  We study GL-equivariant modules over the infinite variable polynomial ring $S
= k[x_1, x_2, ..., x_n, ...]$ with $k$ an infinite field of characteristic $p >
0$. We extend many of Sam--Snowden's far-reaching results from characteristic
zero to this setting. For example, while the Castelnuovo--Mumford regularity of
a finitely generated GL-equivariant $S$-module need not be finite in positive
characteristic, we show that the resolution still has finitely many "linear
strands of higher slope".
  The crux of this paper is two technical results. The first is an extension to
positive characteristic of Snowden's recent linearization of Draisma's
embedding theorem which we use to study the generic category of $S$-modules.
The second is a Nagpal-type "shift theorem" about torsion $S$-modules for which
we introduce certain categorifications of the Hasse derivative. These two
results together allow us to obtain explicit generators for the derived
category. In a follow-up paper, we also use these results to prove finiteness
results for local cohomology modules.
