An Overview of Frobenius Splitting on Finitely-Generated Algebras

Abstract

The Frobenius map is a known and well-studied homomorphism on commutative rings of finite characteristic. In this paper, we examine the splitting behavior of this homomorphism when restricted to certain finitely generated algebras. We utilize a criterion of Fedder to categorize several of these algebras based on the polynomials which we use to define them, developing heuristics for determining which polynomials define Frobenius-split rings. We conclude by examining elliptic curves specifically, using our heuristics to determine Frobenius-splitness and using this property to estimate the number of solutions to a given elliptic curve over a finite field.