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# 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

**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).

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2011-06-24