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March 2022 ${\mathbf G}_a$-actions on the affine line over a non-reduced ring
Motoki Kuroda, Shigeru Kuroda
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Hiroshima Math. J. 52(1): 17-42 (March 2022). DOI: 10.32917/h2021011


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 $m \geq 2$.

Funding Statement

The Second author is supported by JSPS KAKENHI Grant Number 18K03219.


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Motoki Kuroda. Shigeru Kuroda. "${\mathbf G}_a$-actions on the affine line over a non-reduced ring." Hiroshima Math. J. 52 (1) 17 - 42, March 2022.


Received: 26 February 2021; Revised: 11 November 2021; Published: March 2022
First available in Project Euclid: 25 March 2022

Digital Object Identifier: 10.32917/h2021011

Primary: 14R20
Secondary: 13A50

Keywords: additive group action , invariant ring , non-finite generation , positive characteristic

Rights: Copyright © 2022 Hiroshima University, Mathematics Program


Vol.52 • No. 1 • March 2022
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