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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.

MSC:
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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