Stability theorems for positively graded domains and a question of Lindel
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866915430864519168 |
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| author | Banerjee, Sourjya |
| author_facet | Banerjee, Sourjya |
| contents | Given a commutative Noetherian graded domain $R = \bigoplus_{i\ge 0} R_i$ of dimension $d\geq 2$ with $\dim(R_0) \geq 1$, we prove that any unimodular row of length $d+1$ in $R$ can be completed to the first row of an invertible matrix $α$ such that $α$ is homotopic to the identity matrix. Utilizing this result we establish that if $I \subset R$ is an ideal satisfying $μ(I/I^2) = \text{ht}(I) = d$, then any set of generators of $I/I^2$ lifts to a set of generators of $I$, where $μ(-)$ denotes the minimal number of generators. Consequently, any projective $R$-module of rank $d$ with trivial determinant splits into a free factor of rank one. This provides an affirmative answer to an old question of Lindel. Finally, we prove that for any projective $R$-module $P$ of rank $d$, if the Quillen ideal of $P$ is non-zero, then $P$ is cancellative. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_12778 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Stability theorems for positively graded domains and a question of Lindel Banerjee, Sourjya Commutative Algebra 19A13, 13C10, 19A15, 13A10 Given a commutative Noetherian graded domain $R = \bigoplus_{i\ge 0} R_i$ of dimension $d\geq 2$ with $\dim(R_0) \geq 1$, we prove that any unimodular row of length $d+1$ in $R$ can be completed to the first row of an invertible matrix $α$ such that $α$ is homotopic to the identity matrix. Utilizing this result we establish that if $I \subset R$ is an ideal satisfying $μ(I/I^2) = \text{ht}(I) = d$, then any set of generators of $I/I^2$ lifts to a set of generators of $I$, where $μ(-)$ denotes the minimal number of generators. Consequently, any projective $R$-module of rank $d$ with trivial determinant splits into a free factor of rank one. This provides an affirmative answer to an old question of Lindel. Finally, we prove that for any projective $R$-module $P$ of rank $d$, if the Quillen ideal of $P$ is non-zero, then $P$ is cancellative. |
| title | Stability theorems for positively graded domains and a question of Lindel |
| topic | Commutative Algebra 19A13, 13C10, 19A15, 13A10 |
| url | https://arxiv.org/abs/2306.12778 |