Section 1.1

Subsection Chain Complexes and Short Exact Sequences

Homological algebra first appeared in the study of topological spaces. Roughly speaking, homology is a way of associating a sequence of abelian groups (or modules, or other more sophisticated algebraic objects) to another object, for example a topological space. The homology of a topological space encodes topological information about the space in algebraic language - this is what algebraic topology is all about.

More formally, we will study complexes and their homology from a more abstract perspective. While algebraic topologists are often concerned with complexes of abelian groups, we will work a bit more generally with complexes of \(R\)-modules. The basic assumptions and notation about rings and modules we will use in this class can be found in Appendix A. As an appetizer, we begin with some basic homological algebra definitions.

Definition 1.1. Chain Complex.

A chain complex of \(R\)-modules \(\left(C_{\bullet}, \partial_{\bullet}\right)\text{,}\) also referred to simply as a complex, is a sequence of \(R\)-modules \(C_{i}\) and \(R\)-module homomorphisms

\begin{equation*} \cdots \longrightarrow C_{n+1} \stackrel{\partial_{n+1}}{\longrightarrow} C_{n} \stackrel{\partial_{n}}{\longrightarrow} C_{n-1} \longrightarrow \cdots \end{equation*}

such that \(\partial_{n} \partial_{n+1}=0\) for all \(n\text{.}\) The maps \(\partial_{n}\) are the differentials of our complex.

Convention 1.2.

We may sometimes omit the differentials \(\partial_{n}\) and simply refer to the complex \(C_{\bullet}\) or even \(C\text{;}\) we may also sometimes refer to \(\partial_{\bullet}\) as the differential of \(C_{\bullet}\text{.}\)

Definition 1.3. Exact Sequences.

The complex \(\left(C_{\bullet}, \partial_{\bullet}\right)\) is exact at \(n\) if im \(\partial_{n+1}=\operatorname{ker} \partial_{n}\text{.}\) An exact sequence is a complex that is exact everywhere. More precisely, an exact sequence of \(R\)-modules is a sequence

\begin{equation*} \cdots \stackrel{f_{n-1}}{\longrightarrow} C_{n} \stackrel{f_{n}}{\longrightarrow} C_{n+1} \stackrel{f_{n+1}}{\longrightarrow} \cdots \end{equation*}

of \(R\)-modules and \(R\)-module homomorphisms such that \(\operatorname{im} f_{n}=\operatorname{ker} f_{n+1}\) for all \(n\text{.}\) An exact sequence of the form

\begin{equation*} 0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0 \end{equation*}

is a short exact sequence, sometimes written ses.

Exercise 1.4.

The condition that \(\partial_{n} \partial_{n+1}=0\) for all \(n\) implies that \(\im \partial_{n+1} \subseteq \operatorname{ker} \partial_{n}\text{.}\)

Theorem 1.5.

The sequence

\begin{equation*} 0 \longrightarrow M \stackrel{f}{\longrightarrow} N \end{equation*}

is exact if and only if \(f\) is injective. Similarly,

\begin{equation*} M \stackrel{f}{\longrightarrow} N \longrightarrow 0 \end{equation*}

is exact if and only if \(f\) is surjective. So

\begin{equation*} 0 \longrightarrow A \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C \longrightarrow 0 \end{equation*}

is a short exact sequence if and only if

\(f\) is injective
\(g\) is surjective
\(\operatorname{im} f=\operatorname{ker} g\text{.}\)

When this is indeed a short exact sequence, we can identify \(A\) with its image \(f(A)\text{,}\) and \(A=\operatorname{ker} g\text{.}\) Moreover, since \(g\) is surjective, by the First Isomorphism Theorem we conclude that \(C \cong B / f(A)\text{,}\) so we might abuse notation and identify \(C\) with \(B / A\text{.}\)

Convention 1.6.

We write \(A \rightarrow B\) to denote a surjective map, and \(A \hookrightarrow B\) to denote an injective map.

Definition 1.7. Cokernel.

The cokernel of a map of \(R\)-modules \(A \stackrel{f}{\rightarrow} B\) is the module

\begin{equation*} \text { coker } f:=B / \operatorname{im}(f). \end{equation*}

Exercise 1.8.

We can rephrase Theorem 1.5 in a fancier language: if

\begin{equation*} 0 \longrightarrow A \stackrel{f}{\longrightarrow} B \stackrel{g}{\longrightarrow} C \longrightarrow 0 \end{equation*}

is a short exact sequence, then \(A=\operatorname{ker} g\) and \(C=\operatorname{coker} f\text{.}\)

Example 1.9.

Let \(\pi\) be the canonical projection \(\mathbb{Z} \longrightarrow \mathbb{Z} / 2 \mathbb{Z}\text{.}\) The following is a short exact sequence:

\begin{equation*} 0 \longrightarrow \mathbb{Z} \stackrel{2}{\longrightarrow} \mathbb{Z} \stackrel{\pi}{\longrightarrow} \mathbb{Z} / 2 \mathbb{Z} \longrightarrow 0 \end{equation*}

Example 1.10.

Let \(R=k[x]\) be a polynomial ring over the field \(k\text{.}\) The following is a short exact sequence:

\begin{equation*} 0 \longrightarrow R \stackrel{\cdot x}{\longrightarrow} R \stackrel{\pi}{\longrightarrow} R /\igen x \longrightarrow 0 \text {. } \end{equation*}

The first map is multiplication by \(x\text{,}\) and the second map is the canonical projection.

Example 1.11.

Given an ideal \(I\) in a ring \(R\text{,}\) the inclusion map \(\iota: I \rightarrow R\) and the canonical projection \(\pi: R \rightarrow R / I\) give us the following short exact sequence:

\begin{equation*} 0 \longrightarrow I \stackrel{\iota}{\longrightarrow} R \stackrel{\pi}{\longrightarrow} R / I \longrightarrow 0 \end{equation*}

Example 1.12.

Let \(R=k[x] /\igen x\text{.}\) The following complex is exact:

\begin{equation*} \cdots \longrightarrow R \stackrel{\cdot x}{\longrightarrow} R \stackrel{\cdot x}{\longrightarrow} R \longrightarrow \cdots \end{equation*}

Indeed, the image and the kernel of multiplication by \(x\) are both \(\igen x\text{.}\)

Sometimes we can show that certain modules vanish or compute them explicitly when they do not vanish by seeing that they fit in some naturally constructed exact sequence involving other modules we understand better. We will discuss this in more detail when we talk about long exact sequences.

Remark 1.13.

The complex \(0 \longrightarrow M \stackrel{f}{\longrightarrow} N \longrightarrow 0\) is exact if and only if \(f\) is an isomorphism.

Remark 1.14.

The complex \(0 \longrightarrow M \longrightarrow 0\) is exact if and only if \(M=0\text{.}\)

Subsection Homology

Historically, chain complexes first appeared in topology. To study a topological space, one constructs a particular chain complex that arises naturally from information from the space, and then calculates its homology, which ends up encoding important topological information in the form of a sequence of abelian groups.

Definition 1.15. Homology.

The homology of the complex \(\left(C_{\bullet}, \partial_{\bullet}\right)\) is the sequence of \(R\)-modules

\begin{equation*} \mathrm{H}_{n}\left(C_{\bullet}\right)=\mathrm{H}_{n}(C):=\frac{\operatorname{ker} \partial_{n}}{\operatorname{im} \partial_{n+1}} . \end{equation*}

The \(n\)th homology of \(\left(C_{\bullet}, \partial_{\bullet}\right)\) is \(\mathrm{H}_{n}(C)\text{.}\)

Convention 1.16.

The submodules \(Z_{n}\left(C_{\bullet}\right)=Z_{n}(C):=\operatorname{ker} \partial_{n} \subseteq C_{n}\) are sometimes called cycles, while the submodules \(B_{n}\left(C_{\bullet}\right)=B_{n}(C):=\operatorname{im} \partial_{n+1} \subseteq C_{n}\) are sometimes called boundaries. One sometimes uses the word boundary to refer an element of \(B_{n}(C)\) (an \(n\)-boundary), and the word cycle to refer to an element of \(Z_{n}(C)\) (an \(n\)-cycle).

Remark 1.17.

The homology of a complex measures how far our complex is from being exact at each point. Again, we can talk about the cohomology of a cochain complex instead, which we write as \(\mathrm{H}^{n}(C)\text{;}\) we will for now not worry about the distinction.

Exercise 1.18.

Note that \(\left(C_{\bullet}, \partial_{\bullet}\right)\) is exact at \(n\) if and only if \(\mathrm{H}_{n}\left(C_{\bullet}\right)=0\text{.}\)

Example 1.19.

Let \(R=k[x] /\igen{x^3}\text{.}\) Consider the following complex:

\begin{equation*} F_{\bullet}=\cdots \longrightarrow R \stackrel{\cdot x^{2}}{\longrightarrow} R \stackrel{\cdot x^{2}}{\longrightarrow} R \longrightarrow \cdots . \end{equation*}

The image of multiplication by \(x^{2}\) is \(\igen{x^2}\text{,}\) while the the kernel of multiplication by \(x^{2}\) is \(\igen{x} \supseteq\igen{x^2}\text{.}\) For all \(n\text{,}\)

\begin{equation*} \mathrm{H}_{n}\left(F_{\bullet}\right)=\igen{x} /\igen{x^2} \cong R /\igen{x} \end{equation*}

Example 1.20.

Let \(\mathbb{Z} \stackrel{\pi}{\longrightarrow} \mathbb{Z} / 2 \mathbb{Z}\) be the canonical projection map. Then

is a complex of abelian groups, since the image of multiplication by \(4\) is \(4 \mathbb{Z}\text{,}\) and that is certainly contained in \(\operatorname{ker} \pi=2 \mathbb{Z}\text{.}\) The homology of \(C\) is

\begin{equation*} \begin{array}{ll} \mathrm{H}_{n}(C)=0 & \text { for } n \geqslant 3 \\ \mathrm{H}_{2}(C)=\frac{\operatorname{ker}(\mathbb{Z} \stackrel{4}{\rightarrow} \mathbb{Z})}{\operatorname{im}(0 \longrightarrow \mathbb{Z})}=\frac{0}{0}=0 & \\ \mathrm{H}_{1}(C)=\frac{\operatorname{ker}(\mathbb{Z} \stackrel{\pi}{\rightarrow} \mathbb{Z} / 2 \mathbb{Z})}{\operatorname{im}(\mathbb{Z} \stackrel{4}{\rightarrow} \mathbb{Z})}=\frac{2 \mathbb{Z}}{4 \mathbb{Z}} \cong \mathbb{Z} / 2 \mathbb{Z} & \\ \mathrm{H}_{0}(C)=\frac{\operatorname{ker}(\mathbb{Z} / 2 \mathbb{Z} \longrightarrow 0)}{\operatorname{im}(\mathbb{Z} \longrightarrow \mathbb{Z} / 2 \mathbb{Z})}=\frac{\mathbb{Z} / 2 \mathbb{Z}}{\mathbb{Z} / 2 \mathbb{Z}}=0 & \text { for } n<0 \\ \mathrm{H}_{n}(C)=0 & \end{array} \end{equation*}

Notice that our complex is exact at \(2\) and \(0\text{.}\) The exactness at \(2\) says that the map \(\mathbb{Z} \stackrel{4}{\rightarrow} \mathbb{Z}\) is injective, while exactness at \(0\) says that \(\pi\) is surjective.

Before we can continue any further into the world of homological algebra, we will need some categorical language. We will take a short break to introduce category theory, and then armed with that knowledge we will be ready to study homological algebra.

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