Canonical Representatives in Strict Isomorphism Classes of Formal Groups

The aim of the present paper is to explicitly construct canonical representatives in every strict isomorphism class of commutative formal groups over an arbitrary torsion-free ring. The case of an $\mathbb Z_{(p)}$-algebra is treated separately. We prove that, under natural conditions on a subring, the canonical representatives of formal groups over the subring agree with the representatives for the ring. Necessary and sufficient conditions for a mapping induced on strict isomorphism classes of formal groups by a homomorphism of torsion-free rings to be injective and surjective are established.