DEFINITION 03.1
Let
be a ring. A ring
is called an
-algebra if
there is given a distinguished ring homomorphism
(which is called the structure morphism)
from
to
.

DEFINITION 03.2
Let
be a commutative ring. Let
be an
-algebra.
Let
be an
-module.
A map
is called an
-derivation if it satisfies the
following conditions.

is
-linear.

.

PROPOSITION 03.3For any algebra
over a ring
,
There exists an universal derivation
.

DEFINITION 03.4
The module
is called the module of differentials of
over
.

DEFINITION 03.5
An algebra
over a ring
is called unramified over
if
.
is called étale over
if it is unramified and flat.

LEMMA 03.6Let
be a commutative ring. Let
be its multiplicative subset.
Then
is étale over
.