CATEGORIES, ABELIAN CATEGORIES AND COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

Injective and projective objects

Definition 5.1. (1) A morphism f : X → Y in a category is said to be monic if for any object Z of 𝒞 and for any morphism g_{1}, g_{2} : Z → X, we have

f ∘ g_{1} = f ∘ g_{2} ⟹ g_{1} = g_{2}

(2) A morphism f : X → Y in a category is said to be epic if for any object Z of 𝒞 and for any morphism g_{1}, g_{2} : Y → Z, we have

g_{1} ∘ f = g_{2} ∘ f ⟹ g_{1} = g_{2}

Proposition 5.2. Let 𝒞 be an \underline{abelian category}. Then for any morphism f in 𝒞, we have:

(1) f:monic ⟺ Ker(f) = 0.

(2) f:epic ⟺ Coker(f) = 0.

Definition 5.3. Let 𝒞 be an abelian category.

(1) An object I in 𝒞 is said to be injective if it satisfies the following condition: For any morphism f : M → I and for any monic morphism ι : N → M, f “extends” to a morphism f̂: M → I.

\begin{matrix}M & ⟶^{f̂} & I \\ ^{ι}↑ & & ∥ \\ N & ⟶^{f} & I \end{matrix}

(2) An object P in 𝒞 is said to be projective if it satisfies the following condition: For any morphism f : P → N and for any epic morphism π : M → N, f “lifts” to a morphism f̂: M → I.

\begin{matrix}P & ⟶^{f̂} & M \\ ∥ & & ^{π}↓ \\ P & ⟶^{f} & N \end{matrix}

Lemma 5.4. 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 5.5. 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 an epic morphism f : P → M.

YOSHIFUMI TSUCHIMOTO

Definition 5.6. 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 5.7. Let R be a (commutative) principal ideal domain (PID). Then an R-module I is injective if and only if it is divisible.

Proposition 5.8. 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.