  In this paper we study n-inverse pairs of operators on the tensor product of
Banach spaces. In particular we show that an n-inverse pair of elementary
tensors of operators on the tensor product of two Banach spaces can arise only
from l- and m-inverse pairs of operators on the individual spaces. This gives a
converse to a result of Duggal and M\"uller, and proves a conjecture of the
second named author. Our proof uses techniques from algebraic geometry, which
generalize to other relations among operators in a tensor product. We apply
this theory to obtain results for n-symmetries in a tensor product as well.
