The total rank conjecture in characteristic 2

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 article, we prove a stronger form of the Total Rank Conjecture for rings of characteristic 2. 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 2. We also formulate a meaningful extension of the Total Rank Conjecture over non-Cohen–Macaulay rings and prove that this conjecture holds for any algebra over a field (independent of characteristic).