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The big de Rham-Witt complex. (English) Zbl 1316.13028
The author provides a new construction of the big re Rham-Witt complex, a tool he introduced jointly with Madsen to analyse topological Hochschild spectra of rings [L. Hesselholt and I. Madsen, Contemp. Math. 271, 127–140 (2001; Zbl 0992.19002)]. The new construction is more explicit than the original one, and also incorporates a corrected account of the 2-torsion.
The constructions rests on a theory of modules and derivations over \(\lambda\)-rings, developed in §2 of the paper. It is shown that for every \(\lambda\)-ring there exists a universal \(\lambda\)-derivation, and that the resulting module of differentials is the usual one when neglecting the \(\lambda\)-structure (Theorem A).
In §3–§4 the author presents the actual construction. The notion of a Witt complex is introduced, and the big de Rham-Witt complex of a ring is defined to be the initial Witt complex of a ring. Existence of such a universal object is established in Theorem B.
The paper finishes with an analysis of how the big de-Rham Witt complex behaves with respect to étale maps (Theorem C, §5), and with an explicit computation of the big de Rham-Witt complex of the ring of integers in §6. As promised in the introduction the description is very explicit, but too long to be repeated here.
The paper also contains a section on Witt vectors (§1), and a very helpful introduction. While the material is necessarily rather technical, the presentation is very clear throughout, and supported by helpful recollections of material from other publications.

13F35 Witt vectors and related rings
16W50 Graded rings and modules (associative rings and algebras)
13N05 Modules of differentials
19D50 Computations of higher \(K\)-theory of rings
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