In algebra, a commutative k-algebra A is said to be 0-smooth if it satisfies the following lifting property: given a k-algebra C, an ideal N of C whose square is zero and a k-algebra map
u
:
A
→
C
/
N
, there exists a k-algebra map
v
:
A
→
C
such that u is v followed by the canonical map. If there exists at most one such lifting v, then A is said to be 0-unramified (or 0-neat). A is said to be 0-étale if it is 0-smooth and 0-unramified.
A finitely generated k-algebra A is 0-smooth over k if and only if Spec A is a smooth scheme over k.
A separable algebraic field extension L of k is 0-étale over k. The formal power series ring
k
[
[
t
1
,
…
,
t
n
]
]
is 0-smooth only when
char
k
=
p
>
0
and
[
k
:
k
p
]
<
∞
(i.e., k has a finite p-basis.)
Let B be an A-algebra and suppose B is given the I-adic topology, I an ideal of B. We say B is I-smooth over A if it satisfies the lifting property: given an A-algebra C, an ideal N of C whose square is zero and an A-algebra map
u
:
B
→
C
/
N
that is continuous when
C
/
N
is given the discrete topology, there exists an A-algebra map
v
:
B
→
C
such that u is v followed by the canonical map. As before, if there exists at most one such lift v, then B is said to be I-unramified over A (or I-neat). B is said to be I-étale if it is I-smooth and I-unramified. If I is the zero ideal and A is a field, these notions coincide with 0-smooth etc. as defined above.
A standard example is this: let A be a ring,
B
=
A
[
[
t
1
,
…
,
t
n
]
]
and
I
=
(
t
1
,
…
,
t
n
)
.
Then B is I-smooth over A.
Let A be a noetherian local k-algebra with maximal ideal
m
. Then A is
m
-smooth over k if and only if
A
⊗
k
k
′
is a regular ring for any finite extension field
k
′
of k.
