Revisiting the de Rham-Witt complex

The goal of this paper is to offer a new construction the de Rham-Witt complex smooth varieties over perfect fields characteristic $p$. We introduce category cochain complexes equipped with an endomorphism $F$ (of underlying graded abelian groups) satisfying $dF = pFd$, whose homological algebra we study in detail. To any such object abstract analog Cartier isomorphism, elementary process associates generalization construction. Abstractly, can be viewed as calculation fixed points Berthelot-Ogus operator $L \eta_p$ on $p$-complete derived category. We give various applications approach, including simplification crystalline comparison $A \Omega$-cohomology theory.