Let
(
A
,
m
,
k
)
(A, \mathfrak {m}, k)
be a complete intersection of codimension
c
c
, and let
k
~
\tilde {k}
be the algebraic closure of
k
k
. We show that every homogeneous algebraic subset of
k
~
c
\tilde {k}^c
is the cohomological support variety of an
A
A
-module, and that the projective variety of a complete indecomposable maximal Cohen–Macaulay
A
A
-module is connected.