${\mathbf G}_a$-actions on the affine line over a non-reduced ring

Abstract

In this paper, we study ${\mathbf G}_a$-actions on the affine spaces over a commutative ring of characteristic $p^e$, where $p$ is a prime number and $e\geq 2$. We say that a ${\mathbf G}_a$-action is red-nontrivial (resp. red-trivial) if it is nontrivial (resp. trivial) modulo $p$. We give a structure theorem for red-nontrivial ${\mathbf G}_a$-actions on the affine lines under some mild assumptions. Interestingly, the invariant ring for such an action is either the ring of constants or non-finitely generated. We show that every red-trivial ${\mathbf G}_a$-action on the affine space over a certain class of commutative rings is uniquely determined by two derivations, whose invariant ring is finitely generated if the base ring is noetherian. By combining these results, we completely determine the ${\mathbf G}_a$-actions on the affine lines over a certain class of commutative rings of positive characteristic, including ${\mathbf Z}/m{\mathbf Z}$ for any $m \geq 2$.