Class invariants: functional properties and application to kernel studies
The motivation of the present study goes back to \(E.\) Noether's well known ``normal basis'' theorem on the projectivity of the \({\mathcal O}_K[\Gamma]\)-module \({\mathcal O}_F\) for a tamely ramified Galois extension \(F/K\) of number fields, \(\Gamma =\;\text{Gal}\;(F/K).\) In the wildly ramified case, it is more convenient to replace the group algebra \({\mathcal O}_k [\Gamma]\) by a certain Hopf algebra \({\mathcal H}.\) If \(\Gamma\) is commutative, \({\mathcal H}\) is an \({\mathcal O}_k\)-Hopf order in \(K [\Gamma],\) and the original Noether problem boils down to studying the kernel of a certain canonical homomorphism \(\pi_0\;:\;H^1 (\text{Spec}\, {\mathcal O}_k, \;\text{Spec}\, {\mathcal H}^\ast) \to C\!\ell ({\mathcal H}), \, {\mathcal N} \to ({\mathcal N}) \ldotp ({\mathcal H}^\ast)^{-1}\) where \({\mathcal H}^\ast\) is the dual Hopf algebra of \({\mathcal H}.\) In this setting, \({\mathcal N}\) can be viewed as an \({\mathcal O}_K\)-order in \(F\) which is an \({\mathcal H}\)-Galois extension, and \({\mathcal N}\) ``admits a normal basis'' iff \(\pi_0 ({\mathcal N}) = 0,\) iff \({\mathcal N} \simeq {\mathcal H}^\ast.\) \noindent This morphism \(\pi_0\) has been generalized by Whitehead to a morphism \(\pi\) which ``measures'' the Galois structure of torsors : for a noetherian scheme \(S\) and a commutative \(S\)-group scheme \(G,\) finite and flat, Whitehead's morphism is a certain canonical \(\pi\;:\;H^1 (S, G) \to \;\text{Pic} (G^D),\) \(G^D\) being the Cartier dual of \(G,\) and a \(G\)-torsor ``admits a normal basis'' iff its image by \(\pi\) is null. To go further, consider an exact sequence \(0 \to {\mathcal N}(f) \to F_1 \displaystyle{\buildrel f \over\rightarrow} F_2 \to 0,\) where \(F_1\) and \(F_2\) are two abelian sheaves over \(S\) (for the \(fppf\) topology) such that the kernel \(N(f)\) can be represented by a finite flat \(S\)-group scheme. One can construct a morphism \(\psi_f\;:\;F_2(S) \displaystyle{\buildrel \delta \over\rightarrow}\;H^1(S, N(f))\;\displaystyle{\buildrel \pi \over\rightarrow}\;\text{Pic} (N(f)^D)\) which is usually called the class homomorphism associated to the above exact sequence. \noindent In this paper, the author first gives functorial properties of \(\pi\) and \(\psi_f\) with respect to base changes, to products, etc \(\ldots\) He then applies these to the study of Ker \(\psi_f\) when \(f\) is multiplication by \(n\) in three situations : \smallskip\noindent - when \(G\) is an \(S\)-torus, \smallskip\noindent - when \(G\) is an \(S\)-abelian scheme which is isogenous to a product of elliptic curves, \smallskip\noindent - when \(G\) is an \(S\)-abelian scheme, considered as the image of the natural projection of an extension of \(G\) by \({\mathbb C}_m.\)
scientific article; zbMATH DE number 5302783
scientific article