On a separation of orbits in the module variety for string special biserial algebras

Given a pair M,M′M,M′ of finite-dimensional modules over a string special biserial algebra ΛΛ, a fully verifiable criterion, expressed in terms of a finite set of simple linear algebra invariants, deciding if MM and M′M′ lie in the same orbit in module variety, equivalently, if MM and M′M′ are isomorphic, is formulated and proved.