Rearding notation of (Relatively)Projective/ (Relatively)Injective in Group cohomology I am reading Group cohomology from Serre's Local Fields.
I got confused with the notation he used...
We know that : 


*

*$A$ is Projective module if $Hom_R(A, \_)$ is exact

*$A$ is Injective module if $Hom_R(\_,A)$ is exact
Now, He is defining : 


*

*Induced Module to be module which is of the form $\Lambda\otimes X$ for some abelian Group $X$ where $\Lambda$ is group algebra $\mathbb{Z}[G]$ and $G$ is written multiplicatively..

*Coinduced Module to be module which is of the form $Hom _R(\Lambda,X)$ for some abelian Group $X$


I have no confusion with this notation but then, he defines :


*

*Relatively Projective module to be direct summand of an induced module

*Relatively Injective module to be direct summand of an coinduced module.


But then Projective should come with Hom where as relatively projective is coming with $\otimes$ where as relatively injective is clear as it is coming from coinduced module which is Hom.
Please help me to come out of this confusion..
Is there any better way to understand this more clearly..
 A: Yes, the relative projectivity in question, although implicit, has something to do with $\operatorname{Hom}$. It seems better to interpret in the framework of comonad cohomology (a reference: Weibel's Homological Algebra, section 8.6), but we prefer to explain it in more concrete terms.
Let $\mathcal A,\mathcal C$ be two categories and let $U\colon\mathcal A\to\mathcal C$ be a functor with a left adjoint $F\colon\mathcal C\to\mathcal A$ (In your case, the category $\mathcal A$ is the category of $\mathbb ZG$-modules and $\mathcal C$ is the category of $\mathbb Z$-modules, the functor $U$ is the forgetful functor which maps a $\mathbb ZG$-module to its underlying $\mathbb Z$-module, and the functor $F$ is the induced module functor $-\otimes_{\mathbb Z}\mathbb ZG$). Let $C=F\circ U\colon\mathcal A\to\mathcal A$ be the composite functor, called the comonad associated to the adjunction $(F,U)$ in modern literature. By definition of adjoint functors, there exists a natural transformation $\varepsilon\colon C\to\operatorname{id}$.
Definition An object $A\in\mathcal A$ is called $C$-projective if the canonical morphism $\varepsilon_A\colon C(A)\to A$ admits a section $A\to C(A)$, that is, a morphism $A\to C(A)$ such that the composition $A\to C(A)\xrightarrow{\epsilon_A}A$ coincides with the identity morphism $\operatorname{id}_A\colon A\to A$.
Now you can check the following two properties:
Lifting property An object $P\in\mathcal A$ is $C$-projective if and only if it satisfies the following lifting property: given a map $g\colon A_1\to A_2$ in $\mathcal A$ such that $U(g)\colon U(A_1)\to U(A_2)$ is a split surjection and a map $\gamma\colon P\to A_2$, there exists a map $\beta\colon P\to A_1$ such that $\gamma=g\circ\beta$.
(You can read the lifting property as a kind of a property of $\operatorname{Hom}$ that you want)
"Direct sum" characterization An object $P\in\mathcal A$ if and only if there exists an object $A\in\mathcal A$ such that $P$ is a retract of $C(A)$, that is, there exist maps $i\colon P\to C(A)$ and $r\colon C(A)\to P$ such that the composition $r\circ i\colon P\to C(A)\to P$ is the identity morphism $\operatorname{id}_P\colon P\to P$.
(This corresponds to the Definition that you quoted. In fact, for any $\mathbb Z$-module $A$, we can endow it with trivial $G$-action, then the $\mathbb ZG$-modules $C(A)$ enumerates all induced $\mathbb ZG$-modules when $A$ runs through all $\mathbb Z$-modules $A$.)
There is a relative cohomology theory associated to the comonad $C$ (when $\mathcal A$ is abelian) where $C$-projective objects play a similar role as projective objects in ordinary homological algebra. For details, see Weibel's book.
