\(\varepsilon\)-constants and the Galois structure of de Rham cohomology. (English) Zbl 0939.14009

Chinburg, Ted; Erez, Boas; Pappas, Georgios; Taylor, Martin J.

Ann. Math. (2) 146, No. 2, 411-473 (1997).

Let \(X\) be a regular projective scheme of dimension \(d\) defined over a ring \(\Lambda\). Assume that a finite group \(G\) acts on \(X\) and it is given a \(G\)-equivariant sheaf \({\mathcal F}\). The authors study the problem of the structure of the derived complex \(R\Gamma(X,{\mathcal F})\). If \(\Lambda =\mathbb{Q}\) and the action is free then the complex is equivalent a complex of free \(\mathbb{Z}[G]\)-modules. If \(\Lambda\) is a ring of integers in a Galois extension with the group \(G\) over \(\mathbb{Q}\) then there is an obstruction for the \(G\)-module \(\Lambda\) to be free. The obstruction belongs to the class group of the ring \(\mathbb{Z}[G]\) and can be expressed in terms of the \(\epsilon\)-constants entering into the functional equation of the \(L\)-functions associated with the Galois extension. This result was conjectured (and was proved under some conditions) by A. Fröhlich and was obtained in general case by M. Taylor.
The paper contains a generalization of this result to the schemes of dimension \(d > 0\). The authors consider the truncated De Rham complex \(DR(\leq d)\) and defines an obstruction to it being quasi-isomorphic to a bounded complex of finitely generated free \(\mathbb{Z}[G]\)-modules. The obstruction is defined in terms of the \(\varepsilon\)-constants from the functional equation for the \(L\)-function of the covering \(X\) over \(X/G\). A main restriction for the definition and the next theorem is that the covering must be tame. The functional equation for the \(L\)-functions is still conjectural but the \(\varepsilon\)-constants can be defined rigorously without any unproven statements. The important step to define the obstruction is to introduce an Euler class \(\chi(X, {\mathcal F})\) in the same group. The Euler class is a sum of a root number and some ramification number, both coming from the \(\varepsilon\)-constants.
The obstruction theorem is proved under the following restrictions: either
(1) \(d = 1\) and \(X/G\) has semi-stable reduction over \(\mathbb{Z}\), or
(2) \(d \leq 1\), \(G\) acts freely and \(X/G\) has semi-stable reduction.
The authors conjecture that their formula is valid for all regular \(X\) with tame \(G\)-action. [For part II of this paper see T. Chinburg, G. Pappas and M. J. Taylor, J. Reine Angew. Math. 519, 201-230 (2000)].

Reviewer: A.N.Parshin (Moskva)

MSC:

14F40	de Rham cohomology and algebraic geometry
11R33	Integral representations related to algebraic numbers; Galois module structure of rings of integers
11G09	Drinfel’d modules; higher-dimensional motives, etc.

Keywords:

tame action of finite group; \(L\)-functions; de Rham homology; \(\varepsilon\)-constants; Galois extension; Euler class

Citations:

Zbl 0938.14010