Several recent proofs of unimodality theorems in combinatorics rely on unexpected identities satisfied by the volume map on combinatorially defined algebras over fields of characteristic p. The proof of unimodality for the h-vector of simplicial spheres by Papadakis–Petrotou and for the h^*-vector of IDP and Gorenstein lattice polytopes by Adiprasito–Papadakis–Petrotou both fit into this framework. In this talk, we reinterpret the volume map from the perspective of commutative algebra and explain the origin of these identities. This viewpoint yields short proofs and suggests new research directions. This is joint work with Adiprasito, Oba, Papadakis, and Petrotou.