Commutative algebra

Yoshifumi Tsuchimoto

DEFINITION 01.1   A (unital associative) ring is a set $R$ equipped with two binary operations (addition (``+'') and multiplication (``$\cdot$ '')) 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

$$a(b+c)=ab + bc,\quad (a+b)c=ac+bc \qquad (\forall a,\forall b,\forall c\in R).$$

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.

• $a + 0_R =a$ ,      $0_R + a =a$ $\forall a \in R$ .
• $a \cdot 1_R =a$ ,      $1_R \cdot a =a$ $\forall a \in R$ .

When no confusion arises, we omit the subscript `${}_R$ ' and write $0,1$ instead of $0_R,1_R$ .

DEFINITION 01.2   A map $R\to S$ 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 01.3   Let $R$ be a unital associative ring. An $R$ -module $M$ is an additive group $M$ with $R$ -action

$$R\times M\to M$$

which satisfies

Mod-1.

$(r_1 r_2). m= r_1.(r_2.m)$ $\quad (\forall r_1, \forall r_2\in R, \forall m\in M)$

Mod-2.

$1.m=m$ $\quad (\forall m \in M)$

Mod-3.

$(r_1+r_2).m=r_1.m+r_2.m$ $\quad (\forall r_1,\forall r_2\in R, \forall m \in M)$ .

Mod-4.

$r.(m_1+m_2)=r.m_1+r.m_2$ $\quad (\forall r\in R, \forall m_1,\forall m_2 \in M)$ .

EXAMPLE 01.4   Let us give some examples of $R$ -modules.

1. If $k$ is a field, then the concepts ``$k$ -vector space" and ``$k$ -module'' are identical.
2. Every abelian group is a module over the ring of integers $\mathbb{Z}$ in a unique way.

DEFINITION 01.5   Let $M,N$ be modules over a ring $R$ . Then a map $f:M\to N$ 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 $\operatorname{Hom}_R (M,N)$ . 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 01.6   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 $j:M\to N$ is an $R$ -module homomorphism.

DEFINITION 01.7   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 $j:N\to M$ is an $R$ -module homomorphism.

DEFINITION 01.8   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

$$M/N=M/\sim_N$$

where the equivalence relation $\sim_N$ is defined as follows:

$$m_1 \sim_N m_2 \quad \iff \quad m_1-m_2\in N.$$

It may be shown that the quotient $M/N$ so defined is actually an $R$ -module and that the natural projection

$$\pi: M\to M/N$$

is an $R$ -module homomorphism.

DEFINITION 01.9   Let $f:M\to N$ be an $R$ -module homomorphism between $R$ -modules. Then we define its kernel as follows.

$$\operatorname{Ker}(f)=f^{-1}(0)=\{ m\in M; f(m)=0\}.$$

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

THEOREM 01.10   Let $f:M\to N$ be an $R$ -module homomorphism between $R$ -modules. Then

$$M/\operatorname{Ker}(f) \cong f(N).$$

DEFINITION 01.11   Let $R$ be a ring. An ``sequence''

$$M_1\overset {f}{\to} M_2 \overset{g}{\to} M_3$$

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.

$\operatorname{Ker}(g)=\operatorname{Image}(f)$ .

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

$$0\to N\to M\to M/N \to 0$$

EXERCISE 01.1   Compute the following modules.

1. $\operatorname{Hom}_\mathbb{Z}(\mathbb{Z}/3\mathbb{Z},\mathbb{Z})$ .
2. $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Z})$ .
3. $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Z}/5\mathbb{Z})$ .

DEFINITION 01.12   Let $A$ be an associative unital (but not necessarily commutative) ring. Let $L$ be a right $A$ -module. Let $M$ be a left $A$ -module. For any ( $\mathbb{Z}$ -)module $N$ , an map

$$\varphi: L\times M \to N$$

is called an $A$ -balanced biadditive map if

1. $\varphi(x_1+x_2,y)=\varphi(x_1,y)+\varphi(x_2,y)$      $(\forall x_1,\forall x_2\in L, \forall y\in M)$ .
2. $\varphi(x,y_1+y_2)=\varphi(x,y_1)+\varphi(x,y_2)$      $(\forall x\in L, \forall y_1,\forall y_2\in M)$ .
3. $\varphi(x a, y)=\varphi(x, a y)$      $(\forall x\in L, \forall y\in M, \forall a\in A)$ .

PROPOSITION 01.13   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 ( $\mathbb{Z}$ -)module $X_{L,M}$ together with a $A$ -balanced map

$$\varphi_0: L\times M \to X_{L,M}$$

which is universal amoung $A$ -balanced maps.

DEFINITION 01.14   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

$$L\otimes_A M.$$

LEMMA 01.15   Let $A$ be an associative unital ring. Then:

1. $A\otimes_A M \cong M$ .
2. $(L_1\oplus L_2) \otimes_A M \cong (L_1 \otimes M) \oplus (L_2 \otimes_A M ).$
3. For any left $A$ -module $M$ , the functor $L\mapsto L \otimes_A M$ is a right exact functor. Namely, for any exact sequence
   $$0\to L_1\to L_2 \to L_3 \to 0,$$
   the sequence
   $$L_1 \otimes_A M\to L_2 \otimes_A M \to L_3 \otimes_A M\to 0,$$
   is also exact.
4. For any right ideal $J$ of $A$ and for any $A$ -module $M$ , we have
   $$(A/J) \otimes_A M \cong M/ J.M$$

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

$$(A/I) \otimes_A (A/J) \cong A/ (I+ J)$$

EXERCISE 01.2   Compute $(\mathbb{Z}/3 \mathbb{Z}) \otimes_\mathbb{Z}(\mathbb{Z}/4\mathbb{Z})$ and $\mathbb{Q}\otimes _\mathbb{Z}(\mathbb{Z}/3\mathbb{Z})$ .

DEFINITION 01.16   A left $A$ -module $M$ is said to be flat if $L\mapsto L \otimes_A M$ is an exact functor. Namely, for any exact sequence

$$0\to L_1\to L_2 \to L_3 \to 0,$$

of left $A$ -modules, the sequence

$$0\to L_1 \otimes_A M\to L_2 \otimes_A M \to L_3 \otimes_A M\to 0,$$

is also exact.

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

THEOREM 01.17   If $A$ is a Noetherian ring and $M$ is a finitely-generated $R$ -module, then $M$ is flat over $A$ if and only if the associated sheaf $\tilde{M}$ on $\operatorname{Spec}(A)$ is locally free.

THEOREM 01.18   [1, Theorem 23.1+Theorem 15.1] Let $(A,\mathfrak{m}_A)$ be a regular local ring. Let $(B,\mathfrak{m}_B)$ be a Cohen-Macaulay local ring. Let $\varphi: A\to B$ be a local ring homomorphism.We set

$$F=B\otimes_A k(\mathfrak{m}_A) =B/\mathfrak{m}_A B$$

for the fiber ring of $\varphi$ over $\mathfrak{m}_A$ . Then an equality

$$\dim B=\dim A+ \dim F$$

holds if and only if $B$ is flat over $A$ .

**