PERIODIC FLAT MODULES, AND FLAT MODULES FOR FINITE GROUPS

D. J. BENSON AND K. R. GOODEARL

Abstract. The main theorem says that if R is a ring of coefficients and G a finite group, then a flat RG-module M which is projective as an R-module is necessarily projective as an RG-module. This is proved using the following theorem about flat modules over an arbitrary ring R. If a flat R-module M sits in a short exact sequence 0 → M → P → M → 0 with P projective, then M is projective.

1. Introduction

In the representation theory of finite groups, a great deal of attention has been given to the problem of determining whether a module over the group ring is projective. For example, a well known theorem of Chouinard [9] states that a module is projective if and only if its restriction to each elementary abelian subgroup is projective. A theorem of Dade [10] states that over an algebraically closed field of characteristic p, a finitely generated module for an elementary abelian p-group is projective if and only if its restriction to each cyclic shifted subgroup is projective, where a cyclic shifted subgroup is a certain kind of cyclic subgroup of the group algebra. For an infinitely generated module, the statement is no longer valid, but in [6] it is proved that an infinitely generated module is projective if and only if its restriction to each cyclic shifted subgroup defined over each extension field is projective. These theorems have formed the basis for the development of the theory of varieties for modules [2, 5, 6, 7].

The main theorem of this paper is the following characterization of projectivity for flat modules over group rings. The proof can be found in Section 5.

Theorem 1.1. Let R be a ring of coefficients (not necessarily commutative) and G a finite group. If M is a flat RG-module which is projective as an R-module, then M is projective as an RGmodule.

This is proved using the following interesting property of flat modules over an arbitrary ring. The following theorem is proved in Section 2.

Theorem 1.2. Let R be a ring and M a flat R-module. If there exists a short exact sequence 0 → M → P → M → 0

Date: November 1998; version 1.2.

Both authors are partly supported by grants from the NSF.

2 D. J. BENSON AND K. R. GOODEARL

with P a projective R-module, then M itself is projective.

Also in order to reach the main theorem, we need some theorems about reduction modulo primes, which seem to be of independent interest. The following results occupy Sections 3 and 4.

Theorem 1.3. Let R be a ring of coefficients and G a finite group. An RG-module M is projective (respectively, flat) if and only if

(i) M is projective (respectively, flat) as an R-module, and

(ii) M/pM is projective (respectively, flat) as an (R/pR)G-module for each prime number p dividing |G|.

On the basis of this, it is tempting to suppose that it might be true that an RG-module M is projective if and only if it is projective as an R-module, and K ⊗_{R} M is projective as a KG-module for every field K containing R/𝔭 for some prime ideal 𝔭 ⊆ R. In Section 6, we give an example to show that this is false in general, even if R is a discrete valuation ring. In Section 7 we show it is true if R is a Dedekind domain whose field of fractions has characteristic prime to |G|, such as a ring of algebraic integers, or 𝔭-local integers, or 𝔭-adic integers.

2. Periodic flat modules

The purpose of this section is to prove that periodic flat modules do not exist, over any ring. Throughout this section, all modules are modules over a ring R which has an identity element, but which is not necessarily commutative.

Lemma 2.1 (Villamayor). Let M be a submodule of a projective R-module P. The following conditions are equivalent:

(a) P/M is flat.

(b) For each x ∈ M, there exists f ∈ Hom_{R}(P, M) such that f(x) = x.

(c) For each x_{1}, …, x_{n} ∈ M, there exists f ∈ Hom_{R}(P, M) such that f(x_{i}) = x_{i} for i = 1, …, n.

Proof. See Rotman [12], Theorem 3.57.

Proposition 2.2. Let M be a countably generated submodule of a projective R-module P. If P/M is flat, then M is projective.

Proof. Let x_{1}, x_{2}, … be a countable sequence of generators for M. By induction and Lemma 2.1, there exist f_{1}, f_{2}, … in Hom_{R}(P, M) such that

(1 − f_{n})(1 − f_{n−1}) ⋯ (1 − f_{1})(x_{i}) = 0

for n ≥ i. It follows that the union of the kernels of the homomorphisms g_{n} = (1 − f_{n})(1 − f_{n−1}) ⋯ (1 − f_{1})|_{M}

PERIODIC FLAT MODULES, AND FLAT MODULES FOR FINITE GROUPS 3

equals M. Hence, there is a homomorphism g : M → P^{(ω)} = P ⊕ P ⊕ ⋯ given by the rule

g(x) = (x, g_{1}(x), g_{2}(x), …).

Let f : P^{(ω)} → M be the homomorphism given by

f(a_{1}, a_{2}, …) = f_{1}(a_{1}) + f_{2}(a_{2}) + · · · .

For all n, we have g_{n} = (1 − f_{n})g_{n−1} (where g_{0} = 1), and so f_{n}g_{n−1} = g_{n−1} − g_{n}. Hence, for x ∈ M we see that

fg(x) = f_{1}(x) + f_{2}g_{1}(x) + f_{3}g_{2}(x) + …
= x − g_{1}(x) + g_{1}(x) − g_{2}(x) + g_{2}(x) − g_{3}(x) + ⋯ = x.

Thus fg = id_{M}, and therefore M is projective.

In particular, Proposition 2.2 implies the well known fact that countably related flat modules have projective dimension at most one.

Lemma 2.3. Let M be a submodule of a projective R-module P, and suppose that P/M is flat. Then any countably generated submodule of M is contained in a countably generated submodule K of M such that P/K is flat.

Proof. Since it is harmless to replace M and P by M ⊕ 0 and P ⊕ Q for any projective module Q, there is no loss of generality in assuming that P is free, say with basis X.

Given a countably generated submodule K_{0} ⊆ M, choose a countable subset Y ⊆ X such that K_{0} is contained in the submodule P_{0} of P generated by Y. Then P_{0} is a direct summand of P. Let π : P → P_{0} be a projection, so that π is an idempotent endomorphism of P with countably generated image containing K_{0}.

We claim that M has a countably generated submodule K_{1} ⊇ K_{0} such that for each x ∈ K_{0}, there exists f ∈ Hom_{R}(P, K_{1}) with f(x) = x.

Let x_{1}, x_{2}, … be a countable sequence of generators for K_{0}. By Lemma 2.1, there exist f_{1}, f_{2}, … in Hom_{R}(P, M) such that f_{n}(x_{i}) = x_{i} for n ≥ i. Since π(x_{i}) = x_{i} for all i, we may replace f_{n} by f_{n}π. Now each f_{n}P is countably generated, and so the module

K_{1} = K_{0} + ∑_{n=1}^{∞} f_{n}P

is a countably generated submodule of M. Any x ∈ K_{0} lies in the submodule generated by x_{1}, …, x_{n} for some n, whence f_{n}(x) = x. Since f_{n}P ⊆ K_{1}, this establishes the claim.

Now repeat this process ω times, obtaining countably generated submodules K_{0} ⊆ K_{1} ⊆ … of M such that for each x ∈ K_{t}, there is some f ∈ Hom_{R}(P, K_{t+1}) with f(x) = x. Then K = ⋃_{t=0}^{∞} K_{t} is a countably generated submodule of M, and P/K is flat by Lemma 2.1.

Lemma 2.4. Let 0 → M ⟶^{⊆} ⨁_{i∈I} P_{i} ⟶^{g} M → 0 be a short exact sequence with M flat and the P_{i} countably generated projective. Given any countable subset J_{0} ⊆ I and any countably generated submodule K_{0} ⊆ M, there exists a countable subset J ⊆ I such that J_{0} ⊆ J, the submodule P_{J} = ⨁_{j∈J} P_{j} contains K_{0}, the image gP_{J} equals M ∩ P_{J}, and gP_{J} is projective.

Proof. Set P_{J} = ⨁_{j∈J} P_{j} and M_{J} = M ∩ P_{J} for all subsets J ⊆ I.

We first show that given J_{0} and K_{0}, there exists a countable subset J ⊆ I, containing J_{0}, such that K_{0} ⊆ gP_{J} and gP_{J} is projective.

In view of Lemma 2.3, after enlarging K_{0} we may assume that P_{I}/K_{0} is flat. Then, after enlarging J_{0}, we may also assume that K_{0} ⊆ gP_{J_{0}}. Now apply Lemma 2.3 again to obtain a countably generated submodule K_{1} ⊆ M such that gP_{J_{0}} ⊆ K_{1} and P_{I}/K_{1} is flat. Continuing in this manner, we obtain countable subsets J_{0} ⊆ J_{1} ⊆ … in I and countably generated submodules K_{0} ⊆ K_{1} ⊆ … in M such that K_{i} ⊆ gP_{J_{i}} ⊆ K_{i+1} and P_{I}/K_{i} is flat for all i. Then J = ⋃_{i=0}^{∞} J_{i} is a countable subset of I, and K_{0} ⊆ gP_{J} = ⋃_{i=0}^{∞} K_{i}. Since the P_{I}/K_{i} are flat, so is P_{I}/gP_{J}, and thus gP_{J} is projective by Proposition 2.2. This establishes the first claim.

We next observe that any countable subset J_{0} of I is contained in a countable subset J such that M_{J} is countably generated. To see this, use the first claim to find a countable subset J of I such that J_{0} ⊆ J and gP_{J} is projective. Hence, the short exact sequence

0 → M_{J} → P_{J} → gP_{J} → 0

splits, and so M_{J} is countably generated.

Now return to arbitrary J_{0} and K_{0} as in the hypotheses of the lemma. After enlarging J_{0}, we may assume that K_{0} ⊆ P_{J_{0}}. By the second claim above, there exists a countable subset J_{1} ⊆ I such that J_{0} ⊆ J_{1} and M_{J_{1}} is countably generated. By the first claim, there exists a countable subset J_{2} ⊆ J, containing J_{1}, such that M_{J_{1}} ⊆ gP_{J_{2}} and gP_{J_{2}} is projective. Since gP_{J_{2}} is countably generated, it is contained in P_{J_{3}} for some countable subset J_{3} ⊆ I, and thus gP_{J_{2}} ⊆ M_{J_{3}}. It is harmless to enlarge J_{3} enough to contain J_{2}, and the second claim allows us, after a further enlargement, to assume that M_{J_{3}} is countably generated.

Continuing in this manner, we obtain countable subsets J_{0} ⊆ J_{1} ⊆ … in I such that

M_{J_{2i−1}} ⊆ gP_{J_{2i}} ⊆ M_{J_{2i+1}}

and gP_{J_{2i}} is projective for all i. Then J = ⋃_{i=1}^{∞} J_{i} is a countable subset of I such that K_{0} ⊆ P_{J} and gP_{J} = M_{J}. Since M_{J} is the ascending union of the projectives gP_{J_{2i}}, it must be flat. In view of the short exact sequence

0 → M_{J} → P_{J} → M_{J} → 0,

we conclude from Proposition 2.2 that M_{J} is projective.

Theorem 2.5. If 0 → M → P → M → 0 is a short exact sequence of R-modules with M flat and P projective, then M must be projective.

Proof. We may assume that the homomorphism M → P is an inclusion map, and we let g denote the homomorphism P → M.

By a theorem of Kaplansky [11], the projective module P is a direct sum of countably generated submodules, say P = ⨁_{i∈I} P_{i}. Set P_{J} = ⨁_{j∈J} P_{j} and M_{J} = M ∩ P_{J} for all subsets J ⊆ I.

Consider the set P of all pairs (J, L) where J is a subset of I such that gP_{J} = M_{J} and L is a submodule of P_{J} such that P_{J} = M_{J} ⊕ L. In particular, (∅, 0) ∈ P. Order P by componentwise inclusion:

(J_{1}, L_{1}) ≤ (J_{2}, L_{2}) ⟺ J_{1} ⊆ J_{2} and L_{1} ⊆ L_{2}.

If {(J_{α}, L_{α})} is any nonempty chain in P, then (⋃_{α} J_{α}, ⋃_{α} L_{α}) clearly belongs to P. Therefore Zorn’s Lemma gives us a maximal element of P, say (J, L).

If J = I, then P = M ⊕ L and we are done. Hence, assume that J ≠ I.

Since gP_{J} = M_{J}, it follows from the Nine Lemma or the Snake Lemma that there is a short exact sequence

0 → M/M_{J} ⟶^{f} P/P_{J} ⟶^{\overline{g}} M/M_{J} → 0

where f is induced by the inclusion map M → P and \overline{g} is induced by g. Then f maps M/M_{J} isomorphically onto the submodule \overline{M} = (M + P_{J})/P_{J} ⊆ P/P_{J}, and we obtain a short exact sequence

0 → \overline{M} ⟶^{⊆} P/P_{J} ⟶^{h} \overline{M} → 0

where h = f\overline{g}. Note that P/P_{J} ≅ P_{I\J}; in particular, P/P_{J} is projective, and is a direct sum of countably generated submodules (P_{i} + P_{J})/P_{J} for i ∈ I \ J. Since M_{J} is a direct summand of P_{J}, it is also a direct summand of P. Then from M_{J} ⊆ M ⊆ P it follows that M_{J} is a direct summand of M. Hence, M/M_{J}, and thus \overline{M}, is flat.

Now apply Lemma 2.4 to the situation above, with initial data corresponding to a nonempty subset of I \ J. The lemma then gives us a subset K ⊆ I, properly containing J, such that h(P_{K}/P_{J}) = \overline{M} ∩ (P_{K}/P_{J}) and h(P_{K}/P_{J}) is projective. Since

h(P_{K}/P_{J}) = f(gP_{K}/M_{J}) = (gP_{K} + P_{J})/P_{J}
\overline{M} ∩ (P_{K}/P_{J}) = [(M + P_{J}) ∩ P_{K}]/P_{J} = (M_{K} + P_{J})/P_{J},

we obtain gP_{K} + P_{J} = M_{K} + P_{J}. Intersecting with M and using the modular law, we see that gP_{K} = M_{K}.

Since the kernel of h|_{P_{K}/P_{J}} is the submodule \overline{M} ∩ (P_{K}/P_{J}) = (M_{K} + P_{J})/P_{J}, we find that h(P_{K}/P_{J}) ≅ P_{K}/(M_{K} + P_{J}). The latter quotient is thus projective, and so P_{K} = (M_{K} + P_{J}) ⊕ T for some T. Since P_{J} = M_{J} ⊕ L and

M_{K} ∩ L = M_{K} ∩ P_{J} ∩ L = M_{J} ∩ L = 0,

we have M_{K} + P_{J} = M_{K} ⊕ L and so P_{K} = M_{K} ⊕ L ⊕ T. But then (K, L ⊕ T) ∈ P, contradicting the maximality of (J, L).

Therefore J = I and P = M ⊕ L.

3. Reduction modulo p and projectivity

In this section, we prove that projectivity for modules over a group ring is determined by the reduction modulo each prime dividing the group order. The proof works in the context of a group and a subgroup of finite index, so this is the context in which we present it. We are mostly interested in the case H = {1} of the following theorem.

Theorem 3.1. Let R be a ring of coefficients and H a subgroup of finite index in a group G. An RG-module M is projective if and only if

(i) M is projective as an RH-module, and

(ii) M/pM is projective as an (R/pR)G-module for each prime number p dividing [G : H].

Proof. It is easy to see that a projective RG-module satisfies (i) and (ii). Conversely, we suppose that M is an RG-module satisfying (i) and (ii), and we shall prove that M is projective. Since M is projective as an RH-module, it suffices to prove that M is projective relative to H. By D. G. Higman’s criterion (see for example Proposition 3.6.4 of [4]), this amounts to showing that the identity endomorphism can be written as

1_{M} = Tr_{H,G}(α)

for some α ∈ End_{RH}(M), where Tr_{H,G}(α) is the RG-endomorphism defined by

Tr_{H,G}(α)(m) = ∑_{g∈G/H} g(α(g^{−1}(m))).

The sum runs over a set of left coset representatives of H in G.

Let [G : H] = ∏_{i} p_{i}^{γ_{i}} where the p_{i} are distinct primes. Since M is projective as an RH-module, for each p_{i}, the natural map

End_{RH}(M) → End_{(R/p_{i}R)H}(M/p_{i}M)

given by reduction modulo p_{i} is surjective, and the kernel of this map equals p_{i}End_{RH}(M). Since M/p_{i}M is a projective (R/p_{i}R)G-module, its identity endomorphism can be written as Tr_{H,G}(\overline{α}_{i})

PERIODIC FLAT MODULES, AND FLAT MODULES FOR FINITE GROUPS 7

for some \overline{α}_{i} ∈ End_{(R/p_{i}R)H}(M/p_{i}M). In view of the observations above, this implies that there exist elements α_{i}, β_{i} ∈ End_{RH}(M) such that

1_{M} = Tr_{H,G}(α_{i}) + p_{i}β_{i}.

Note that p_{i}β_{i} = 1_{M} − Tr_{H,G}(α_{i}) is an RG-homomorphism. Multiplying the above expressions, we get

1_{M} = ∏_{i}(Tr_{H,G}(α_{i}) + p_{i}β_{i})^{γ_{i}} = Tr_{H,G}(α′) + [G : H]β′

for some α′, β′ ∈ End_{RH}(M). (Here, we have used the fact that if a ∈ End_{RH}(M) and b ∈ End_{RG}(M), then Tr_{H,G}(a)b = Tr_{H,G}(ab) and bTr_{H,G}(a) = Tr_{H,G}(ba).) Now the map [G : H]β′ = 1_{M} − Tr_{H,G}(α′) is an RG-homomorphism, so applying Tr_{H,G} we get

[G : H]Tr_{H,G}(β′) = [G : H]^{2}β′,

and hence δ = Tr_{H,G}(β′) − [G : H]β′ satisfies [G : H]δ = 0. Hence,

δ = δ.1_{M} = δ(Tr_{H,G}(α′) + [G : H]β′) = Tr_{H,G}(δα′),

and so

[G : H]β′ = Tr_{H,G}(β′) − δ = Tr_{H,G}(β′ − δα′).

Finally,

1_{M} = Tr_{H,G}(α′) + [G : H]β′ = Tr_{H,G}(α′ + β′ − δα′).

4. Reduction modulo p and flatness

In this section, we prove that flatness for modules over a group ring is determined by the reduction modulo each prime dividing the group order. Again, we prove the theorem in the generality of a subgroup of finite index.

Lemma 4.1. Let R be a ring of coefficients and H a subgroup of finite index in a group G. If M is a left RG-module and N a right RG-module with M flat as a left RH-module, then [G : H] annihilates Tor_{n}^{RG}(N, M) for all n > 0.

Proof. Let g_{1}, …, g_{l} be a set of left coset representatives of H in G, and let

0 → M′ ⟶^{ϕ} F → M → 0

be a short exact sequence with F a free left RG-module. Then we have the long exact sequence

0 → Tor_{1}^{RG}(N, M) → N ⊗_{RG} M′ ⟶^{ϕ_{∗}} N ⊗_{RG} F → N ⊗_{RG} M → 0.

8 D. J. BENSON AND K. R. GOODEARL

Consider an element ∑_{i=1}^{k} n_{i}⊗m′_{i} in the kernel of ϕ_{∗}. By Lemma 2.1, there is an RH-homomorphism ρ : F → M′ such that ρϕ(g_{j}^{−1}m′_{i}) = g_{j}^{−1}m′_{i} for 1 ≤ j ≤ l and 1 ≤ i ≤ k. Let θ = Tr_{H,G}(ρ). Then θ : F → M′ is an RG-homomorphism such that θϕ(m′_{i}) = [G : H]m′_{i} for 1 ≤ i ≤ k. So we have

0 = θ_{∗}ϕ_{∗}(∑_{i=1}^{k} n_{i} ⊗ m′_{i}) = ∑_{i=1}^{k} n_{i} ⊗ θϕ(m′_{i}) = [G : H] ∑_{i=1}^{k} n_{i} ⊗ m′_{i}.

This proves that [G : H] annihilates Tor_{1}^{RG}(N, M). Since M′ is also flat as an RH-module and

Tor_{n}^{RG}(N, M′) ≅ Tor_{n+1}^{RG}(N, M)

for n ≥ 1, an induction on n (i.e., dimension shifting) shows that

[G : H]Tor_{n}^{RG}(N, M) = 0

for n ≥ 1.

Lemma 4.2. Let S be a ring, p ∈ S a central element, and M a left S-module. Assume that M/pM is flat as a left S/pS-module, and that Tor_{n}^{S}(S/pS, M) = 0 for all n > 0. If N is any right S-module, then Tor_{n}^{S}(N, M) is p-torsionfree for all n > 0 and p-divisible for all n > 1.

Proof. For any right S/pS-module L, we have

Tor_{n}^{S}(L, M) ≅ Tor_{n}^{S/pS}(L, M/pM) = 0

for all n > 0 by [8], Proposition VI.4.1.2. In particular,

Tor_{n}^{S}(N/pN, M) = Tor_{n}^{S}(N[p], M) = 0

for all n > 0, where N[p] = ann_{N}(p). Consider the short exact sequences

0 → pN ⟶^{ϕ} N → N/pN → 0

0 → N[p] → N ⟶^{π} pN → 0

where ϕ is the inclusion map and π is multiplication by p. From the long exact sequences for Tor_{∗}^{S}(−, M), we see that Tor_{n}^{S}(ϕ, M) is an isomorphism for all n > 0, while Tor_{n}^{S}(π, M) is an isomorphism for all n > 1 and a monomorphism for n = 1.

Therefore Tor_{n}^{S}(ϕπ, M) is an isomorphism for all n > 1 and a monomorphism for n = 1. Since ϕπ : N → N is just multiplication by p, so is Tor_{n}^{S}(ϕπ, M) (e.g., [12], Theorem 8.13), and the lemma follows.

Theorem 4.3. Let R be a ring of coefficients and H a subgroup of finite index in a group G. An RG-module M is flat if and only if

(i) M is flat as an RH-module, and

(ii) M/pM is flat as an (R/pR)G-module for each prime p dividing [G : H].

PERIODIC FLAT MODULES, AND FLAT MODULES FOR FINITE GROUPS 9

Proof. That flatness of M as an RG-module implies (i) and (ii) is well known and easy to see. Condition (i) holds because (−) ⊗_{RH} M is naturally equivalent to the composition of the exact functors (−) ⊗_{RH} RG and (−) ⊗_{RG} M. Condition (ii) holds because (−) ⊗_{(R/pR)G} (M/pM) is naturally equivalent to the restruction of (−) ⊗_{RG} M to the category of right (R/pR)G-modules.

For the converse, we begin by observing that (i) implies that M is flat as an R-module. Since RG is a flat R-module, it now follows from [8], Proposition VI.4.1.1 that

Tor_{n}^{RG}(RG/pRG, M) = Tor_{n}^{RG}((R/pR) ⊗_{R} RG, M) ≅ Tor_{n}^{R}(R/pR, M) = 0

for all n > 0 and any prime p.

Now let N be an arbitrary right RG-module. By (ii) and Lemma 4.2, Tor_{n}^{RG}(N, M) is p-torsion-free for all n > 0 and any prime p dividing [G : H]. On the other hand, Lemma 4.1 shows that [G : H] annihilates Tor_{n}^{RG}(N, M), and therefore Tor_{n}^{RG}(N, M) = 0 for all n > 0.

5. R-projective and RG-flat imply RG-projective

In this section, we prove the main theorem of the paper. This is a matter of putting together what we have proved so far with the results of [3].

Theorem 5.1. Let R be a ring of coefficients and G a finite group. If M is a flat RG-module which is projective as an R-module, then M is projective as an RG-module.

Proof. Suppose that M is a flat RG-module which is projective as an R-module, but not as an RG-module. Then by Theorem 4.3 (with H = {1}), for each prime p dividing |G|, M/pM is a flat (R/pR)G-module. By Theorem 3.1 (again with H = {1}), there exists a prime p dividing |G| such that M/pM is not a projective (R/pR)G-module. By Chouinard’s theorem [9], we can find an elementary abelian p-subgroup E ≅ (ℤ/p)^{r} of G such that M/pM is flat but not projective as an (R/pR)E-module. Choose a chain of subgroups

1 = E_{0} < E_{1} < ⋯ < E_{r−1} < E_{r} = E

so that each E_{i−1} has index p in E_{i}. Then there is a unique index j in the range 0 ≤ j ≤ r − 1 so that M/pM is projective as an (R/pR)E_{j}-module and flat but not projective as an (R/pR)E_{j+1}- module. Set R′ = (R/pR)E_{j}. Then we can regard the group ring (R/pR)E_{j+1} as R′(ℤ/p). We are now in a position to be able to apply theorem 2.4 of [3] to deduce the existence of a flat R′-module M′ which is not projective, but which sits in a short exact sequence 0 → M′ → P → M′ → 0 with P projective.^{1} This contradicts Theorem 2.5, and so the theorem is proved.

1Theorem 2.4 of [3] is stated under the hypothesis that the coefficient ring is commutative, but this hypothesis is not used in the proof.

10 D. J. BENSON AND K. R. GOODEARL

6. Passage to fields of coefficients

Let R be a commutative ring of coefficients and G a finite group. On the basis of what we have proved, it is tempting to suppose that it might be true that an RG-module M is projective if and only if

(i) M is projective as an R-module, and (ii) K ⊗_{R} M is projective as a KG-module for every field K containing R/𝔭 for some prime ideal 𝔭 ⊆ R. In this section, we give an example to show that this is false, and in the next section we show that it is true if R is a Dedekind domain of characteristic zero.

Let k be an algebraically closed field of characteristic 2 and let G = ℤ/2 = ⟨g | g^{2} = 1⟩. Let R = k[[t]] be the ring of formal power series over k in a variable t. Then X = k[t^{−1}] can be considered as an R-module in an obvious way. In fact, X is the injective hull of k as an R-module. Since R has global dimension one, there is an R-module projective resolution of X of the form

0 → Q_{1} ⟶^{ρ} Q_{0} → X → 0.

Let M be the RG-module whose underlying R-module is Q_{0} ⊕ Q_{1}, with the group element g acting as the matrix ( \begin{matrix}1 & ρ \\ 0 & 1 \end{matrix} ). Then M is not even flat, let alone projective, since the image of 1 − g does not coincide with the kernel of 1 + g (see Theorem 2.1 of [3]). Since Q_{0} and Q_{1} are projective as R-modules, so is M, and so (i) is satisfied. To check condition (ii), we note that any candidate for the field K is an extension of either k = R/(t) or of k((t)), the field of fractions of R, so it suffices to examine these two fields. Now k ⊗_{R} X = 0 because every element of X is in the image of multiplication by t, so k ⊗_{R} M is a projective kG-module. (Here we use the criterion of [3], Theorem 2.1 again.) Similarly, k((t)) ⊗_{R} X = 0 because every element is killed by a suitably high power of t, so k((t)) ⊗_{R} M is a projective k((t))G-module.

7. Dedekind domains of coefficients

In this section, we examine what happens if R is a Dedekind domain of coefficients whose characteristic (i.e., the characteristic of the field of fractions of R) is prime to the order of the group. In this case, if n is a nonzero integer prime to the characteristic of R, then the principal ideal nR is nonzero, and so it can be written as a finite product of maximal ideals (cf. Atiyah and Macdonald [1]). All that we actually need is that nR contain a finite product of maximal ideals, and for this to hold it suffices that R be a commutative noetherian domain of Krull dimension one whose characteristic is prime to n.

PERIODIC FLAT MODULES, AND FLAT MODULES FOR FINITE GROUPS 11

Theorem 7.1. Let H be a subgroup of finite index in a group G and R a commutative noetherian ring of coefficients such that the principal ideal generated by [G : H] (times the identity element) contains a finite product of maximal ideals that contain [G : H]. Then an RG-module M is projective if and only if

(i) M is projective as an RH-module, and (ii) (R/𝔪) ⊗_{R} M is projective as an (R/𝔪)G-module for each maximal ideal 𝔪 ⊆ R containing [G : H].

Proof. We modify the proof of Theorem 3.1 as follows.

Let M be an R-module satisfying (i) and (ii), and let 𝔪_{1}, …, 𝔪_{t} be maximal ideals of R, containing [G : H], such that ∏_{i} 𝔪_{i} is contained in the ideal [G : H]R. Since R is noetherian, each 𝔪_{i} is finitely generated. Thus, since M/𝔪_{i}M is a projective (R/𝔪_{i})G-module, it follows as in the previous proof that

1_{M} = Tr_{H,G}(α_{i}) + β_{i}

for some α_{i} ∈ End_{RH}(M) and some β_{i} ∈ 𝔪_{i}End_{RH}(M). Multiplying these expressions, and noting that β_{i} is an RG-homomorphism, we get

1_{M} = Tr_{H,G}(α′) + ∏_{i} β_{i}

for some α′ ∈ End_{RH}(M). Since R is commutative and ∏_{i} 𝔪_{i} ⊆ [G : H]R, we get ∏_{i} β_{i} = [G : H]β′ for some β′ ∈ End_{RH}(M).

The remainder of the proof now proceeds as before.

Theorem 7.2. Let R be a commutative noetherian domain of coefficients of Krull dimension one, and G a finite group whose order is relatively prime to the characteristic of R. Then an RG-module M is projective if and only if

(i) M is projective as an R-module, and (ii) M/𝔪M is projective as an (R/𝔪)G-module for each maximal ideal 𝔪 ⊆ R containing |G|.

Proof. The ideal I = |G|R is nonzero by hypothesis. Well known results then imply that I contains a finite product of maximal ideals that contain I. For instance, R/I has Krull dimension zero and so is artinian (e.g., [1], Theorem 8.5), whence R/I has only finitely many prime ideals, all of which are maximal (e.g., [1], Propositions 8.1 and 8.3). Thus the prime radical (nilradical) of R/I contains a finite product of maximal ideals. Since this radical is nilpotent (e.g., [1], Proposition 8.4), the desired conditions on I hold. Therefore the theorem follows from Theorem 7.1.

Remark This theorem seems to be well known for finitely generated ℤG-modules, but does not seem to be well known in the generality described here.

References

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6. , Complexity and varieties for infinitely generated modules, II, Math. Proc. Cambridge Phil. Soc. 120 (1996), 597–615.

7. J. F. Carlson, The varieties and cohomology ring of a module, J. Algebra 85 (1983), 104–143.

8. H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, 1956.

9. L. Chouinard, Projectivity and relative projectivity over group rings, J. Pure Appl. Algebra 7 (1976), 278–302.

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11. I. Kaplansky, Projective modules, Annals of Math. 68 (1958), 372–377.

12. J. J. Rotman, An Introduction to Homological Algebra, Academic Press, 1979.

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

  E-mail address: djb@byrd.math.uga.edu

Department of Mathematics, University of California, Santa Barbara, CA 93106, USA

  E-mail address: goodearl@math.ucsb.edu