With enough enthusiasm, I would try to learn about crystalline cohomology and the de-Rham-Witt complex from the homonymous article by Illusie:
Illusie, Luc. Complexe de deRham-Witt et cohomologie cristalline. (French) Ann. Sci. École Norm. Sup. (4) 12 (1979), no. 4, 501--661. MR0565469 (82d:14013)
Fortunately, it is publicly available at:
But this is most usefully read as needed after one is acquainted with the following also relevant references (perhaps in this order):
S. Bloch, Algebraic K-theory and crystalline cohomology, Publ. Math. Inst. Hautes Etudes Sci. 47 (1977), 187–268.
O. Hyodo and K. Kato, Semi-stable reduction and crystalline cohomology with logarithmic poles, in Periodes p-adiques (Bures-sur-Yvette, 1988), Asterisque 223 (1994), 221–268.
O. Hyodo, On the de Rham–Witt complex attached to a semi-stable family, Compositio Math. 78 (1991), 241–260.
O. Hyodo, A cohomological construction of Swan representations over the Witt ring. I, Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), 300–303.