Let
k
k
be a field and let
G
G
be a finite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology
γ
G
∈
H
H
3
,
−
1
H
^
∗
(
G
,
k
)
\gamma _G\in H\!H^{3,-1}\hat H^*(G,k)
with the following property. Given a graded
H
^
∗
(
G
,
k
)
\hat H^*(G,k)
-module
X
X
, the image of
γ
G
\gamma _G
in
Ext
H
^
∗
(
G
,
k
)
3
,
−
1
(
X
,
X
)
\operatorname {Ext}^{3,-1}_{\hat H^*(G,k)}(X,X)
vanishes if and only if
X
X
is isomorphic to a direct summand of
H
^
∗
(
G
,
M
)
\hat H^*(G,M)
for some
k
G
kG
-module
M
M
. The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a differential graded algebra
A
A
, there is also a canonical element of Hochschild cohomology
H
H
3
,
−
1
H
∗
(
A
)
H\!H^{3,-1}H^*(A)
which is a predecessor for these obstructions.