Let
R
R
be a commutative ring,
S
S
an
R
R
-algebra,
H
H
a Hopf
R
R
algebra, both finitely generated and projective as
R
R
-modules, and suppose
S
S
is an
H
H
-object, so that
H
∗
=
Hom
R
(
H
,
R
)
{H^{\ast }} = {\operatorname {Hom} _R}(H,R)
acts on
S
S
via a measuring. Let
I
I
be the space of left integrals of
H
∗
{H^{\ast }}
. We say
S
S
has normal basis if
S
≅
H
S \cong H
as
H
∗
{H^{\ast }}
modules, and
S
S
has local normal bases if
S
p
≅
H
p
{S_p} \cong {H_p}
as
H
p
∗
H_p^{\ast }
-modules for all prime ideals
p
p
of
R
R
. When
R
R
is a perfect field,
H
H
is commutative and cocommutative, and certain obvious necessary conditions on
S
S
hold, then
S
S
has normal basis if and only if
I
S
=
R
=
S
H
∗
IS = R = {S^{{H^{\ast }}}}
. If
R
R
is a domain with quotient field
K
K
,
H
H
is cocommutative, and
L
=
S
⊗
R
K
L = S \otimes {}_RK
has normal basis as
(
H
∗
⊗
K
)
({H^{\ast }} \otimes K)
-module, then
S
S
has local normal bases if and only if
I
S
=
R
=
S
H
∗
IS = R = {S^{{H^{\ast }}}}
.