf{C}). Indeed, we know that in this case \text{im }\partial_{n+1} is just the honest to God image of \partial_{n+1} sitting inside C_n and that \ker\partial_n is the similarly honest literal kernel, also sitting inside C_n. We also know then (since \mathbf{C} is a chain complex) that \text{im }\partial_{n+1}\subseteq\ker\partial_n and that the map \text{im }\partial_{n+1}\to\ker\partial_n is just the inclusion. We see then that the cokernel \text{coker}(\text{im }\partial_{n+1}\to\ker\partial_n) is nothing more than the quotient module \ker\partial_n/\text{im }\partial_{n+1}.

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Our first order of business is to prove that n^{\text{th}} homology is actually an additive functor \mathbf{Ch}(\mathscr{A})\to\mathscr{A}. To do this though we need to specify what the action of H_n should be on chain maps between complexes. To do this let \mathbf{C}\xrightarrow{\{f_n\}}\mathbf{D} be a chain map.

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Let’s start with the case when everything is in R\text{-}\mathbf{Mod} and see if we can extrapolate this to the general case. So,  to prove we get a well-defined map H_n(\mathbf{C})\to H_n(\mathbf{D}) we need to show that we can get a well-defined map \ker\partial_n\to\ker d_n and that the map sends \text{im }\partial_{n+1} to \text{im }d_{n+1}. Of course we get a map \ker\partial_n\to D_n just by restricting f_n to \ker\partial_n. Thus, the first step is to check that f_n(\ker\partial_n)\subseteq\ker d_n. To see this we merely note that for each x\in\text{im }\partial_{n+1}, d_n(f_n(x))=f_{n-1}(\partial_n(x))=0 where the last step is because \text{im }\partial_{n+1}\subseteq\ker\partial_n. Thus, we see that f_n restricts to a well-defined map \ker\partial_n\to\ker d_n and so evidently, by composing with the canonical projection, a map \ker\partial_n\to H_n(\mathbf{D}). To show that this descends to a well-defined map H_n(\mathbf{C})\to H_n(\mathbf{D}) what we must show is that f_n sends \text{im }\partial_{n+1} to zero, or equivalently that f_n(\text{im }\partial_{n+1})\subseteq \text{im }d_{n+1}. Once again, the fact that f_n is a chain map saves us since f_n(\partial_{n+1}(x))=d_{n+1}(f_{n+1}(x)) from where the desired containment follows. Thus, f_n restricts and then descends to a well-defined map H_n(\mathbf{C})\to H_n(\mathbf{D}) which takes x+\text{im }\partial_{n+1} to f_n(x)+\text{im }d_{n+1}.

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[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.


April 10, 2012 - Posted by | Algebra, Homological Algebra | , , , , , ,


  1. […] Exact Sequences and Homology (Pt. IV) Point of Post: This is a continuation of this post. […]

    Pingback by Exact Sequences and Homology (Pt. IV) « Abstract Nonsense | April 10, 2012 | Reply

  2. […] is a simple example of why in general categories aren’t balanced. Indeed, the monos and epis in are the injective and surjective continuous maps. Thus, we see […]

    Pingback by The Exponential and Trigonometric Functions (Pt. II) « Abstract Nonsense | May 5, 2012 | Reply

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