Commutative algebra

Yoshifumi Tsuchimoto

DEFINITION 05.1   Let $A$ be a ring. Let $I$ be an ideal of $A$ . The $I$ -adic topology on $A$ is a topology defined by introducing for each $a\in A$ , $\{a+I^n\}_{n=1}^\infty$ as the neighbourhood base.

It is easy to see that the $I$ -adic topology is given by a quasi-metric defined by

$$d(a,b)=\inf\{ \frac{1}{2^n} ; a-b \in I^n\}$$

PROPOSITION 05.2   Let $A$ be a ring. Let $I$ be an ideal of $A$ . We equip $A$ with the $I$ -adic topology. Then $A$ is Hausdorff if and only if

$$\bigcap_n I^n =0.$$

If this is the case, the completion of $A$ is equal to $\varprojlim A/I^n$ . Thus $A$ is complete Hausdorff if and only if a cannonically defined map

$$A \to \varprojlim A/I^n$$

is an isomorphism.

EXAMPLE 05.3   Let $p$ be a prime number. The ring $\mathbb{Z}$ of rational integers equipped with the $p \mathbb{Z}$ -adic topology is Hausdorff. Its completion is denoted by $\mathbb{Z}_p$ and is called the ring of $p$ -adic integers.

DEFINITION 05.4   Let $A$ be a ring. Let $B$ be an $A$ -algebra. Let $I$ be an ideal of $B$ . We equip $B$ with the $I$ -adic topology. $B$ is $I$ -smooth over $A$ if for any $A$ -algebra $C$ , any ideal $N$ of $C$ satisfying $N^2=0$ and any $A$ -algebra homomorphism $u:B\to C/N$ which is continuous with respect to the discrete topology of $C/N$ , there exists a lifting $v:B\to C$ of $u$ to $C$ , as an $A$ -algebra homormophism.

DEFINITION 05.5   Let $A$ be a ring. Let $B$ be an $A$ -algebra. Let $I$ be an ideal of $B$ . We equip $B$ with the $I$ -adic topology. $A$ -algebra $B$ is $I$ -unramified over $A$ if for any $A$ -algebra $C$ , any ideal $N$ of $C$ satisfying $N^2=0$ and any $A$ -algebra homomorphism $u:B\to C/N$ which is continuous with respect to the discrete topology of $C/N$ , there is at most one lifting $v:B\to C$ of $u$ to $C$ , as an $A$ -algebra homormophism.

DEFINITION 05.6   An $A$ -algebra $B$ is $I$ -étale over $A$ if it is both $I$ -smooth and $I$ -unramified.

Note that the conditions $I$ -smooth/unramified/étale become weaker if we take $I$ larger.

In the ``strongest'' case where $I=0$ , the continuity of the homomorphism $u$ is automatic (any homomorphism is continuous.) 0 -smoothness (respectively, 0 -unramifiedness, respectively, 0 -étale-ness) is also refered to as formal smoothness (respectively, formal unramifiedness, respectively, formal étale-ness).