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# Commutative algebra

**Yoshifumi Tsuchimoto**

Let us recall the universality of polynomial algebras.

As a corollary, we see:

**LEMMA 08.3**
*Let
be a ring. Let
be an
-algebra.
Let
be a finitely generated ideal of
.
Let us denote by
(respectively,
)
the completion of
(respectively,
) with respect to the
-adic
topology.
Then
is
-smooth over
if and only
if
is
-smooth over
.*

**COROLLARY 08.4**
*Let
be a ring. Then
is
-smooth over
for
.*
Note.
In general,
is not 0
-smooth over
. See [1] and
the literatures cited there.

The following theorem says that the converse is true if the ring
is 0
-smooth.

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2011-07-21