I'm studying group homology and cohomology following the book An introduction to homological algebra from Weibel and I have difficulties in understanding a statement regarding projective resolutions.
From now on, let $G$ a group, $H\subset G$ a subgroup, and $\mathbb{Z}G$ the integral group ring.
At the beginning of the proof of Shapiro's lemma (6.3.2) its said the following:
Note that $\mathbb{Z}G$ is a free $\mathbb{Z}H$-module. Hence any projective right $\mathbb{Z}G$-module resolution $P\rightarrow\mathbb{Z}$ is also a projective $\mathbb{Z}H$-module resolution.
I have no problems in proving that $\mathbb{Z}G$ is free as $\mathbb{Z}H$-module.
However, I am not able to undestand why any projective right $\mathbb{Z}G$-module resolution $P\rightarrow\mathbb{Z}$ has also to be a projective $\mathbb{Z}H$-module resolution.
I would be very grateful if someone could help me with this.
Thanks in advance!
Best Answer
We have a subring $S$ ($= ℤ[H]$) of a ring $R$ ($= ℤ[G]$) such that $R$ is free as an $S$-module, and we have for some $R$-module $M$ a projective resolution $$ \dotsb \xrightarrow{\enspace d_2 \enspace} P_2 \xrightarrow{\enspace d_1 \enspace} P_1 \xrightarrow{\enspace d_0 \enspace} P_0 \xrightarrow{\enspace ε \enspace} M \longrightarrow 0 \,. \tag{$\ast$} $$ By restriction, both $M$ and each $P_i$ become $S$-modules. To show that $(\ast)$ is also a projective resolution of $M$ as an $S$-module, we need to show that
the sequence $(\ast)$ is also exact as $S$-modules, and
each $P_i$ is projective as an $S$-module.
Part 1 is true because exactness can be checked on the underlying level of abelian groups. Part 2 can be seen in three steps:
We know that $R$ is free as an $S$-module.
Consequently, every free $R$-module $F$ is also free as an $S$-module. Indeed, we have $R ≅ S^{⊕ I}$ as $S$-modules for some set $I$, as well as $F ≅ R^{⊕ J}$ as $R$-modules for some set $J$. Therefore $F ≅ R^{⊕ J} ≅ ((S^{⊕ I})^{⊕ J} = S^{⊕ (I × J)}$ as $S$-modules.
Let $P$ be a projective $R$-module. This means that there exists another $R$-module $P'$ such that $P ⊕ P'$ is free as an $R$-module. As seen above, this means that $P ⊕ P'$ is also free as an $S$-module. The existence of $P'$ thus shows that $P$ is also projective as an $S$-module.