Change of Rings

Executive summary

Move the computation to a ring where it is easier

A ring homomorphism Λ → Γ gives three functors between the module categories: restriction, and its left and right adjoints, extension and coinduction. Comparing derived functors across these produces the change-of-rings theorems, of which Shapiro's lemma is the most used. The general statement is a spectral sequence, and the useful special cases are those where it collapses.

Learning objectives

Identify the three change-of-rings functors and their adjunctions.
State Shapiro's lemma and its use in group cohomology.
Give conditions under which the comparison is an isomorphism.
Recognise the general change-of-rings spectral sequence.

Section 01The three functors

Functors associated with Λ → Γ
Functor	Direction	Adjointness
Restriction Res	Γ-modules → Λ-modules	Middle of the triple
Extension Γ ⊗_Λ −	Λ → Γ	Left adjoint to restriction — right exact
Coinduction Hom_Λ(Γ, −)	Λ → Γ	Right adjoint to restriction — left exact

Restriction is exact

It changes nothing but the acting ring, so it preserves exactness in both directions. That is what makes its adjoints well behaved: the left adjoint preserves projectives and the right adjoint preserves injectives, which is exactly what comparison of derived functors needs.

Section 02Shapiro's lemma

Extⁿ_Γ(Γ ⊗_Λ M, N) ≅ Extⁿ_Λ(M, Res N)

and dually for coinduction. In group cohomology, with Λ = ℤ[H] and Γ = ℤ[G] for a subgroup H ≤ G:

Hⁿ(G, Coind^G_H A) ≅ Hⁿ(H, A)

The workhorse of group cohomology

Shapiro's lemma converts a computation over a large group with an induced module into one over a small subgroup with the original module. Cohomology of a permutation module reduces to the cohomology of a point stabiliser, which is how most explicit computations actually proceed.

Section 03When the comparison is clean

Change-of-rings results
Situation	Result
Γ flat over Λ	Tor^Λ ⊗ Γ ≅ Tor^Γ — flat base change
Γ projective over Λ	Restriction preserves projective resolutions
Γ = Λ/(x), x a non-zero-divisor acting as zero	A long exact sequence relating Ext over Λ and over Γ
Localisation Λ → S⁻¹Λ	Ext and Tor localise for finitely presented modules
General Λ → Γ	A spectral sequence, not an isomorphism

E₂^p,q = Ext^p_Γ(M, Ext^q_Λ(Γ, N)) ⇒ Ext^p+q_Λ(M, N)

Collapse requires a hypothesis

The spectral sequence gives an isomorphism only when it degenerates — typically because Γ is flat or projective over Λ, killing all but one row. Asserting an isomorphism without such a hypothesis is a common error, and the resulting statements are false in general.

ReferenceFrequently asked questions

Why are there two adjoints to restriction?

Because restriction is exact, so it can have adjoints on both sides. The left adjoint is extension of scalars and the right is coinduction; for a finite index subgroup of a group they coincide, which is why induction and coinduction are often conflated in that setting.

Does Shapiro's lemma need finiteness?

The coinduced version holds in general. The induced version agrees with it when the index is finite; for infinite index induction and coinduction differ and only one of the two statements is available.

What is the most common use of change of rings?

Reducing a cohomology computation over a group ring to one over a subgroup or a quotient. The Lyndon–Hochschild–Serre spectral sequence is the change-of-rings sequence for a normal subgroup, and it is the single most used tool in group cohomology.

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