Automorphisms of $\mathbb{A}^1_R$

by Unit   Last Updated June 29, 2017 23:20 PM - source

When $R$ is an integral domain, the automorphisms of the affine line are all of the form $X \mapsto aX + b$ with $a \in R^\times$ and $b \in R$; the proof is the same as in the case of $R$ a field, see this answer. It is easy to see that this group of maps is isomorphic to the semidirect product $R^\times \! \rtimes R^+$.

This paper by Dupuy studies the case where $R$ is not reduced. In particular, he proves that the group $\text{Aut}(\mathbb{A}^1_{\mathbb{Z}/m})$ is solvable, for arbitrary m. What other properties does it have? Is there an explicit description of this group?

For example, with a bit of playing around I found that $\varphi(X) = X + 2X^2$ is an involutive automorphism of $(\mathbb{Z}/4)[X]$: $$\varphi^2(X) = (X + 2X^2) + 2(X + 2X)^2 = X + 4X^2 + 4X^3 + 8X^4$$ which reduces to $X$ when we take the coefficients mod 4. But I could not generalize this example to the ring $R = \mathbb{Z}/p^2$.

Also, what can be said in the case that $R$ is reduced but not integral? E.g. $\mathbb{Z}[X,Y]/(XY)$ or $\mathbb{Z}/6$.

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