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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.11278 |
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| _version_ | 1866909680033333248 |
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| author | Amano, Michio |
| author_facet | Amano, Michio |
| contents | Let $\widehat{\mathcal{G}}^{(λ)}$ be a formal group scheme which deforms $\widehat{\mathbb{G}}_a$ to $\widehat{\mathbb{G}}_m$. And let $ψ^{(l)}:\widehat{\mathcal{G}}^{(λ)}\rightarrow\widehat{\mathcal{G}}^{(λ^{p^l})}$ be the $l$-th Frobenius-type homomorphism determined by $λ$. We show that the homomorphism $(ψ^{(l)})^\ast:H^2_0(\widehat{\mathcal{G}}^{(λ^{p^l})},\widehat{\mathbb{G}}_m)\rightarrow H^2_0(\widehat{\mathcal{G}}^{(λ)},\widehat{\mathbb{G}}_m)$ induced by $ψ^{(l)}$ is injective over a $\mathbb{Z}_{(p)}$-algebra under a suitable restriction on $λ$. In this situation, the Cartier dual of $\mathrm{Ker}(ψ^{(l)})$, which is a finite group scheme of order $p^l$, is described over a $\mathbb{Z}/(p^n)$-algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11278 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the injectivity of certain homomorphisms between extensions of $\hat{\mathcal{G}}^{(λ)}$ by $\hat{\mathbb{G}}_m$ over a $\mathbb{Z}_{(p)}$-algebra Amano, Michio Algebraic Geometry Let $\widehat{\mathcal{G}}^{(λ)}$ be a formal group scheme which deforms $\widehat{\mathbb{G}}_a$ to $\widehat{\mathbb{G}}_m$. And let $ψ^{(l)}:\widehat{\mathcal{G}}^{(λ)}\rightarrow\widehat{\mathcal{G}}^{(λ^{p^l})}$ be the $l$-th Frobenius-type homomorphism determined by $λ$. We show that the homomorphism $(ψ^{(l)})^\ast:H^2_0(\widehat{\mathcal{G}}^{(λ^{p^l})},\widehat{\mathbb{G}}_m)\rightarrow H^2_0(\widehat{\mathcal{G}}^{(λ)},\widehat{\mathbb{G}}_m)$ induced by $ψ^{(l)}$ is injective over a $\mathbb{Z}_{(p)}$-algebra under a suitable restriction on $λ$. In this situation, the Cartier dual of $\mathrm{Ker}(ψ^{(l)})$, which is a finite group scheme of order $p^l$, is described over a $\mathbb{Z}/(p^n)$-algebra. |
| title | On the injectivity of certain homomorphisms between extensions of $\hat{\mathcal{G}}^{(λ)}$ by $\hat{\mathbb{G}}_m$ over a $\mathbb{Z}_{(p)}$-algebra |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2405.11278 |