Representation-theoretic properties of balanced big Cohen-Macaulay modules
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arXiv
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| Format: | Preprint |
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2018
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| author | Bahlekeh, Abdolnaser Fotouhi, Fahimeh Sadat Salarian, Shokrollah |
| author_facet | Bahlekeh, Abdolnaser Fotouhi, Fahimeh Sadat Salarian, Shokrollah |
| contents | Let $(R, \m, k)$ be a complete Cohen-Macaulay local ring. In this paper, we assign a numerical invariant, for any balanced big Cohen-Macaulay module, called $\uh$-length. Among other results, it is proved that, for a given balanced big Cohen-Macaulay $R$-module $M$ with an $\m$-primary cohomological annihilator, if there is a bound on the $\uh$-length of all modules appearing in $\CM$-support of $M$, then it is fully decomposable, i.e. it is a direct sum of finitely generated modules. While the first Brauer-Thrall conjecture fails in general by a counterexample of Dieterich dealing with multiplicities to measure the size of maximal Cohen-Macaulay modules, our formalism establishes the validity of the conjecture for complete Cohen-Macaulay local rings. In addition, the pure-semisimplicity of a subcategory of balanced big Cohen-Macaulay modules is settled. Namely, it is shown that $R$ is of finite $\CM$-type if and only if the category of all fully decomposable balanced big Cohen-Macaulay modules is closed under kernels of epimorphisms. Finally, we examine the mentioned results in the context of Cohen-Macaulay artin algebras admitting a dualizing bimodule $ω$, as defined by Auslander and Reiten. It will turn out that, $ω$-Gorenstein projective modules with bounded $\CM$-support are fully decomposable. In particular, a Cohen-Macaulay algebra $Λ$ is of finite $\CM$-type if and only if every $ω$-Gorenstein projective module is of finite $\CM$-type, which generalizes a result of Chen for Gorenstein algebras. Our main tool in the proof of results is Gabriel-Roiter (co)measure, an invariant assigned to modules of finite length, and defined by Gabriel and Ringel. This, in fact, provides an application of the Gabriel-Roiter (co)measure in the category of maximal Cohen-Macaulay modules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1807_04508 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Representation-theoretic properties of balanced big Cohen-Macaulay modules Bahlekeh, Abdolnaser Fotouhi, Fahimeh Sadat Salarian, Shokrollah Commutative Algebra Let $(R, \m, k)$ be a complete Cohen-Macaulay local ring. In this paper, we assign a numerical invariant, for any balanced big Cohen-Macaulay module, called $\uh$-length. Among other results, it is proved that, for a given balanced big Cohen-Macaulay $R$-module $M$ with an $\m$-primary cohomological annihilator, if there is a bound on the $\uh$-length of all modules appearing in $\CM$-support of $M$, then it is fully decomposable, i.e. it is a direct sum of finitely generated modules. While the first Brauer-Thrall conjecture fails in general by a counterexample of Dieterich dealing with multiplicities to measure the size of maximal Cohen-Macaulay modules, our formalism establishes the validity of the conjecture for complete Cohen-Macaulay local rings. In addition, the pure-semisimplicity of a subcategory of balanced big Cohen-Macaulay modules is settled. Namely, it is shown that $R$ is of finite $\CM$-type if and only if the category of all fully decomposable balanced big Cohen-Macaulay modules is closed under kernels of epimorphisms. Finally, we examine the mentioned results in the context of Cohen-Macaulay artin algebras admitting a dualizing bimodule $ω$, as defined by Auslander and Reiten. It will turn out that, $ω$-Gorenstein projective modules with bounded $\CM$-support are fully decomposable. In particular, a Cohen-Macaulay algebra $Λ$ is of finite $\CM$-type if and only if every $ω$-Gorenstein projective module is of finite $\CM$-type, which generalizes a result of Chen for Gorenstein algebras. Our main tool in the proof of results is Gabriel-Roiter (co)measure, an invariant assigned to modules of finite length, and defined by Gabriel and Ringel. This, in fact, provides an application of the Gabriel-Roiter (co)measure in the category of maximal Cohen-Macaulay modules. |
| title | Representation-theoretic properties of balanced big Cohen-Macaulay modules |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/1807.04508 |