  Determining the matrix multiplication exponent $\omega$ is one of the
greatest open problems in theoretical computer science. We show that it is
impossible to prove $\omega = 2$ by starting with structure tensors of modules
of fixed degree and using arbitrary restrictions. It implies that the same is
impossible by starting with $1_A$-generic non-diagonal tensors of fixed size
with minimal border rank. This generalizes the work of Bl\"aser and Lysikov
[3]. Our methods come from both commutative algebra and complexity theory.
