A cohomological construction of swan representations over the Witt ring, I

A De Rham–Witt approach to crystalline rational homotopy theory

“…We will work in the setting of log schemes, and make extensive use of the theory developed by Kato [Kat89], Hyodo [Hyo91,Hyo88], and Hyodo-Kato [HK94]. The reader should consult these articles for precise notions and notation.…”

“…K denotes the fraction field of W . We wish to present a relatively direct approach to the unipotent crystalline fundamental group of a variety over k using the De Rham-Witt (DRW) complex of Bloch and Illusie as complemented by Hyodo and Kato ([3], [19], [17], [18], [16]) and constructions that come from rational homotopy theory ( [34], [29], [4]) and its Hodge-De Rham realizations ( [24], [12], [13], [26]). In the process, to a smooth connected proper fine log scheme Y over k of Cartier type, we will associate a canonical commutative differential graded algebra that deserves to be called the unipotent crystalline rational homotopy type.…”