Projective modules over Rees-like algebras and its monoid extensions
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| Format: | Preprint |
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2024
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| author | Bhaumik, Chandan Raihan, Md Abu Sarwar, Husney Parvez |
| author_facet | Bhaumik, Chandan Raihan, Md Abu Sarwar, Husney Parvez |
| contents | Let $A$ be a Rees-like algebra of dimension $d$ and $N$ a commutative partially cancellative torsion-free seminormal monoid. We prove the following results. \begin{enumerate}
\item Let $P$ be a finitely generated projective $A$-module of $\rank\geq d$. Then $(i)$ $P$ has a unimodular element; $(ii)$ The action of $\EL(A\oplus P)$ on $\Um(A\oplus P)$ is transitive.
\item Let $P$ be a finitely generated projective $A[N]$-module of $\rank~r$. Then $(i)$ $P$ has a unimodular element for $r\geq\max\{3,d\}$; $(ii)$ The action of $\EL(A[N]\oplus P)$ on $\Um(A[N]\oplus P)$ is transitive for $r\geq\max\{2,d\}$. \end{enumerate} These improve the classical results of Serre \cite{Se58} and Bass \cite{Ba64}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_17096 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Projective modules over Rees-like algebras and its monoid extensions Bhaumik, Chandan Raihan, Md Abu Sarwar, Husney Parvez Commutative Algebra Primary 13C10, Secondary 19A13, 19A15 Let $A$ be a Rees-like algebra of dimension $d$ and $N$ a commutative partially cancellative torsion-free seminormal monoid. We prove the following results. \begin{enumerate} \item Let $P$ be a finitely generated projective $A$-module of $\rank\geq d$. Then $(i)$ $P$ has a unimodular element; $(ii)$ The action of $\EL(A\oplus P)$ on $\Um(A\oplus P)$ is transitive. \item Let $P$ be a finitely generated projective $A[N]$-module of $\rank~r$. Then $(i)$ $P$ has a unimodular element for $r\geq\max\{3,d\}$; $(ii)$ The action of $\EL(A[N]\oplus P)$ on $\Um(A[N]\oplus P)$ is transitive for $r\geq\max\{2,d\}$. \end{enumerate} These improve the classical results of Serre \cite{Se58} and Bass \cite{Ba64}. |
| title | Projective modules over Rees-like algebras and its monoid extensions |
| topic | Commutative Algebra Primary 13C10, Secondary 19A13, 19A15 |
| url | https://arxiv.org/abs/2405.17096 |