Approximate Fiber Products of Schemes and Their \'Etale Homotopical Invariants

Abstract

The classical fiber product in algebraic geometry provides a powerful tool for studying loci where two morphisms to a base scheme, $\phi: X \to S$ and $\psi: Y \to S$, coincide exactly. This condition of strict equality, however, is insufficient for describing many real-world applications. This paper develops a new framework to study and quantify the nature of approximate alignment between morphisms in the context of arithmetic geometry, using the tools of {\'e}tale homotopy theory. We introduce the central object of our work, the \textit{{\'e}tale mismatch torsor}, a sheaf of torsors over the product scheme $X \times_S Y$. The structure of this sheaf serves as a rich, intrinsic, and purely algebraic object amenable to both qualitative classification and quantitative analysis of the global relationship between the two morphisms. Our main results are twofold. First, we provide a complete classification of these structures, establishing a bijection between their isomorphism classes and the first {\'e}tale cohomology group $H_{\text{\'e}t}^1(X \times_S Y, \underline{\pi_1^{\text{\'e}t}(S)})$. Second, we construct a canonical filtration on this classifying cohomology group based on the theory of infinitesimal neighborhoods. This filtration induces a new invariant, which we term the \textit{order of mismatch}, providing a hierarchical, algebraic measure for the degree of approximation between the morphisms. We apply this framework to the concrete case of generalized Howe curves over finite fields, demonstrating how both the characteristic class and its order reveal subtle arithmetic properties.