The interesting case is when the characteristic divides the order
Maschke's theorem makes k[G] semisimple when the characteristic of k does not divide |G|, so every module is projective and all higher cohomology vanishes. When the characteristic does divide the order, the group algebra has infinite global dimension, modules have infinite resolutions, and the cohomology ring becomes a genuine geometric object. Support varieties turn that ring into a space, and the representation theory is organised by geometry.
Learning objectives
- State Maschke's theorem and its failure.
- Explain why global dimension becomes infinite.
- Define complexity and support varieties.
- State the Quillen stratification and its consequence.
Section 01The two regimes
k[G] is semisimple by Maschke's theorem. Every module is projective and injective; Hn(G, M) = 0 for n ≥ 1. Representation theory is character theory.
k[G] is not semisimple. Global dimension is infinite, resolutions never terminate, and cohomology is non-zero in infinitely many degrees.
The averaging idempotent (1/|G|)∑g requires |G| to be invertible. Without it there is no canonical projection onto a submodule, and short exact sequences need not split. The failure is exactly the failure of an averaging argument, as with the Casimir in the Lie algebra case.
| Setting | Behaviour |
|---|---|
| k[ℤ/p], char k = p | k[x]/(xp) — a local ring |
| Indecomposable modules | k[x]/(xi) for 1 ≤ i ≤ p |
| Projectives | Only the free module k[G] itself |
| Cohomology H*(ℤ/p, k) | Non-zero in every degree, periodic |
| Global dimension | Infinite |
Section 02Complexity and support varieties
The complexity of a module is the polynomial rate of growth of the ranks in its minimal projective resolution. It is zero exactly for projective modules, and bounded above by the p-rank of the group.
- Stage 01Cohomology ringH*(G, k) is a finitely generated graded-commutative k-algebra — the Evens–Venkov theorem.
- Stage 02Its spectrumThe maximal ideal spectrum is an affine variety, the support variety of the trivial module.
- Stage 03Module supportEach module M has a support variety, the subvariety defined by the annihilator of Ext*(M, M).
- Stage 04Dimension = complexityThe dimension of the support variety equals the complexity of the module, tying geometry to resolution growth.
The variety is stratified by contributions from the elementary abelian p-subgroups, with the maps induced by restriction. So the geometry of the cohomology ring is assembled from the elementary abelian subgroups — the geometric counterpart of Sylow detection.
Section 03Structural tools
Relative projectivity and vertices
Every indecomposable module is relatively projective with respect to a p-subgroup, its vertex, unique up to conjugacy. This is relative homological algebra doing structural work.
Green correspondence
A bijection between indecomposable modules with a given vertex for G and for the normaliser of that vertex, reducing questions to smaller groups.
Blocks
The group algebra decomposes into blocks; each has a defect group measuring how far it is from semisimple. Blocks of defect zero behave like the ordinary case.
Stable module category
Quotient the module category by projectives. It is triangulated, and support varieties classify its thick subcategories.
Tate cohomology
Extends cohomology to negative degrees for finite groups, and is the natural home for the stable category.
Periodicity
Modules of complexity 1 are exactly those with periodic resolutions — the representation-theoretic face of periodic cohomology.
ReferenceFrequently asked questions
Why is the cohomology ring finitely generated?
By the Evens–Venkov theorem, proved using the norm map and an induction over subgroups. Finite generation is what makes the spectrum a genuine algebraic variety and the whole geometric approach possible.
Is modular representation theory relative homological algebra?
In large part. Vertices and sources are defined by relative projectivity with respect to subgroups, so the projective-class machinery of this collection is exactly the right framework.
What does complexity zero mean?
That the minimal resolution terminates, so the module is projective. Complexity 1 means periodic resolutions; higher complexity means polynomially growing ranks, and the exponent is the dimension of the support variety.
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