COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

02. “Hom” modules.

Lemma 2.1. Let R be a ring. Let f : M → N be a homomorphism of R-modules. Then for any R-module L we may define:

(1) A homomorphism Hom_{R}(L, f) : Hom_{R}(L, M) → Hom_{R}(L, N) defined by Hom_{R}(L, f)(g) = f ∘ g.

(2) A homomorphism Hom_{R}(f, L) : Hom_{R}(N, L) → Hom_{R}(M, L) defined by Hom_{R}(f, L)(h) = h ∘ f.

Proposition 2.2. Let R be a ring. Let

0 → M_{1} →^{f} M_{2} →^{g} M_{3} → 0

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

(1)

0 → Hom_{R}(N, M_{1}) →^{Hom_{R}(N,f)} Hom_{R}(N, M_{2}) →^{Hom_{R}(N,g)} Hom_{R}(N, M_{3})

is exact. The third arrow Hom_{R}(N, g) need not be surjective. (2)

0 → Hom_{R}(M_{3}, N) →^{Hom_{R}(g,N)} Hom_{R}(M_{2}, N) →^{Hom_{R}(f,N)} Hom_{R}(M_{1}, N)

is exact. The third arrow Hom_{R}(f, N) need not be surjective.

Exercise 2.1. We consider an exact sequence

0 → 3ℤ →^{i} ℤ → ℤ/3ℤ → 0

where i is the inclusion map. Show that

Hom_{ℤ}(ℤ, ℤ) →^{Hom(i,ℤ)} Hom_{ℤ}(3ℤ, ℤ)

is not surjective

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