In Integral Grothendieck-Riemann-Roch theorem, Pappas mentions that Mumford computed the determinant of cohomology of $f:X\to S$ a relative curve integrally, and thus proved an integral version of GRR in that setting over a base of arbitrary characteristic.
Does anyone know where this is written down? I asked Pappas himself, but he'd misplaced the reference. I'm also interested in whether there are other known calculations of the determinant of cohomology; I'd guess, based on this result of Mumford, that maybe for $f$ a (principally polarized?) abelian scheme, something similar is possibly in that setting. Without knowing what Mumford's original argument is, however, I'm at a loss.
