  The Total Rank Conjecture is a coarser version of the
Buchsbaum-Eisenbud-Horrocks conjecture which, loosely stated, predicts that
modules with large annihilators must also have ``large'' syzygies. This
conjecture was proved by the second author for rings of odd characteristic by
taking advantage of the Adams operations on the category of perfect complexes
with finite length homology. In this paper, we prove a stronger form of the
Total Rank Conjecture for rings of characteristic two. This result may be seen
as evidence that the Generalized Total Rank Conjecture (which is known to be
false in odd characteristic) actually holds in characteristic two. We also
formulate a meaningful extension of the Total Rank Conjecture over non
Cohen-Macaulay rings and prove that this conjecture holds for any ring
containing a field (independent of characteristic).
