For a commutative noetherian algebra
B
B
over a perfect field
A
A
regularity is equivalent to the flatness of
Ω
B
|
A
{\Omega _{B|A}}
plus
H
1
(
A
,
B
,
B
)
=
0
{H_1}(A,B,B) = 0
(simplicial homology). In characteristic 0 the homological condition is superfluous.