Loop torsors and Abhyankar's lemma
Philippe Gille
TL;DR
The paper develops a comprehensive theory of tame loop torsors for locally algebraic groups over the Abhyankar-type localization $A_D$, extending loop-torsor techniques from Laurent polynomials to more general base rings. It builds a robust framework using ind-quasi-affine schemes, twisted constant group schemes, and normalizer structures to obtain a Galois-cohomological criterion for classifying tame loop torsors, along with an acyclicity result linking loop torsors to tame Galois cohomology over fields of iterated Laurent series. A key methodological pillar is Abhyankar-type tamely ramified covers, the fixed-point method, and blow-up techniques, enabling local-global principles and transfer between $A_D$-torsors and $K_v$-torsors. The paper culminates with explicit examples for $ ext{GL}_N$ and $ ext{O}(N)$, illustrating diagonalizability and Witt-theoretic classifications of tame loop forms, with potential implications for ramification, isotropy, and cohomological rigidity in broad settings.
Abstract
We define the notion of loop torsors under certain group schemes defined over the localization of a regular henselian ring A at a strict normal crossing divisor D. We provide a Galois cohomological criterion for classifying those torsors. We revisit also the related theory of loop torsors on Laurent polynomial rings.