Total rank conjecture for short complexes

For a Noetherian commutative ring $R$, let $P$ be a short complex, meaning a non-exact complex of finite-rank projective $R$-modules concentrated in degrees $[0,c]$, where

$$c=\operatorname{codim}_R(H(P)):=\max_j\{\operatorname{codim}_R H_j(P)\}.$$

For a local ring $R$, a tiny complex is a short complex with $\operatorname{codim}_R(H(P))=\operatorname{dim}(R)$. Total rank conjecture for short complexes. For every short complex $P$ over $R$,

$$\sum_i\operatorname{rank}_R(P_i)\geq 2^{\operatorname{codim}_R(H(P))}.$$

In particular, for every tiny complex over a local ring,

$$\sum_i\operatorname{rank}_R(P_i)\geq 2^{\operatorname{dim}R}.$$

This generalizes the original total rank conjecture to non-Cohen–Macaulay settings. The paper proves it for rings containing a field; it remains open in general mixed characteristic.