Homological Algebra I: Complexes and Resolutions introduces the basic language and methods used to study algebraic objects through chains of modules and maps between them. The course focuses on chain complexes, exactness, and the idea that many structural questions can be reformulated in terms of how kernels, images, and cokernels fit together inside diagrams. From there, it develops the standard diagram-chasing tools that make homological arguments effective, and it shows how homotopy and quasi-isomorphisms capture when two complexes should be regarded as equivalent from a homological point of view.

A central theme of the course is extracting information from exact sequences and using it to compare algebraic structures. The early chapters build the formal foundation: complexes, exactness, the snake lemma, the five lemma, and related results. The middle chapters then move to short exact sequences of complexes, chain homotopy, and mapping cones, which together explain how maps of complexes encode homological data. The final chapters turn to projective and injective modules, then to projective and injective resolutions, which provide systematic ways to compute and organize derived information.

The chapters are arranged to move from local manipulations of diagrams to global constructions of resolutions. First, students learn to work confidently with exact sequences and commutative diagrams; then they learn to compare complexes up to homotopy and quasi-isomorphism; finally, they apply these ideas to projective and injective objects, where resolutions become the main tool for homological computation. By the end of the course, the basic formalism of homological algebra is in place, preparing for derived functors and more advanced homological methods.

# Introduction

This opening chapter fixes the scope and expectations for the course. Homological algebra studies algebraic objects by replacing difficult questions about objects with more flexible questions about diagrams, complexes, and resolutions. The course starts from modules and exact sequences, then builds the language needed to discuss derived functors in a subsequent course without developing $\operatorname{Ext}$, $\operatorname{Tor}$, or spectral sequences here.

The central theme is that exactness records whether information is lost, created, or transported correctly through a sequence of homomorphisms. Chain complexes organise many exactness questions at once, while resolutions replace an arbitrary module by a long exact algebraic model assembled from projective or injective modules. The purpose of this course is to make these replacements precise and to develop the diagram techniques that make them usable.

## What Homological Algebra Measures

A first problem in algebra is that many natural constructions preserve only part of the information in a sequence. Tensoring, taking invariants, applying Hom, and passing to quotients often interact well with some maps but not with all exact sequences. Homological algebra begins by measuring this failure rather than hiding it.

[explanation: Exactness As Information Control] A sequence of module homomorphisms records a chain of algebraic operations. Exactness at one term says that everything killed by the outgoing map was already produced by the incoming map. Thus exactness is a bookkeeping device for distinguishing genuine new obstructions from elements that have already been accounted for.

This viewpoint appears throughout the course. Homology will measure the discrepancy between cycles and boundaries in a complex. Connecting homomorphisms will explain how a failure at one degree is transported into the next. Resolutions will replace a module by a complex whose failure of exactness is concentrated at a chosen location. [/explanation]

This explanation turns a familiar condition into a guiding question: if exactness fails, can we name the failure and move it through a diagram? To make that question precise, all kernels, images, quotients, and exact sequences must live in one category where homomorphisms have compatible domains and codomains. The course therefore fixes the side of the module action before any homological constructions begin.

[definition: Module Category Setting] Let $R$ be a ring with multiplicative identity. The course works in the category $R\text{-}\mathrm{Mod}$ of left $R$-modules and $R$-linear maps. [/definition]

For commutative rings the distinction between left and right modules is often suppressed, but in homological algebra it is useful to keep sidedness visible. Many constructions, especially tensor products and Hom functors, depend on whether the module action is on the left or on the right.

[example: A First Exact Sequence] Let $n \ge 2$. In the category of abelian groups, equivalently $\mathbb Z$-modules, define $i:n\mathbb Z\to \mathbb Z$ by $i(nk)=nk$ and define $q:\mathbb Z\to \mathbb Z/n\mathbb Z$ by $q(m)=m+n\mathbb Z$. We show that \begin{align*} 0 \longrightarrow n\mathbb Z \xrightarrow{i} \mathbb Z \xrightarrow{q} \mathbb Z/n\mathbb Z \longrightarrow 0 \end{align*} is exact by checking equality of image and kernel at each nonzero term. At $n\mathbb Z$, exactness means that $\ker i=\operatorname{im}(0\to n\mathbb Z)$. If $x\in n\mathbb Z$ and $i(x)=0$, then $x=0$ because $i$ is inclusion into $\mathbb Z$. Hence \begin{align*} \ker i=\{0\}=\operatorname{im}(0\to n\mathbb Z). \end{align*} At $\mathbb Z$, exactness means $\operatorname{im} i=\ker q$. The image of $i$ is \begin{align*} \operatorname{im} i=\{i(nk):k\in\mathbb Z\}=\{nk:k\in\mathbb Z\}=n\mathbb Z. \end{align*} For the kernel of $q$, \begin{align*} m\in\ker q &\Longleftrightarrow q(m)=0+n\mathbb Z\\ &\Longleftrightarrow m+n\mathbb Z=0+n\mathbb Z\\ &\Longleftrightarrow m\in n\mathbb Z. \end{align*} Thus $\ker q=n\mathbb Z=\operatorname{im} i$. At $\mathbb Z/n\mathbb Z$, exactness means that $q$ is surjective, since the next map is the zero map to $0$ and its kernel is all of $\mathbb Z/n\mathbb Z$. If $m+n\mathbb Z\in \mathbb Z/n\mathbb Z$, then \begin{align*} q(m)=m+n\mathbb Z, \end{align*} so every coset is in $\operatorname{im}q$. Therefore \begin{align*} \operatorname{im}q=\mathbb Z/n\mathbb Z=\ker(\mathbb Z/n\mathbb Z\to 0). \end{align*} The sequence is exact because, at each term, the elements killed by the outgoing map are exactly the elements produced by the incoming map. [/example]

This example is elementary, but it already contains the pattern used later: identify kernels, identify images, and compare them at each position. Diagram lemmas systematise this comparison when several such sequences interact.

## The Objects Built in the Course

The next problem is that a single exact sequence is too short to remember repeated algebraic structure. Many constructions naturally produce long strings of maps, and the condition that consecutive maps compose to zero is the minimal compatibility needed for homology.

[definition: Chain Complex] A chain complex $C_\bullet$ of $R$-modules is a sequence of $R$-modules and $R$-linear maps \begin{align*} \cdots \xrightarrow{d_{n+2}} C_{n+1} \xrightarrow{d_{n+1}} C_n \xrightarrow{d_n} C_{n-1} \xrightarrow{d_{n-1}} \cdots \end{align*} such that $d_n \circ d_{n+1} = 0$ for every $n \in \mathbb Z$. [/definition]

The maps $d_n$ are called differentials. The relation $d_n d_{n+1}=0$ ensures that every boundary is a cycle.

The remaining question is how much larger the cycle module is than the boundary module in each degree. That excess is exactly the obstruction to exactness at $C_n$, so it is recorded as a quotient rather than as a raw set of cycles.