  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 cristalline cohomology. In this first paper we
mainly discuss the theory for smooth projective varieties over perfect fields
of charatceristic p. We also compare our theory with other existing structures
like the F-zips of Moonen and Wedhorn and the displays of Langer and Zink.
