Length complexity of tensor products

In this paper we introduce techniques to gauge the torsion of the tensor product $A{\otimes }_{R}B$ of two finitely generated modules over a Noetherian ring $R$. The outlook is very different from the study of the rigidity of Tor carried out in the work of Auslander and other authors. Here the emphasis in on the search for bounds for the torsion part of $A{\otimes }_{R}B$ in terms of global invariants of $A$ and of $B$ in special classes of modules: vector bundles and modules of dimension at most three.