Given a compact Lie group
G
G
and a commutative orthogonal ring spectrum
R
R
such that
R
[
G
]
∗
=
π
∗
(
R
∧
G
+
)
R[G]_* = \pi _*(R \wedge G_+)
is finitely generated and projective over
π
∗
(
R
)
\pi _*(R)
, we construct a multiplicative
G
G
-Tate spectral sequence for each
R
R
-module
X
X
in orthogonal
G
G
-spectra, with
E
2
E^2
-page given by the Hopf algebra Tate cohomology of
R
[
G
]
∗
R[G]_*
with coefficients in
π
∗
(
X
)
\pi _*(X)
. Under mild hypotheses, such as
X
X
being bounded below and the derived page
R
E
∞
RE^\infty
vanishing, this spectral sequence converges strongly to the homotopy
π
∗
(
X
t
G
)
\pi _*(X^{tG})
of the
G
G
-Tate construction
X
t
G
=
[
E
G
~
∧
F
(
E
G
+
,
X
)
]
G
X^{tG} = [\widetilde {EG}\wedge F(EG_+, X)]^G
.