Linear growth of homological regularity for FI^2-modules

We study total-degree homological regularity of modules over the product of two categories of finite sets and injections. For an arbitrary commutative coefficient ring, we prove that a module generated in degrees at most d and related in degrees at most r >= 1 has regularity at most 2d+2r-1. No finiteness of the number of generators or noetherian hypothesis is imposed. A family consisting of a principal projective and a rectangular truncation has regularity max{d,2r-2}. Consequently the optimal uniform upper bound has linear growth in d+r. The coefficient in our upper bound is not asserted to be optimal. The main step estimates the remaining one-variable regularity of the kernel of a coordinate shift before estimating its total regularity. We give proofs of the homological tools and compute the lower-bound family over every nonzero commutative ring.