Billybug大哥
2021-5-1 13:23:14
Theorem III.3.4. Suppose that Λ is a discrete valuation ring with residue field k and maximal
ideal m.
1. If P is a Λ[G]-module that is free of finite type4 over Λ, then P is a projective Λ[G]-module
if and only if P := P ⊗Λ k is a projective k[G]-module.
2. If P and P
0 are projective Λ[G]-modules, then P ' P
0 as Λ[G]-modules if and only if
P ⊗Λ k ' P
0 ⊗Λ k as k[G]-modules.
3. Suppose that the discrete valuation ring Λ is complete. (See problem VII.3.3.) If P is a
projective k[G]-module, then there exists a unique (up to isomorphism) projective Λ[G]-
module P such that P ' P ⊗Λ k.