Symplectic completion over smooth affine algebras
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912988430073856 |
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| author | Sharma, Gopal Sharma, Sampat |
| author_facet | Sharma, Gopal Sharma, Sampat |
| contents | In this article, we prove the following results:\\ \noindent \text{(1).} Let $R$ be a smooth affine algebra of dimension $3$ over an algebraically closed field $K$ with $3!\in K$, then we show that $\Um_4(R)=e_1\Sp_4(R)$ and $\Um_4(R [X])=e_1\Sp_4(R[X])$.
\noindent \text{(2).} We also show that if $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $K$ with $4!\in K$, and assume that $\W_E(R)$ is divisible, then $\Um_3(R)=e_1\SL_3(R)$. As a consequence it is shown that if $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $K$ with $4!\in K$, and assume that $\W_E(R)$ is divisible, then $\Um_4(R)=e_1\Sp_4(R)$.
\noindent \text{(3).} We show that if $R$ is a local ring of dimension $3$ with $\frac{1}{3!}\in R$. Then $\Um_4(R[X])=e_1\Sp_4(R[X])$.
\noindent \text{(4).} We also show that if $R=\oplus_{i\geq 0}R_i$ is a graded ring over a local ring of dimension $3$ with $\frac{1}{3!}\in R$. Then $\Um_4(R)=e_1\Sp_4(R)$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_28293 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Symplectic completion over smooth affine algebras Sharma, Gopal Sharma, Sampat Commutative Algebra In this article, we prove the following results:\\ \noindent \text{(1).} Let $R$ be a smooth affine algebra of dimension $3$ over an algebraically closed field $K$ with $3!\in K$, then we show that $\Um_4(R)=e_1\Sp_4(R)$ and $\Um_4(R [X])=e_1\Sp_4(R[X])$. \noindent \text{(2).} We also show that if $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $K$ with $4!\in K$, and assume that $\W_E(R)$ is divisible, then $\Um_3(R)=e_1\SL_3(R)$. As a consequence it is shown that if $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $K$ with $4!\in K$, and assume that $\W_E(R)$ is divisible, then $\Um_4(R)=e_1\Sp_4(R)$. \noindent \text{(3).} We show that if $R$ is a local ring of dimension $3$ with $\frac{1}{3!}\in R$. Then $\Um_4(R[X])=e_1\Sp_4(R[X])$. \noindent \text{(4).} We also show that if $R=\oplus_{i\geq 0}R_i$ is a graded ring over a local ring of dimension $3$ with $\frac{1}{3!}\in R$. Then $\Um_4(R)=e_1\Sp_4(R)$. |
| title | Symplectic completion over smooth affine algebras |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2603.28293 |