Cohomologies.

Yoshifumi Tsuchimoto

LEMMA 02.1   Let $R$ be a ring. Let $f:M\to N$ be a homomorphism of $R$ -modules. Then for any $R$ -module $L$ we may define:

1. A homomorphism $\operatorname{Hom}_R(L,f): \operatorname{Hom}_R(L,M)\to \operatorname{Hom}_R(L,N)$ defined by $\operatorname{Hom}_R(L,f)(g)=f\circ g$ .
2. A homomorphism $\operatorname{Hom}_R(f,L): \operatorname{Hom}_R(N,L)\to \operatorname{Hom}_R(M,L)$ defined by $\operatorname{Hom}_R(f,L)(h)=h\circ f$ .

PROPOSITION 02.2   Let $R$ be a ring. Let

$$0 \to M_1 \overset{f}{\to} M_2 \overset{g}{\to} M_3 \to 0$$

be an exact sequence of $R$ -modules. Then for any $R$ -module $N$ , we have:

1. $$0 \to \operatorname{Hom}_R (N,M_1) \overset{\operatorname{Hom}_R... ...(N,M_2) \overset{\operatorname{Hom}_R(N,g)}{\to} \operatorname{Hom}_R(N,M_3)$$
   is exact. The third arrow $\operatorname{Hom}_R(N,g)$ need not be surjective.

2. $$0 \to \operatorname{Hom}_R (M_3,N) \overset{\operatorname{Hom}_R... ...(M_2,N) \overset{\operatorname{Hom}_R(f,N)}{\to} \operatorname{Hom}_R(M_1,N)$$
   is exact. The third arrow $\operatorname{Hom}_R(f,N)$ need not be surjective.

EXERCISE 02.1   We consider an exact sequence

$$0\to 3\mathbb{Z}\overset{i}{\to}\mathbb{Z}\to \mathbb{Z}/3 \mathbb{Z}\to 0$$

where $i$ is the inclusion map. Show that

$$\operatorname{Hom}_\mathbb{Z}(\mathbb{Z},\mathbb{Z}) \overset{\op... ...Hom}(i,\mathbb{Z})}{\to} \operatorname{Hom}_\mathbb{Z}(3\mathbb{Z},\mathbb{Z})$$

is not surjective

EXERCISE 02.2   Assume $R$ is a field. Then show that the third arrow which appear in the sequence (1) in Proposition 2.2 is surjective.