COMMUTATIVE ALGEBRA

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.

(Ring-1) R is an additive group with respect to the addition. (Ring-2) distributive law holds. Namely, we have

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

(Ring-3) The multiplcation is associative. (Ring-4) R has a multiplicative unit.

In this lectuer we are mainly interested in commutative rings, that means, rings on which the multiplication satisfies the commutativity law.

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. A map R → S from a unital associative ring R to another unital associative ring S is said to be ring homomorphism if it satisfies the following conditions.

(Ringhom-1) f(a + b) = f(a) + f(b) (Ringhom-2) f(ab) = f(a)f(b) (Ringhom-3) f(1_{R}) = 1_{S}

Definition 1.3. Let R be a unital associative ring. An R-module M is an additive group M with R-action

R × M → M

which satisfies (Mod-1) (r_{1}r_{2}).m = r_{1}.(r_{2}.m) (∀r_{1}, ∀r_{2} ∈ R, ∀m ∈ M) (Mod-2) 1.m = m (∀m ∈ M) (Mod-3) (r_{1} + r_{2}).m = r_{1}.m + r_{2}.m (∀r_{1}, ∀r_{2} ∈ R, ∀m ∈ M). (Mod-4) r.(m_{1} + m_{2}) = r.m_{1} + r.m_{2} (∀r ∈ R, ∀m_{1}, ∀m_{2} ∈ M).

Example 1.4. 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 1.5. 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. Furthermore, when R is a commutative ring, then it has a structure of an R-module.

Definition 1.6. 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 1.7. 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.

Definition 1.8. 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 1.9. 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 1.10. Let f : M → N be an R-module homomorphism between R-modules. Then

M/ Ker(f) ≅ f(N).

Definition 1.11. 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).

Definition 1.12. Let A be an associative unital (but not necessarily commutative) ring. Let L be a right A-module. Let M be a left Amodule. For any (ℤ-)module N, an map

φ : L × M → N

is called an A-balanced biadditive map if (1) φ(x_{1} + x_{2}, y) = φ(x_{1}, y) + φ(x_{2}, y) (∀x_{1}, ∀x_{2} ∈ L, ∀y ∈ M).

(2) φ(x, y_{1} + y_{2}) = φ(x, y_{1}) + φ(x, y_{2}) (∀x ∈ L, ∀y_{1}, ∀y_{2} ∈ M).

(3) φ(xa, y) = φ(x, ay) (∀x ∈ L, ∀y ∈ M, ∀a ∈ A).

Proposition 1.13. Let A be an associative unital (but not necessarily commutative) ring. Then for any right A-module L and for any left A-module M, there exists a (ℤ-)module X_{L,M} together with a Abalanced map

φ_{0} : L × M → X_{L,M}

which is universal amoung A-balanced maps.

Definition 1.14. We employ the assumption of the proposition above. By a standard argument on universal objects, we see that such object is unique up to a unique isomorphism. We call it the tensor product of L and M and denote it by

L ⊗_{A} M.

Lemma 1.15. Let A be an associative unital ring. Then:

(1) A ⊗_{A} M ≅ M.

(2) (L_{1} ⊕ L_{2}) ⊗_{A} M ≅ (L_{1} ⊗ M) ⊕ (L_{2} ⊗_{A} M).

(3) For any left A-module M, the functor L ↦ L ⊗_{A} M is a right exact functor. Namely, for any exact sequence

0 → L_{1} → L_{2} → L_{3} → 0,

the sequence

L_{1} ⊗_{A} M → L_{2} ⊗_{A} M → L_{3} ⊗_{A} M → 0,

is also exact.

(4) For any right ideal J of A and for any A-module M, we have

(A/J) ⊗_{A} M ≅ M/J.M

In particular, if the ring A is commutative, then for any ideals I, J of A, we have

(A/I) ⊗_{A} (A/J) ≅ A/(I + J)

Exercise 1.2. Compute (ℤ/3ℤ) ⊗_{ℤ} (ℤ/4ℤ) and ℚ ⊗_{ℤ} (ℤ/3ℤ).

Definition 1.16. A left A-module M is said to be flat if L ↦ L ⊗_{A} M is an exact functor. Namely, for any exact sequence

0 → L_{1} → L_{2} → L_{3} → 0,

of left A-modules, the sequence

0 → L_{1} ⊗_{A} M → L_{2} ⊗_{A} M → L_{3} ⊗_{A} M → 0,

is also exact.

**The following two facts may give some intuitive idea of what flatness means.

Theorem 1.17. If A is a Noetherian ring and M is a finitelygenerated R-module, then M is flat over A if and only if the associated sheaf M̃ on Spec(A) is locally free.

Theorem 1.18. [1, Theorem 23.1+Theorem 15.1] Let (A, 𝔪_{A}) be a regular local ring. Let (B, 𝔪_{B}) be a Cohen-Macaulay local ring. Let φ : A → B be a local ring homomorphism.We set

F = B ⊗_{A} k(𝔪_{A}) = B/𝔪_{A}B

for the fiber ring of φ over 𝔪_{A}. Then an equality

dim B = dim A + dim F

holds if and only if B is flat over A.

**

References

[1] H. Matsumura, Commutative ring theory, Cambridge university press, 1986.