2025/01/08 by Luis Manuel Navas Vicente, Vicente, Luis Manuel Navas, Francisco J. Plaza Martín +1
Mathematics · #advanced mathematical theories #Rings, Modules, and Algebras #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.2501.04365
We consider projective, irreducible, non-singular curves over an algebraically closed field \k. A cover Y → X of such curves corresponds to an extension Ω/Σ of their function fields and yields an isomorphism \AY ≃ \AX ⊗Σ Ω of their geometric adele rings. The primitive element theorem shows that \AY is a quotient of \AX[T] by a polynomial. In general, we may look at quotient algebras \AXp\p = \AX[T]/(\p(T)) where \p(T) ∈ \AX[T] is monic and separable over \AX, and try to characterize the field extensions Ω/Σ lying in \AXp\p which arise from covers as above. We achieve this topologically, namely, as those Ω which embed discretely in \AXp\p, and in terms of an additive analog of the product formula for global fields, a result which is reminiscent of classical work of Artin-Whaples and Iwasawa. The technical machinery requires studying which topology on \AXp\p is natural for this problem. Local compactness no longer holds, but instead we have linear topologies defined by commensurability of \k-subspaces which coincide with the restricted direct product topology with respect to integral closures. The content function is given as an index measuring the discrepancy in commensurable subspaces.