Algebrization of some complete modules
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866918162012831744 |
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| author | Asgharzadeh, Mohsen |
| author_facet | Asgharzadeh, Mohsen |
| contents | Let $(R,\mathfrak{m})$ be a Noetherian local ring and $\widehat{R}$ its $\mathfrak{m}$-adic completion. We study the problem of determining when a finitely generated $\widehat{R}$-module arises from an $R$-module, i.e., when it is algebraic. We introduce and investigate the class of \emph{strongly algebraic} modules, those complete modules all of whose direct summands are algebraic. Our approach unifies and extends several known results of Levy--Odenthal, Weston, Peskine--Szpiro, Puthenpurakal, and several others, and provides new examples and homological criteria for algebrization. Applications include a computation of the Grothendieck group $G_0(R)$ in dimension one and new algebrization results for generalized Cohen--Macaulay modules and vector bundles along with a connection to local cohomology modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1805_03587 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Algebrization of some complete modules Asgharzadeh, Mohsen Commutative Algebra Let $(R,\mathfrak{m})$ be a Noetherian local ring and $\widehat{R}$ its $\mathfrak{m}$-adic completion. We study the problem of determining when a finitely generated $\widehat{R}$-module arises from an $R$-module, i.e., when it is algebraic. We introduce and investigate the class of \emph{strongly algebraic} modules, those complete modules all of whose direct summands are algebraic. Our approach unifies and extends several known results of Levy--Odenthal, Weston, Peskine--Szpiro, Puthenpurakal, and several others, and provides new examples and homological criteria for algebrization. Applications include a computation of the Grothendieck group $G_0(R)$ in dimension one and new algebrization results for generalized Cohen--Macaulay modules and vector bundles along with a connection to local cohomology modules. |
| title | Algebrization of some complete modules |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/1805.03587 |