Homological growth of Artin kernels in positive characteristic

We prove an analogue of the L\"uck Approximation Theorem in positive characteristic for certain residually finite rationally soluble (RFRS) groups including right-angled Artin and Bestvina--Brady groups. Specifically, we that mod $p$ homology growth equals dimension group with coefficients a universal division ring this is independent choice residual chain. For general RFRS obtain inequality between invariants. also consider number applications to fibring, amenable category, minimal volume entropy.