# On stably trivial spin torsors over low-dimensional schemes

## TL;DR

This paper classifies stably trivial spin torsors over low-dimensional smooth schemes using $A^1$-homotopy theory, providing a complete set of invariants and examples for schemes of dimension up to 3.

## Contribution

It offers a comprehensive description of invariants for classifying stably trivial spin torsors over low-dimensional schemes, leveraging $A^1$-representability and homotopy sheaves.

## Key findings

- Complete classification of invariants for schemes of dimension ≤ 3.
- Explicit examples illustrating the classification.
- Application of $A^1$-homotopy techniques to spin torsors.

## Abstract

The paper discusses stably trivial torsors for spin and orthogonal groups over smooth affine schemes over infinite perfect fields of characteristic unequal to 2. We give a complete description of all the invariants relevant for the classification of such objects over schemes of dimension at most $3$, along with many examples. The results are based on the $\mathbb{A}^1$-representability theorem for torsors and transfer of known computations of $\mathbb{A}^1$-homotopy sheaves along the sporadic isomorphisms to spin groups.