COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

01.Review of elementary definitions on modules.

Definition 1.1. A (unital associative) ring is a set R equipped with two binary operations (addition (“+”) and multiplication (“·”)) such that the following axioms are satisfied. (Ring1) R is an additive group with respect to the addition. (Ring2) distributive law holds. Namely, we have

a(b + c) = ab + bc, (a + b)c = ac + bc (∀a, ∀b, ∀c ∈ R).

(Ring3) The multiplcation is associative. (Ring4) R has a multiplicative unit.

For any ring R, we denote by 0_{R} (respectively, 1_{R}) the zero element of R (respectively, the unit element of R). Namely, 0_{R} and 1_{R} are elements of R characterized by the following rules.

• a + 0_{R} = a, 0_{R} + a = a ∀a ∈ R.

• a · 1_{R} = a, 1_{R} · a = a ∀a ∈ R.

When no confusion arises, we omit the subscript ‘_{R}’ and write 0, 1 instead of 0_{R}, 1_{R}.

Definition 1.2. Let R be a unital associative ring. An R-module M is an additive group M with R-action

R × M → M

which satisfies (Mod1) (r_{1}r_{2}).m = r_{1}.(r_{2}.m) (∀r_{1}, ∀r_{2} ∈ R, ∀m ∈ M) (Mod2) 1.m = m (∀m ∈ M) (Mod3) (r_{1} + r_{2}).m = r_{1}.m + r_{2}.m (∀r_{1}, ∀r_{2} ∈ R, ∀m ∈ M). (Mod4) r.(m_{1} + m_{2}) = r.m_{1} + r.m_{2} (∀r ∈ R, ∀m_{1}, ∀m_{2} ∈ M).

Example 1.3. Let us give some examples of R-modules.

(1) If k is a field, then the concepts “k-vector space” and “kmodule” are identical.

(2) Every abelian group is a module over the ring of integers ℤ in a unique way.

Definition 1.4. An subset M of an R-module N is said to be an R-submodule of N if M itself is an R-module and the inclusion map j : M → N is an R-module homomorphism.

Definition 1.5. Let M, N be modules over a ring R. Then a map f : M → N is called an R-module homomorphism if it is additive and preserves the R-action.

The set of all module homomorphisms from M to N is denoted by Hom_{R}(M, N). It has an structure of an module in an obvious manner.

Definition 1.6. An subset N of an R-module M is said to be an R-submodule of M if N itself is an R-module and the inclusion map j : N → M is an R-module homomorphism.

YOSHIFUMI TSUCHIMOTO

Definition 1.7. Let R be a ring. Let N be an R-submodule of an R-module M. Then we may define the quotient M/N by

M/N = M/ ∼_{N}

where the equivalence relation ∼_{N} is defined as follows:

m_{1} ∼_{N} m_{2} ⟺ m_{1} − m_{2} ∈ N.

It may be shown that the quotient M/N so defined is actually an Rmodule and that the natural projection

π : M → M/N

is an R-module homomorphism.

Definition 1.8. Let f : M → N be an R-module homomorphism between R-modules. Then we define its kernel as follows.

Ker(f) = f^{−1}(0) = {m ∈ M; f(m) = 0}.

The kernel and the image of an R-module homomorphism f are Rmodules.

Theorem 1.9. Let f : M → N be an R-module homomorphism between R-modules. Then

M/ Ker(f) ≅ f(N).

Definition 1.10. Let R be a ring. An “sequence”

M_{1} →^{f} M_{2} →^{g} M_{3}

is said to be an exact sequence of R-modules if the following conditions are satisfied

(Exact1) M_{1}, M_{2} are R-modules. (Exact2) f, g are R-module homomorphisms. (Exact3) Ker(g) = Image(f).

For any R-submodule N of an R-module M, we have the following exact sequence.

0 → N → M → M/N → 0

Exercise 1.1. Compute the following modules.

(1) Hom_{ℤ}(ℤ/3ℤ, ℤ).

(2) Hom_{ℤ}(ℚ, ℤ).

(3) Hom_{ℤ}(ℚ, ℤ/5ℤ).