We show that the additive higher Chow groups of regular schemes over a field induce a Zariski sheaf of pro-differential graded algebras, the Milnor range of which is isomorphic to the Zariski sheaf of big de Rham–Witt complexes. This provides an explicit cycle-theoretic description of the big de Rham–Witt sheaves. Several applications are derived.

we compare the absolute and relative de rham–witt complexes considered by hesselholt and madsen (ann. sci. école norm. sup. 37 (2004), 1–43; ann. of math. (2) 158 (2003), 1–113) and by langer and zink (j. inst. math. jussieu 3 (2004), 231–314), which both generalize the classical de rham–witt complex of bloch, deligne, and illusie (ann. sci. école norm. sup. (4) 12 (1979), 501–661) from $\mathbb{f}_p$-schemes to $\mathbb{z}_{(p)}$-schemes. from this comparison, we derive a gauss–manin connection on the crystalline cohomology of x/wn(s) for a smooth family x/s.

We give a definition of the crystalline fundamental group of suitable log schemes in positive characteristic using the techniques of rational homotopy theory applied to the De Rham–Witt complex.

A Pfaffian equation on $X$ is an invertible subsheaf $L \hookrightarrow \Omega^1_X$ of the sheaf of differential 1-forms on $X$. It can thus be thought of as a global section of $\Omega^1_X \otimes L^{-1}$ or, by choosing an isomorphism $L^{-1} \simeq {\cal O}_X(E) \subset K$, as a meromorphic differential form $\omega$ on $X$. We say that the Pfaffian equation has a first integral if there is a non-empty open subset $U\subset X$ and a smooth map $f: U\rightarrow {\rm P}^1$ such that $L_U \simeq f^* \Omega^1_{{\rm P}^1}$ as subsheaves of $\Omega^1_X$; that is, if $L$ ‘comes from’ a rational map to a curve. In this case, $f$, viewed as a rational function, is called a first integral. Note that as rational differential forms, ${\rm d} f \wedge \omega = 0$.