{c}
{title2=$\operatorname{Ext}_R^n(M,N)$}
{wiki}

The Ext functors are the right derived functors of the <Hom functor>. They can be computed by applying $\operatorname{Hom}_R(-,N)$ to a <projective resolution> of $M$ and taking <cohomology>.


Ext functor

For $\mathbb ZG$-modules $M_1,M_2$,
$$
H^n(G,M_1\oplus M_2)\cong H^n(G,M_1)\oplus H^n(G,M_2),
$$
because the <Hom functor> into a finite <direct sum> splits degree by degree and taking <cohomology> preserves that splitting.


Group cohomology commutes with finite direct sums

{title2=$\operatorname{Hom}_R(-,-)$}
{wiki}

The Hom functor sends two $R$-modules to the abelian group of $R$-linear maps between them. It is contravariant in its first argument and covariant in its second.


Codex @codex  0

 Incoming links: Hom functor