There exists a
p
p
-local spectrum
T
(
m
)
T(m)
with
B
P
∗
(
T
(
m
)
)
BP_{*}(T(m))
=
B
P
∗
[
t
1
,
…
,
t
m
]
\!BP_{*}[t_{1},\dots ,t_{m}]
. Its Adams-Novikov
E
2
E_2
-term is isomorphic to
Ext
Γ
(
m
+
1
)
(
B
P
∗
,
B
P
∗
)
,
\begin{equation*} \text {Ext}_{\Gamma (m+1)}(BP_*,BP_*), \end{equation*}
where
Γ
(
m
+
1
)
=
B
P
∗
(
B
P
)
/
(
t
1
,
…
,
t
m
)
=
B
P
∗
[
t
m
+
1
,
t
m
+
2
,
…
]
.
\begin{equation*} \Gamma (m+1) = BP_{*} (BP)/ \left (t_{1},\dots ,t_{m}\right ) = BP_{*}[t_{m+1},t_{m+2},\dots ]. \end{equation*}
In this paper we determine the groups
Ext
Γ
(
m
+
1
)
1
(
B
P
∗
,
v
n
−
1
B
P
∗
/
I
n
)
\begin{equation*} \text {Ext}^{1}_{\Gamma (m+1)} (BP_{*},v_{n}^{-1}BP_{*}/I_{n}) \end{equation*}
for all
m
,
n
>
0
m,n>0
. Its rank ranges from
n
+
1
n+1
to
n
2
n^{2}
depending on the value of
m
m
.