Commutative algebra

Yoshifumi Tsuchimoto

DEFINITION 02.1   Let $A$ be a commutative ring. Let $S$ be its subset. We say that $S$ is multiplicative if

1. $1\in S$
2. $x,y \in S \implies x y \in S$

holds.

DEFINITION 02.2   Let $S$ be a multiplicative subset of a commutative ring $A$ . Then we define $A[S^{-1}]$ as

$$A[\{X_s ; s \in S\}]/(\{ s X_s -1; s \in S\})$$

where in the above notation $X_s$ is a indeterminate prepared for each element $s \in S$ .) We denote by $\iota_S$ a canonical map $A\to A[S^{-1}]$ .

LEMMA 02.3   Let $S$ be a multiplicative subset of a commutative ring $A$ . Then the ring $B=A[S^{-1}]$ is characterized by the following property:

Let $C$ be a ring, $\varphi:A\to C$ be a ring homomorphism such that $\varphi(s)$ is invertible in $C$ for any $s \in S$ . Then there exists a unique ring homomorphism $\psi=\phi[S^{-1}]:B\to C$ such that

$$\varphi=\psi \circ \iota_S$$

holds.

COROLLARY 02.4   Let $S$ be a multiplicative subset of a commutative ring $A$ . Let $I$ be an ideal of $A$ given by

$$I=\{ x \in I; \exists s \in S$$    such that $$s x=0\}$$

Then $I$ is an ideal of $A$ . Let us put $\bar{A}=A/I$ , $\pi:A\to \bar{A}$ the canonical projection. Then:

1. $\bar{S}=\pi(S)$ is multiplicatively closed.

2. We have
   $$A[S^{-1}]\cong\bar{A}[\bar{S}^{-1}]$$

3. $\iota_{\bar{S}}: \bar{A}\to \bar{A}[\bar{S}^{-1}]$ is injective.

There is another description of $A[S^{-1}]$ . Namely, We consider an equivalence relateion $\sim_S$ on a set $S \times A$ by

$$(s_1,a_1)\sim_S (s_2,a_2) \iff t(s_1 a_2 -s_2 a_1)=0 (\exists t \in S)$$

We call the quotient space space $S\times A/\sim_S$ as $S^{-1}A$ . The equivalence class of $(s,a)\in S\times A$ in $S^{-1}A$ is denoted by $s^{-1} a$ . Then it is easy to introduce a ring structure of $S^{-1}A$ and see that $S^{-1}A$ actually satisfies the universal property of $A[S^{-1}]$ . We thus have a canonical isomorphism $S^{-1}A\cong A[S^{-1}]$ .

EXAMPLE 02.5   $A_f=A[S^{-1}]$ for $S=\{1,f,f^2,f^3,f^4,\dots\}$ . The total ring of quotients $Q(A)$ is defined as $A[S^{-1}]$ for

$$S=\{ x \in A; x$$    is not a zero divisor of A$$\}.$$

When $A$ is an integral domain, then $Q(A)$ is the field of quotients of $A$ .

DEFINITION 02.6   Let $A$ be a commutative ring. Let $\mathfrak{p}$ be its prime ideal. Then we define the localization of $A$ with respect to $\mathfrak{p}$ by

$$A_\mathfrak{p}=A[ (A\setminus \mathfrak{p})^{-1}]$$

DEFINITION 02.7   Let $S$ be a multiplicative subset of a commutative ring $A$ . Let $M$ be an $A$ -module we may define $S^{-1}M$ as

$$\{ (m/s); m\in M , s\in S\} / \sim$$

where the equivalence relation $\sim$ is defined by

$$(m_1/s_1)\sim (m_2/s_2) \iff t (m_1 s_2 -m_2 s_1)=0 \quad (\exists t \in S).$$

We may introduce a $S^{-1}A$ -module structure on $S^{-1}M$ in an obvious manner.

$S^{-1}M$ thus constructed satisfies an universality condition which the reader may easily guess.

By a universality argument, we may easily see the following proposition.

PROPOSITION 02.8   Let $A$ be a commutative ring. Let $S$ be a multiplicative subet of $A$ . Let $M$ be an $A$ -module. Then we have an isomorphism

$$S^{-1} A \otimes_A M \cong S^{-1}M$$

of $S^{-1}A$ -modules.

PROPOSITION 02.9   Let $A$ be a commutative ring. Let $S$ be a multiplicative subet of $A$ . Then the natural homomorphism $A\to S^{-1} A$ is flat.