COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

08. Homological algebra

Definition 8.1. Let R be a ring. A cochain complex of Rmodules is a sequence of R-modules

C^{•} : … →^{d^{n−1}} C^{n} →^{d^{n}} C^{n+1} →^{d^{n+1}} …

such that d^{n}∘d^{n−1} = 0 . The n-th cohomology of the cochain complex is defined to be the R-module

H^{n}(C^{•}) = Ker(d^{n})/ Image(d^{n−1}).

Elements of Ker(d^{n}) (respectively, Image(d^{n−1})) are often referred to as cocycles (respectively, coboundaries).

Definition 8.2. Let R be a ring.

(1) An R-module I is said to be injective if it satisfies the following condition: For any R-module homomorphism f : M → I and for any monic R-module homomorphism ι : N → M, f “extends” to an R-module homomorphism f̂: M → I.

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

(2) A R-module P is said to be projective if it satisfies the following condition: For any R-module homomorphism f : P → N and for any epic R-module homomorphism π : M → N, f “lifts” to a morphism f̂: M → I.

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

Exercise 8.1. Let R be a ring. Let

0 → M_{1} → M_{2} → M_{3} → 0

be an exact sequence of R-modules. Assume furthermore that M_{3} is projective. Then show that the sequence

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

is exact.

Lemma 8.3. 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

YOSHIFUMI TSUCHIMOTO

Corollary 8.4. 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 8.5. 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 8.6. 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 8.7. Let R be a (commutative) principal ideal domain (PID). Then an R-module I is injective if and only if it is divisible.

Proposition 8.8. 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.

A bit of category theory:

Definition 8.9. A category 𝒞 is a collection of the following data (1) A collection Ob(𝒞) of objects of 𝒞.

(2) For each pair of objects X, Y ∈ Ob(𝒞), a set

Hom_{𝒞}(X, Y)

of morphisms.

(3) For each triple of objects X, Y, Z ∈ Ob(𝒞), a map(“composition (rule)”)

Hom_{𝒞}(X, Y) × Hom_{𝒞}(Y, Z) → Hom_{𝒞}(X, Z)

satisfying the following axioms (1) Hom(X, Y) ∩ Hom(Z, W) = ∅ unless (X, Y) = (Z, W).

(2) (Existence of an identity) For any X ∈ Ob(𝒞), there exists an element id_{X} ∈ Hom(X, X) such that

id_{X} ∘f = f, g ∘ id_{X} = g

holds for any f ∈ Hom(S, X), g ∈ Hom(X, T) (∀S, T ∈ Ob(𝒞)).

(3) (Associativity) For any objects X, Y, Z, W ∈ Ob(𝒞), and for any morphisms f ∈ Hom(X, Y), g ∈ Hom(Y, Z), h ∈ Hom(Z, W), we have

(f ∘ g) ∘ h = f ∘ (g ∘ h).

Morphisms are the basic actor/actoress in category theory.

An additive category is a category in which one may “add” some morphisms.

Definition 8.10. An additive category 𝒞 is said to be abelian if it satisfies the following axioms.

(A4-1) Every morphism f : X → Y in 𝒞 has a kernel ker(f) : Ker(f) →

X. (A4-2) Every morphism f : X → Y in 𝒞 has a cokernel coker(f) : Y →

Coker(f). (A4-3) For any given morphism f : X → Y, we have a suitably defined

isomorphism

l : Coker(ker(f)) ≅ Ker(coker(f))

in 𝒞. More precisely, l is a morphism which is defined by the following relations:

ker(coker(f)) ∘ \overline{f} = f (∃\overline{f}), \overline{f} = l ∘ coker(ker(f)).

Definition 8.11. A (covariant) functor F from a category 𝒞 to a category 𝒟 consists of the following data:

(1) An function which assigns to each object C of 𝒞 an object F (C) of 𝒟.

(2) An function which assigns to each morphism f of 𝒞 an mor-

phism F (f) of 𝒟. The data must satisfy the following axioms:

(functor-1) F (1_{C}) = 1_{F (C)} for any object C of 𝒞. (functor-2) F (f ∘ g) = F (f) ∘ F (g) for any composable morphisms f, g of

𝒞.

By employing the following axiom instead of the axiom (functor-2) above, we obtain a definition of a contravariant functor: (functor-2′) F (f ∘ g) = F (g) ∘ F (f) for any composable morphisms

Definition 8.12. Let F : 𝒞_{1} → 𝒞_{2} be a functor between additive categories. We call F additive if for any objects M, N in 𝒞_{1},

Hom(M, N) → Hom(F (M), F (N))

is additive.

Definition 8.13. Let F be an additive functor from an abelian category 𝒞_{1} to 𝒞_{2}.

(1) F is said to be left exact (respectively, right exact ) if for any exact sequence

0 → L → M → N → 0,

the corresponding map

0 → F (L) → F (M) → F (N)

(respectively,

F (L) → F (M) → F (N) → 0)

is exact

(2) F is said to be exact if it is both left exact and right exact.

Lemma 8.14. 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 8.15. For any ring R, the category (R -modules) of R-modules 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 8.16. 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 8.17. 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.

Definition 8.18. Let (K^{•}, d_{K}), (L^{•}, d_{L}) be complexes of objects of an additive category 𝒞.

(1) A morphism of complex u : K^{•} → L^{•} is a family

u^{j} : K^{j} → L^{j}

of morphisms in 𝒞 such that u commutes with d. That means,

u^{j+1} ∘ d_{K}^{j} = d_{K}^{j} ∘ u^{j}

holds.

(2) A homotopy between two morphisms u, v : K^{•} → L• of complexes is a family of morphisms

h^{j} : K^{j} → L^{j−1}

such that u − v = d ∘ h + h ∘ d holds.

Lemma 8.19. Let 𝒞 be an abelian category that has enough injectives. Then:

(1) For any object M in 𝒞, there exists an injective resolution of M. That means, there exists an complex I^{•} and a morphism ι_{M} : M → I^{0} such that

H^{j}(I^{•}) = \begin{cases}M (via ι_{M}) & if j = 0 \\ 0 & if j ≠ 0 \end{cases}

(2) For any morphism f : M → N of 𝒞, and for any injective resolutions (I^{•}, ι_{M}), (J^{•}, ι_{N}) of M and N (respectively), There exists a morphism f̄ : I^{•} → J^{•} of complexes which commutes with f. Forthermore, if there are two such morphisms f̄ and f′, then the two are homotopic.

Definition 8.20. Let 𝒞_{1} be an abelian category which has enough injectives. Let F : 𝒞_{1} → 𝒞_{2} be a left exact functor to an abelian category. Then for any object M of 𝒞_{1} we take an injective resolution I_{M}^{•} of M and define

R^{i}F (M) = H^{i}(I_{M}^{•}).

and call it the derived functor of F.

Lemma 8.21. The derived functor is indeed a functor.