# Commutative algebra

Yoshifumi Tsuchimoto

DEFINITION 05.1   Let be a ring. Let be an ideal of . The -adic topology on is a topology defined by introducing for each , as the neighbourhood base.

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

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

If this is the case, the completion of is equal to . Thus is complete Hausdorff if and only if a cannonically defined map

is an isomorphism.

EXAMPLE 05.3   Let be a prime number. The ring of rational integers equipped with the -adic topology is Hausdorff. Its completion is denoted by and is called the ring of -adic integers.

DEFINITION 05.4   Let be a ring. Let be an -algebra. Let be an ideal of . We equip with the -adic topology. is -smooth over if for any -algebra , any ideal of satisfying and any -algebra homomorphism which is continuous with respect to the discrete topology of , there exists a lifting of to , as an -algebra homormophism.

DEFINITION 05.5   Let be a ring. Let be an -algebra. Let be an ideal of . We equip with the -adic topology. -algebra is -unramified over if for any -algebra , any ideal of satisfying and any -algebra homomorphism which is continuous with respect to the discrete topology of , there is at most one lifting of to , as an -algebra homormophism.

DEFINITION 05.6   An -algebra is -étale over if it is both -smooth and -unramified.

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

In the strongest'' case where , the continuity of the homomorphism 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).