# Bijective correspondence between $\mathbb G_a$ actions on affine varieties and exponential maps on affine $k$-domains

Let $A$ be an integral domain which is a finitely generated algebra over an algebraically closed field $k$. Let $\phi :A \to A^{[1]}$ be a $k$-algebra homomorphism and let us write $\phi_t : A \to A[t]$ to emphasize the indeterminate $t$. $\phi$ is called an exponential map if the following two conditions are satisfied:

(i) $\epsilon_0 \phi_t : A \to A$ is the identity map, where $\epsilon_0 : A[t] \to A$ is the evaluation map at $t=0$.

(ii) $\phi_{s} \phi_t=\phi_{s+t}$, where $\phi_s$ is extended to a homomorphism $A[t] \to A[t,s]$ by setting $\phi_s(t)=t$.

Can someone please refer me to some book or lecture note or paper explaining the bijective correspondence between these exponential maps and algebraic actions of $\mathbb G_a=(k,+)$ on $X$ , the affine variety corresponding to $A$ ?

So far I have only been able to locate the bijective correspondence between locally nilpotent derivations on finitely generated domain over algebraically closed field of characteristic zero and the $\mathbb G_a$ actions on affine variety corresponding to the affine $k$-domain, in Gene Freudenberg's book : Algebraic theory Locally nilpotent derivations. But I haven't been able to find explicit detail concerning the correspondence between $\mathbb G_a$-actions and exponential maps (that there is a correspondence is mentioned,for example, in Gupta, N. Invent. math. (2014) 195: 279-288, On the cancellation problem for the affine space $\mathbb A^3$ in characteristic $p$

• I think you are missing main points in Freudenberg's book. He does say explicitly that $\mathbb{G}_a$ actions, locally nilpotent derivations and associated exponential maps are all essentially equivalent (page 32-33, second edition). – Mohan May 15 '18 at 19:10
• @Mohan: Yes, but that is only in characteristic zero (section 1.5 of chapter 1 of Freudenberg is all about characteristic zero); exponential maps and LND are not equivalent in positive characteristic as far as I know ... – users May 15 '18 at 19:17
• Isn't this literally the definition of a $\mathbb G_a$ action on a variety, but with all the arrows reversed and products replaced with coproducts by the contravariant equivalence between affine schemes and rings? – Will Sawin May 15 '18 at 19:18
• In positive characteristics, these are more complicated and you may find some results for example in Dufresne's article in the arxiv. – Mohan May 15 '18 at 20:09
• As Will Sawin says, this is literally the definition of a $\mathbb{G}_a$ action on the affine scheme $\text{Spec } A$, just interpreted in the opposite category. This means none of your hypotheses are necessary; $k$ can be any commutative ring and $A$ can be any commutative $k$-algebra. The thing that replaces locally nilpotent derivations in positive characteristic can be found here: qchu.wordpress.com/2017/11/26/… – Qiaochu Yuan May 15 '18 at 20:28