On triviality of $\mathbb{A}^2$-forms admitting a nontrivial $\mathbb{G}_a$-action
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| Format: | Preprint |
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2026
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| _version_ | 1866914426577223680 |
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| author | Saha, Debojyoti |
| author_facet | Saha, Debojyoti |
| contents | T. Kambayashi had shown that $\mathbb{A}^2$-forms over separable field extensions are necessarily polynomial rings. However, there exist inseparable $\mathbb{A}^2$-forms which are not necessarily polynomial rings. In this paper, we give a structure theorem for $\mathbb{A}^2$-forms over arbitrary field extensions admitting a nontrivial $\mathbb{G}_a$-action. From this structure theorem we derive some conditions under which an $\mathbb{A}^2$-form becomes trivial. In particular, we prove that over a field $k$, a factorial $\mathbb{A}^2$-form having a $k$-rational point and a non-trivial $\mathbb{G}_a$-action is trivial and we also give examples demonstrating that none of these hypotheses can be discarded. As a consequence of the structure theorem, we obtain a generalization of the Zariski Cancellation Theorem for the affine plane over an arbitrary field. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_26289 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On triviality of $\mathbb{A}^2$-forms admitting a nontrivial $\mathbb{G}_a$-action Saha, Debojyoti Commutative Algebra T. Kambayashi had shown that $\mathbb{A}^2$-forms over separable field extensions are necessarily polynomial rings. However, there exist inseparable $\mathbb{A}^2$-forms which are not necessarily polynomial rings. In this paper, we give a structure theorem for $\mathbb{A}^2$-forms over arbitrary field extensions admitting a nontrivial $\mathbb{G}_a$-action. From this structure theorem we derive some conditions under which an $\mathbb{A}^2$-form becomes trivial. In particular, we prove that over a field $k$, a factorial $\mathbb{A}^2$-form having a $k$-rational point and a non-trivial $\mathbb{G}_a$-action is trivial and we also give examples demonstrating that none of these hypotheses can be discarded. As a consequence of the structure theorem, we obtain a generalization of the Zariski Cancellation Theorem for the affine plane over an arbitrary field. |
| title | On triviality of $\mathbb{A}^2$-forms admitting a nontrivial $\mathbb{G}_a$-action |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2603.26289 |