an:06428139
Zbl 1316.13028
Hesselholt, Lars
The big de Rham-Witt complex
EN
Acta Math. 214, No. 1, 135-207 (2015).
00342569
2015
j
13F35 16W50 13N05 19D50
Witt vector; lambda-ring; de Rham-Witt complex
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 [\textit{L. Hesselholt} and \textit{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 \S2 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 \S3--\S4 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 ??tale maps (Theorem C, \S5), and with an explicit computation of the big de Rham-Witt complex of the ring of integers in~\S6. 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 (\S1), 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.
Thomas Huettemann (Belfast)
Zbl 0992.19002