Compute it, don't copy it
Resolutions, Ext modules, Betti tables and group cohomology rings are all computed rather than looked up, and the systems that compute them are actively maintained and versioned. This page records where the computations are done and which databases are authoritative. Consistent with the policy applied across this library, numerical and tabular results are not transcribed here — they carry a version dependency that a static page cannot honour.
Learning objectives
- Identify the appropriate system for a given computation.
- Locate the authoritative databases for group cohomology.
- Understand why this collection carries method rather than tables.
- Record the version and parameters needed to reproduce a computation.
Section 01Systems by task
| Task | System | Note |
|---|---|---|
| Free resolutions, Betti tables, Ext and Tor over polynomial rings | Macaulay2, Singular | Gröbner basis engines; graded Betti numbers are the standard output |
| Group cohomology rings of finite groups | GAP with the HAP package, Magma | Minimal resolutions over group algebras; ring structure and restriction maps |
| General ring-theoretic homological computation | Magma, Singular | Broad coverage including non-commutative cases |
| Lie algebra cohomology | GAP, LiE, custom Chevalley–Eilenberg code | Finite complexes make direct computation feasible |
| Simplicial and topological homology | GAP/HAP, CHomP, Sage | Chain complexes from cell structures |
| Spectral sequence bookkeeping | Sage, custom code | Differentials generally require human input |
| Formal verification of homological arguments | Lean with mathlib | Abelian categories, derived functors and Ext are formalised |
A Betti table depends on the base field, the characteristic, the term order and the software version. A computation quoted without those is not reproducible, and differences between characteristics are frequently the whole point. Record all four alongside any result.
Section 02Databases
Group cohomology rings
Computed cohomology rings for the finite groups of small order are maintained as structured datasets alongside the systems that generated them. Consult the current release rather than a printed table.
Small groups library
The classification of groups of small order, distributed with GAP and Magma, underpins any systematic cohomology computation.
Betti tables and resolutions
Macaulay2 and Singular ship example libraries; published Betti tables should be regenerated rather than copied.
Section 03The sourcing policy
This collection carries the durable method layer: definitions, constructions, theorems, proof strategies and the reasoning that makes a computation correct. It deliberately does not reproduce computed tables — Betti numbers, cohomology ring presentations, resolution ranks — from any source.
- Stage 01ReproducibilityA computed table is valid only for a stated base field, characteristic and software version. A static page cannot carry that dependency reliably.
- Stage 02CurrencyDatabases are revised as algorithms improve and errors are found. A transcribed table ages silently, with no signal that it has become wrong.
- Stage 03IntegrityTranscription of dense numerical data introduces substitution errors that look plausible and are not detectable from the page itself.
- Stage 04ConsequenceMethod here; numbers from the live system, with the parameters recorded.
Durable engineering and mathematical method belongs in the knowledge base. Numeric catalogue data belongs at its authoritative source, cited with the version that produced it. The same policy governs the Computational Algebraic Number Theory collection in this library.
ReferenceFrequently asked questions
Which system should I start with?
For commutative algebra and resolutions over polynomial rings, Macaulay2. For finite group cohomology, GAP with the HAP package. Both are freely available and widely used, so results are easy to have checked by others.
Why not include a table of small Ext computations?
Because the ones worth memorising are already derived in the relevant pages of this collection — Ext over ℤ between cyclic groups, for instance — and anything larger is version-dependent computed data that belongs at its source.
How should a computation be cited?
Record the system, the version, the base field and characteristic, the term order where relevant, and the exact input. That is enough for another person to reproduce it, which a transcribed result is not.
NavigateContinue in this stream
Curated next steps from this page. The site also surfaces algorithmically related reading below.
ProvenanceSources and further reading
This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.