Frobenius gauges and a new theory of $p$-torsion sheaves in characteristic $p$

Abstract

We develop a new cohomology theory in characteristic $p>0$, the so called $F$-gauge cohomology, a cohomology with values in the category of so-called $F$-gauges, which refines the crystalline cohomology. We mainly discuss the theory for smooth projective varieties over perfect fields of characteristic $p$. We also compare our theory with other existing structures like the $F$-zips of B. Moonen and T. Wedhorn [Int. Math. Res. Not. 2004, No. 72, 3855–3903 (2004; Zbl 1084.14023)] and the displays of A. Langer and T. Zink [Doc. Math. 12, 147–191 (2007; Zbl 1122.14016)].