On the Buchstaber--Rees theory of "Frobenius $n$-homomorphisms" and its generalization

Abstract

This is a survey of our results on the theory of $n$-homomorphisms of Buchstaber--Rees and its generalization that we obtained. In short, we are concerned with classes of linear maps between commutative rings that can be described the "next level after ring homomorphisms" with respect to multiplicative properties. Our main tool is a construction which we call the "characteristic function" -- whose functional properties encode algebraic properties of a linear map in question. Namely, if the characteristic function is polynomial of degree $n$, the map is an $n$-homomorphism in the sense of Buchstaber--Rees, and our approach simplifies their theory substantially. If the characteristic function is an irreducible rational fraction with the numerator and denominator of degrees $p$ and $q$ respectively, we arrive at a new notion of a "$p|q$-homomorphism". Examples of $p|q$-homomorphisms are sums and differences of ring homomorphisms. Our construction is motivated by our earlier results in superalgebra/supergeometry concerning Berezinians and super exterior powers.