##
**Galois structure of de Rham cohomology of tame covers of schemes.**
*(English)*
Zbl 0828.14007

The object of this paper is to generalize to tame covers of schemes the theory of the Galois module structure of tamely ramified rings of integers. To motivate this generalization, we recall two theorems of Noether and Taylor, respectively. Let \(N/K\) be a finite Galois extension of number fields with group \(G = \text{Gal} (N/K)\). Let \({\mathfrak O}_N\) be the ring of integers of \(N\). Noether proved that \({\mathfrak O}_N\) is a projective \(\mathbb{Z} [G]\)-module if and only if \(N/K\) is at most tamely ramified: we suppose from now on that is the case. Then \({\mathfrak O}_N\) defines a class \(({\mathfrak O}_N)\) in the Grothendieck group \(K_0 (\mathbb{Z} [G])\) of all finitely generated projective \(\mathbb{Z} [G]\)-modules. The class group \(\text{Cl} (\mathbb{Z} [G])\) of \(\mathbb{Z} [G]\) is defined to be the quotient of \(K_0 (\mathbb{Z} [G])\) by the subgroup generated by the class of \(\mathbb{Z} [G]\). Let \(({\mathfrak O}_N)^{\text{stab}}\) be the image of \(({\mathfrak O}_N)\) in \(\text{Cl} (\mathbb{Z} [G])\). In Invent. Math. 63, 41-79 (1981; Zbl 0469.12003), M. J. Taylor proved Fröhlich’s conjecture that \(({\mathfrak O}_N)^{\text{stab}}\) is equal to another invariant \(W_{N/K}\) in \(\text{Cl} (\mathbb{Z} [G])\) which Cassou-Noguès had defined by means of the root-numbers of symplectic representations of the Galois group \(G\).

To generalize the results above, let \(G\) be an abstract finite group. We consider tame \(G\)-covers \(f : X \mapsto Y\) of schemes of finite type over a Noetherian ring \(A\) with respect to a divisor \(D\) on \(Y\) having normal crossings.

Let \(\text{CT} (A[G])\) be the Grothendieck group of all finitely generated \(A[G]\)-modules which are cohomologically trivial as \(\mathbb{Z}[G]\)- modules. The “forgetful” homomorphism \(K_0 (\mathbb{Z} [G]) \mapsto \text{CT} (\mathbb{Z} [G])\) is an isomorphism. The image of \(({\mathfrak O}_N)\) under this isomorphism is \(\Psi (X/Y)\). – Our main results concern the case in which \(A\) is a finite field of characteristic \(p\).

Theorem: Suppose \(A\) is a finite field of characteristic \(p\). Assume that \(X\) and \(Y\) are projective over \(A\), \(X\) is regular, and that \(D\) has strictly normal crossings. Restriction of operators from \(A[G]\) to \(\mathbb{F}_p [G]\) induces a homomorphism \(\text{Res}_{A \mapsto \mathbb{F}_p} : \text{CT} (A [G]) \mapsto \text{CT} (\mathbb{F}_p [G])\), where \(\mathbb{F}_p\) is the field of order \(p\). The class \(\text{Res}_{A \mapsto \mathbb{F}_p} (\Psi (X/Y))\) in \(\text{CT} (\mathbb{F}_p [G]) = K_0 (\mathbb{F}_p [G])\) both determines and is determined by the set of \(p\)- adic absolute values \(|j_p \varepsilon (Y,V) |_p\) as \(V\) ranges over all of the irreducible complex representations of \(G\).

Our second main result over finite fields is a precise counterpart for regular projective schemes of Taylor’s proof of Fröhlich’s conjecture concerning the ring of integers. To state this, identify the class group \(\text{Cl} (\mathbb{Z} [G])\) of \(\mathbb{Z} [G]\) with \(\text{CT} (\mathbb{Z} [G])/ \langle (\mathbb{Z} [G]) \rangle\). If \(A\) is a finitely generated \(\mathbb{Z}\)- module, one has a homomorphism \(\text{Res}_{A \mapsto \mathbb{Z}}^{\text{stab}} : \text{CT} (A[G]) \mapsto \text{Cl} (\mathbb{Z} [G])\) induced by restriction of operators from \(A[G]\) to \(\mathbb{Z} [G]\). We define in Cl\((\mathbb{Z}[G])\) a symplectic root-number class \(W_{x/y}\) and a ramification class \(R_{x/y}\) which depends only on root-numbers associated to the restriction of \(f:X\mapsto Y\) to the branch locus of \(f\).

Theorem: Under the same hypotheses as the theorem above: \[ \text{Res}_{A \mapsto \mathbb{Z}}^{\text{stab}} \bigl( \Psi (X/Y) \bigr) = W_{X/Y} + R_{X/Y} . \] The class \(W_{X/Y}\) is determined by the signs at infinity of the (totally real) algebraic numbers \(\varepsilon (Y,V)\) as \(V\) ranges over all of the irreducible symplectic representations of \(G\). In particular, \(W_{X/Y}\) has order 1 or2, and \(W_{X/Y}\) is trivial if \(G\) has no irreducible symplectic representation.

To generalize the results above, let \(G\) be an abstract finite group. We consider tame \(G\)-covers \(f : X \mapsto Y\) of schemes of finite type over a Noetherian ring \(A\) with respect to a divisor \(D\) on \(Y\) having normal crossings.

Let \(\text{CT} (A[G])\) be the Grothendieck group of all finitely generated \(A[G]\)-modules which are cohomologically trivial as \(\mathbb{Z}[G]\)- modules. The “forgetful” homomorphism \(K_0 (\mathbb{Z} [G]) \mapsto \text{CT} (\mathbb{Z} [G])\) is an isomorphism. The image of \(({\mathfrak O}_N)\) under this isomorphism is \(\Psi (X/Y)\). – Our main results concern the case in which \(A\) is a finite field of characteristic \(p\).

Theorem: Suppose \(A\) is a finite field of characteristic \(p\). Assume that \(X\) and \(Y\) are projective over \(A\), \(X\) is regular, and that \(D\) has strictly normal crossings. Restriction of operators from \(A[G]\) to \(\mathbb{F}_p [G]\) induces a homomorphism \(\text{Res}_{A \mapsto \mathbb{F}_p} : \text{CT} (A [G]) \mapsto \text{CT} (\mathbb{F}_p [G])\), where \(\mathbb{F}_p\) is the field of order \(p\). The class \(\text{Res}_{A \mapsto \mathbb{F}_p} (\Psi (X/Y))\) in \(\text{CT} (\mathbb{F}_p [G]) = K_0 (\mathbb{F}_p [G])\) both determines and is determined by the set of \(p\)- adic absolute values \(|j_p \varepsilon (Y,V) |_p\) as \(V\) ranges over all of the irreducible complex representations of \(G\).

Our second main result over finite fields is a precise counterpart for regular projective schemes of Taylor’s proof of Fröhlich’s conjecture concerning the ring of integers. To state this, identify the class group \(\text{Cl} (\mathbb{Z} [G])\) of \(\mathbb{Z} [G]\) with \(\text{CT} (\mathbb{Z} [G])/ \langle (\mathbb{Z} [G]) \rangle\). If \(A\) is a finitely generated \(\mathbb{Z}\)- module, one has a homomorphism \(\text{Res}_{A \mapsto \mathbb{Z}}^{\text{stab}} : \text{CT} (A[G]) \mapsto \text{Cl} (\mathbb{Z} [G])\) induced by restriction of operators from \(A[G]\) to \(\mathbb{Z} [G]\). We define in Cl\((\mathbb{Z}[G])\) a symplectic root-number class \(W_{x/y}\) and a ramification class \(R_{x/y}\) which depends only on root-numbers associated to the restriction of \(f:X\mapsto Y\) to the branch locus of \(f\).

Theorem: Under the same hypotheses as the theorem above: \[ \text{Res}_{A \mapsto \mathbb{Z}}^{\text{stab}} \bigl( \Psi (X/Y) \bigr) = W_{X/Y} + R_{X/Y} . \] The class \(W_{X/Y}\) is determined by the signs at infinity of the (totally real) algebraic numbers \(\varepsilon (Y,V)\) as \(V\) ranges over all of the irreducible symplectic representations of \(G\). In particular, \(W_{X/Y}\) has order 1 or2, and \(W_{X/Y}\) is trivial if \(G\) has no irreducible symplectic representation.

### MSC:

14E20 | Coverings in algebraic geometry |

14L30 | Group actions on varieties or schemes (quotients) |

11R33 | Integral representations related to algebraic numbers; Galois module structure of rings of integers |