Homological Algebra

Prerequisite path Preparation & skip guidance · Abstract Algebra & Galois Theory · Linear Algebra

Studying Homological Algebra

Homology measures cycles left over after boundaries have been removed, but mistakes often begin before that quotient is formed. Baker keeps complexes, connecting maps and homotopies close together. The homological sections of Etingof and collaborators give a continuation into resolutions and explicit Ext and Tor calculations. The Stacks Project supplies a detailed tagged reference when a construction's hypotheses need checking. Start with a connected argument, then choose calculations or reference work according to the gap.

What to work on

Compute homology from kernels and images in an explicit complex.
Construct a connecting map for a short exact sequence of complexes.
Build a free resolution of a cyclic module.
Compute first Ext and Tor groups from that resolution.

A starting text

Notes on Basic Homological Algebra — Andrew Baker. Read sections 1-3: Chain complexes and their homology; Short exact sequences of chain complexes; Chain homotopies and quasi-isomorphisms. In section 2 work through the connecting-map construction before proceeding to resolutions in the alternative text.

Choosing what to read

Compute a small complex of abelian groups, checking that consecutive differentials compose to zero. Construct a connecting map by lifting an element and prove independence from the chosen lift. Draw each square when comparing complexes, and distinguish a chain homotopy equivalence from a quasi-isomorphism. Only after those operations are reliable should resolutions become the main object of study. Keep module calculations alongside the categorical language so an exact sequence continues to describe actual kernels and images.

Notes on Basic Homological Algebra

Andrew Baker

Learn connecting-map constructions before resolutions and derived functors.

How it reads: Connected lecture-note introduction through explicit complexes and homotopies.
What it expects: Linear algebra, groups and modules; kernels and cokernels.
What to keep in mind: The opening three sections do not constitute a full Ext, Tor or derived-category course.
Practice: Companion practice at https://math.mit.edu/~etingof/reprbook.pdf: Exercises provided; public solutions not verified. Some problems contain hints. Exercise 8.2.2 on existence of projective resolutions; Problem 8.2.5 on independence of resolution; Problem 8.2.7 on finitely generated modules over Z and k[x].

Complete public PDF.

Edition, reading range and source

Read sections 1-3: Chain complexes and their homology; Short exact sequences of chain complexes; Chain homotopies and quasi-isomorphisms. In section 2 work through the connecting-map construction before proceeding to resolutions in the alternative text.

Source · Checked 2026-10-16

Introduction to Representation Theory

Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob and Elena Yudovina

Continue from Baker's complexes to projective resolutions and explicit Tor and Ext.

How it reads: A focused homological-algebra selection within a representation-theory text.
What it expects: Modules, complexes and exact sequences; the representation chapters are not prerequisites for this selection.
What to keep in mind: This selection does not cover derived categories or spectral sequences.
Practice: Exercise 8.2.2 and Problems 8.2.5 and 8.2.7; existing 2026-10-16 evidence verifies exercises and some hints, not public solutions.

Complete MIT author PDF.

Edition, reading range and source

Sections 7.8-7.9 on complexes and exact functors; 8.1-8.2 on projective modules, resolutions, Tor and Ext, printed pages 205-211 for chapter 8. Range inherited from course-study-guides.json.

Source · Checked 2026-10-16

The Stacks Project — Homological Algebra

The Stacks Project Authors

Check exact hypotheses or compare a construction after attempting it in Baker.

How it reads: Detailed tagged reference with separate homology and derived-category chapters.
What it expects: Basic homological algebra and category theory.
What to keep in mind: Its reference organization gives less guidance about what to learn next.
Practice: Use a tagged statement to reconstruct a proof or compute a module example; these are editorial tasks, not a verified public exercise-and-solution set.

Full online reference with an index and table of contents.

Edition, reading range and source

Part 1, chapter 12, Homological Algebra; chapter 13, Derived Categories, is a later continuation. Chapter names verified in the official contents; individual tags were not checked.

Source · Checked 2026-10-17

Other ways into the subject

Introduction to Representation Theory — Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob and Elena Yudovina. Sections 7.8-7.9 on complexes and exact functors; 8.1-8.2 on projective modules, resolutions, Tor and Ext, printed pages 205-211 for chapter 8. Continue from complexes to explicit derived-functor calculations; representation theory chapters are not prerequisites for these algebra sections.

Practice

Resolutions and Ext/Tor problems. Exercise 8.2.2 on existence of projective resolutions; Problem 8.2.5 on independence of resolution; Problem 8.2.7 on finitely generated modules over Z and k[x].. Exercises provided; public solutions not verified. Some problems contain hints.

Try these before moving on

Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.

Resolve Z/4Z by free Z-modules and compute Tor_1 over Z of (Z/4Z, Z/6Z) and Ext^1 over Z of (Z/4Z, Z/6Z), showing the kernel or cokernel used in each case.
A hint
Use multiplication by 4 in a length-one free resolution. Tensor gives a kernel; applying Hom gives a cokernel.
Work through the solution
The sequence $0\to\mathbb Z\xrightarrow{\times4}\mathbb Z\to\mathbb Z/4\mathbb Z\to0$ is exact: multiplication by $4$ is injective and its image is the kernel of reduction modulo $4$ . It is a free resolution. Set $M=\mathbb Z/6\mathbb Z$ . Tensoring the resolution with $M$ gives the two-term chain complex $M\xrightarrow{\times4}M$ in degrees $1,0$ . Thus

$\operatorname{Tor}_1^{\mathbb Z}(\mathbb Z/4\mathbb Z,M)=\ker(4:M\to M)=\{\bar0,\bar3\}\cong\mathbb Z/2\mathbb Z.$

Applying $\operatorname{Hom}_{\mathbb Z}(-,M)$ gives the cochain complex $M\xrightarrow{\times4}M$ in degrees $0,1$ . Its degree-one cohomology is $M/4M$ . Here $4M=\{\bar0,\bar2,\bar4\}$ , so $\operatorname{Ext}^1_{\mathbb Z}(\mathbb Z/4\mathbb Z,M)\cong\mathbb Z/2\mathbb Z$ . The two answers are isomorphic, but arise from different operations: a kernel and a cokernel.

If this was the sticking point:
- Abstract Algebra & Galois Theory — Review cyclic modules, exactness and quotient groups.
- Linear Algebra — Rehearse kernel/image computations before replacing vector spaces by modules.

Already comfortable with this? Skip the complex review if you can construct and check the connecting homomorphism, including independence of representatives. Skip the resolution selection only if you can compute both Ext and Tor in the cyclic example and justify independence of the chosen resolution.

Review the prerequisites · Compare the full subject collection

Where this can lead

Category Theory — Clarify naturality and universal constructions used in derived-functor arguments.
Algebraic Topology & Homotopy Theory — Apply chain complexes and connecting maps to homology computations of spaces.

Reading-source dates

Notes on Basic Homological Algebra · 2026-10-16
Introduction to Representation Theory · 2026-10-16

Reading choices revised 2026-10-17.

Suggest an edit · Report a correction

Notes on Basic Homological Algebra

PDF

Andrew Baker · Short notes (15 pages) · Introductory

A compact entry into homological methods for readers already comfortable with modules. Useful as course notes; less expansive than a comprehensive graduate textbook.

Preparation: Rings, modules and exact sequences.

Open official PDF About / terms

The Stacks Project — Homological Algebra

Online text

The Stacks Project Authors · Reference work · Advanced

A detailed searchable reference with stable theorem tags. Use the Homology, Derived Categories and related chapters; it is not a gentle sequential first course.

Preparation: Homological algebra and category theory.

Read full reference

Further print reading (1 retained references)

From the original catalogue. These are bibliographic references, not free full-text links; an authorized complete free edition is not currently linked here.

同调代数基础 — 熊锐 (Rui Xiong)

Additional reading

Search the complete catalogue.

Graduate reading

Lectures on Cyclic Homology

Dale Husemoller · Long lecture notes · Graduate · English · PDF

Develops cyclic and Hochschild homology. Historical notation may differ from current treatments. Individual-use access only; do not redistribute.

Preparation: Homological algebra; associative algebras.

Use: Bridge to noncommutative invariants.

Open full PDF Source / terms

Edition and sources

Version: TIFR volume 83; retypeset digital edition (see PDF imprint for publication year) · Checked 2026-10-11

Introduction to A-infinity Algebras and Modules

Bernhard Keller · Survey · Graduate · English · PDF

Introduces A-infinity structures, modules and their derived categories through a compact exposition.

Preparation: Homological algebra; differential graded algebras.

Use: Bridge classical derived categories and higher algebra.

Open full PDF Source / terms

Edition and sources

Version: Expanded version, January 2001; 30 pages · Checked 2026-10-11

Cohomology of Groups

Kenneth S. Brown · Textbook · Graduate · English · Purchase / institutional access

The complete published book is subscription or purchase content. A preview or sample is not the complete book; institutional access depends on your library licence.

Connects group actions and homological constructions through group cohomology.

Preparation: Groups; modules; basic algebraic topology.

Use: Build tools used in topology and geometric group theory.

Publisher / access page

Edition and sources

Version: 1982 · Checked 2026-10-11

Methods of Homological Algebra

Sergei I. Gelfand and Yuri I. Manin · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Develops classical and derived methods together. Use its categorical perspective after practicing chain complexes and derived functors concretely.

Preparation: Algebra; modules; category theory.

Official page / access details

Edition and sources

Version: Second edition, 2003 · Checked 2026-10-11

Homology

Saunders Mac Lane · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A classical homological-algebra reference for complexes, resolutions and derived functors. Compare historical terminology with modern treatments.

Preparation: Rings, modules and elementary category theory.

Official page / access details

Edition and sources

Version: 1995 reprint; original 1963 · Checked 2026-10-11

An Introduction to Homological Algebra

Joseph J. Rotman · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

An extensive course with examples and connections to commutative algebra and topology. Distinct from Weibel’s book with the same title.

Preparation: Abstract algebra, modules and category language.

Official page / access details

Edition and sources

Version: Second edition, 2009 · Checked 2026-10-11

An Introduction to Homological Algebra

Charles A. Weibel · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A course and reference for derived functors, spectral sequences and homological constructions. Start with concrete resolutions before the more specialized applications.

Preparation: Rings, modules, chain complexes and basic category theory.

Official page / access details

Edition and sources

Version: 1994 · Checked 2026-10-11

Computational Homology

Tomasz Kaczynski, Konstantin Mischaikow and Marian Mrozek · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Studies effective homology computations and their mathematical foundations. A computational branch distinct from a general derived-category course.

Preparation: Linear algebra, basic topology and chain complexes.

Official page / access details

Edition and sources

Version: 2004 · Checked 2026-10-11

Henri Cartan and Samuel Eilenberg · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A foundational classical reference for resolutions and derived functors. Historical exposition is useful alongside, rather than in place of, a modern course.

Preparation: Graduate algebra; modules and complexes.

Official page / access details

Edition and sources

Version: Original 1956; Princeton reprint 1999 · Checked 2026-10-11

MATH 8030 Introduction to Homological Algebra

Pete L. Clark · Course notes and exercises · Graduate · English · PDF

Connects tensor and Hom functors, module classes, chain complexes and derived functors.

Preparation: Rings, modules and tensor products.

Use: Work through a first homological algebra course with exercises.

Open full PDF Source / terms

Edition and sources

Version: Author-hosted notes; date not stated · Checked 2026-10-13

Algebra + Homotopy = Operad

Bruno Vallette · Expository survey · Graduate · English · PDF

Complete author-deposited PDF opens on arXiv.

Explains operads, operadic syzygies and the homotopy transfer theorem through algebraic examples.

Preparation: Tensor products, chain complexes and categories.

Use: Bridge elementary homological algebra and operadic methods.

Open full PDF Source / terms

Edition and sources

Version: arXiv v1, February 15, 2012 · Checked 2026-10-13

Homotopical and Homological Algebra

卜辰璟 · Course notes · Graduate · Chinese · Online text

Online lecture text; chapters can be read separately.

Seminar notes motivated by similarities between algebraic topology and homological algebra, leading toward infinity-categorical language and localization.

Preparation: Categories, chain complexes and some algebraic topology.

Use: Use it alongside a homological-algebra course to understand why ordinary categorical constructions need refinement.

Open online text / archive

Edition and sources

Original title: 同伦代数与同调代数

Version: Fall 2019 seminar · Checked 2026-11-03

Research reading

Lectures on Topics in Algebraic K-Theory

Hyman Bass · Long lecture notes · Research · English · PDF

Complete institutional PDF, authorized for individual use only. Copyright remains with TIFR; retypesetting may retain errors. This is not a redistribution licence.

A classical route to algebraic K-theory from rings and projective modules. Compare with modern higher-K-theory references before using later categorical constructions.

Preparation: Rings; projective modules; homological algebra.

Open full PDF Source / terms

Edition and sources

Version: TIFR Lectures on Mathematics 41; authorized retypeset historical edition — original date on title page · Checked 2026-10-11

Operads and PROPs

Martin Markl · Research survey · Research · English · PDF

Complete author-deposited PDF opens on arXiv.

Reviews operads, PROPs and their algebras, including structures with several inputs and outputs.

Preparation: Homological algebra, tensor categories and basic operads.

Use: Compare operadic and PROP formalisms before specialized research.

Open full PDF Source / terms

Edition and sources

Version: arXiv v3, February 11, 2006 · Checked 2026-10-13

On Differential Graded Categories

Bernhard Keller · Research survey · Research · English · PDF

Complete author-deposited PDF opens on arXiv.

Surveys differential graded enhancements, derived Morita theory and invariants beyond triangulated categories.

Preparation: Derived categories and homological algebra.

Use: Enter the language of dg categories and derived invariants.

Open full PDF Source / terms

Edition and sources

Version: arXiv v5, June 19, 2006 · Checked 2026-10-13

Maurer–Cartan Methods in Deformation Theory: The Twisting Procedure

Vladimir Dotsenko, Sergey Shadrin and Bruno Vallette · Author manuscript · Research · English · PDF

Full author manuscript on arXiv; front matter permits personal viewing and download.

Develops deformation and gauge methods for dg Lie algebras, homotopy algebras and operads, including twisting constructions.

Preparation: Homological algebra, Lie algebras and basic operads.

Use: Study deformation theory through a systematic account of twisting.

Open full PDF Source / terms

Edition and sources

Version: arXiv v2, February 23, 2023 · Checked 2026-10-13

Condensed Mathematics: Seminar Notes

唐珑珂、李一笑 · Course notes · Research · Chinese · Online text

Online lecture text; chapters can be read separately.

Chinese seminar notes on condensed mathematics and analytic rings, with corrections and comparisons to the cited versions of the original lecture notes.

Preparation: Sheaves, homological algebra and higher categories.

Use: Read after the basic sheaf and derived-category language; use the original references for version comparisons.

Open online text / archive

Edition and sources

Original title: 凝聚态数学

Version: Spring 2021 seminar · Checked 2026-11-03

K-Theory and Topological Cyclic Homology of Henselian Pairs

Dustin Clausen, Akhil Mathew and Matthew Morrow · Research paper · Research · English · PDF

Public full text; checked against the title and contents.

Relates relative K-theory and topological cyclic homology with finite coefficients for henselian pairs.

Preparation: Algebraic K-theory; spectra; topological cyclic homology.

Use: Study the finite-coefficient TC input before the henselian-pair comparison.

Open full PDF Source / terms

Found via — References: $K$-Theory and Topological Cyclic Homology of Henselian Pairs

Edition and sources

Version: arXiv 1803.10897; inspected electronic version · Checked 2026-11-03