This is joint work (in progress) with Marcin Chałupnik.

Let (X,s) be a difference scheme, by which we mean a scheme X with a fixed morphism s

from X to X. We develop a theory of difference sheaves over (X,s). Actually, we obtain two categories: of left difference sheaves, and of right difference sheaves. These categories are equivalent in the case when the morphism s is an automorphism, but, in general, they behave quite differently. Each of these categories is still a Grothendieck category, in particular they enjoy good cohomology theories (including the Cech versions). Analogously to the classical case, the first difference cohomology group classifies difference torsors, which specializes to the context considered by Bachmayr and Wibmer. Another specialization of our theory is a cohomological description of the (defined in a natural way) difference Picard group.