\(\textbf{GL}\)-algebras in positive characteristic I: the exterior algebra

We study the category of \(\textbf{GL}\) $\mathbf{GL}$ -equivariant modules over the infinite exterior algebra in positive characteristic. Our main structural result is a shift theorem à la Nagpal. Using this, we obtain a Church–Ellenberg type bound for the Castelnuovo–Mumford regularity. We also prove finiteness results for local cohomology.