COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

04. projective and injective modules

Lemma 4.1. Let R be a (unital associative but not necessarily commutative) ring. Then for any R-module M, the following conditions are equivalent.

(1) M is a direct summand of free modules.

(2) M is projective

Corollary 4.2. For any ring R, the category (R -modules) of Rmodules have enough projectives. That means, for any object M ∈ (R -modules), there exists a projective object P and a surjective morphism f : P → M.

Definition 4.3. Let R be a commutative ring. We assume R is a domain (that means, R has no zero-divisors except for 0.)

An R-module M is said to be divisible if for any r ∈ R \ {0}, the multplication map

M →^{r×} M

is surjective.

Definition 4.4. Let R be a commutative ring. We assume R is a domain (that means, R has no zero-divisors except for 0.)

An R-module M is said to be divisible if for any r ∈ R \ {0}, the multplication map

M →^{r×} M

is epic.

Lemma 4.5. Let R be a (commutative) principal ideal domain (PID). Then an R-module I is injective if and only if it is divisible.

Proposition 4.6. For any (not necessarily commutative) ring R, the category (R -modules) of R-modules has enough injectives. That means, for any object M ∈ (R -modules), there exists an injective object I and an monic morphism f : M → I.

For the proof of the proposition above, we need the followin lemmas.

Lemma 4.7. For any ℤ-module M, let us denote by M̂ the module Hom_{ℤ}(M, 𝕋_{1}) where T_{1} = ℝ/ℤ. Then:

(1) For any free ℤ-module F, F̂ is divisible (hence is ℤ-injective).

(2) For any ℤ-module M, there is a canonical injective ℤ-homomorphism

M → ˆ(M̂).

(3) Any Z-module M may be embeded in a divisible module T.

Lemma 4.8. Let T be a divisible module. Then for any ring A,

Hom_{ℤ}(A, T)

is A-injective.