Adic finiteness: Bounding homology and applications

We prove a versions of amplitude inequalities of Iversen, Foxby and Iyengar, and Frankild and Sather-Wagstaff that replace finite generation conditions with adic finiteness conditions. As an application, we prove that a local ring $R$ of prime characteristic is regular if and only if for some proper ideal $\mathfrak b$ the derived local cohomology complex $\mathbf{R}\Gamma_{\mathfrak{b}}(R)$ has finite flat dimension when viewed through some positive power of the Frobenius endomorphism.
Comment: 25 pages. part 3 of a series with arXiv:1401.6925, arXiv:1506.07052, arXiv:1602.03224, arXiv:1602.03226, and arXiv:1602.03227. comments welcome. v.2 has updated references and updated URL for Sather-Wagstaff