Colocalization of generalized local homology modules
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Let $R$ be a commutative Noetherian ring and $I$ an ideal of $R$. In this paper, we study colocalization of generalized local homology modules. We intend to establish a dual case of local-global principle for the finiteness of generalized local cohomology modules. Let $M$ be a finitely generated $R$-module and $N$ a representable $R$-module. We introduce the notions of the representation dimension $r^I (M, N)$ and artinianness dimension $a^I (M, N)$ of $M,N$ with respect to $I$ by $r^I (M, N)= \inf\{i\in \mathbb{N}_0 : H^I_i(M,N) \text{ is not representable}\}$ and $a^I (M, N)= \inf\{i\in \mathbb{N}_0 : H^I_i(M,N)\text{ is not artinian}\}$ and we show that $a^I (M, N)=r^I (M, N)$ $=\inf\{r^{IR_{\fp}}(M_{\fp},_{\fp}N) : \fp\in \Spec(R)\}\geq \inf\{a^{IR_{\fp}}(M_{\fp},_{\fp}N) : \fp\in \Spec(R)\}$. Also, in the case where $R$ is semi-local and $N$ a semi discrete linearly compact $R$-module such that $N/\bigcap_{t>0} I^tN$ is artinian we prove that $\inf\{i: H^I_i(M,N) \text{ is not minimax}\}\!=\!\inf\{r^{IR_{\fp}}(M_{\fp},_{\fp}N) : \fp\in \Spec(R)\setminus\Max(R)\}.$
Abstract
Let $R$ be a commutative Noetherian ring and $I$ an ideal of $R$. In this paper, we study colocalization of generalized local homology modules. We intend to establish a dual case of local-global principle for the finiteness of generalized local cohomology modules. Let $M$ be a finitely generated $R$-module and $N$ a representable $R$-module. We introduce the notions of the representation dimension $r^I (M, N)$ and artinianness dimension $a^I (M, N)$ of $M,N$ with respect to $I$ by $r^I (M, N)= \inf\{i\in \mathbb{N}_0 : H^I_i(M,N) \text{ is not representable}\}$ and $a^I (M, N)= \inf\{i\in \mathbb{N}_0 : H^I_i(M,N)\text{ is not artinian}\}$ and we show that $a^I (M, N)=r^I (M, N)$ $=\inf\{r^{IR_{\fp}}(M_{\fp},_{\fp}N) : \fp\in \Spec(R)\}\geq \inf\{a^{IR_{\fp}}(M_{\fp},_{\fp}N) : \fp\in \Spec(R)\}$. Also, in the case where $R$ is semi-local and $N$ a semi discrete linearly compact $R$-module such that $N/\bigcap_{t>0} I^tN$ is artinian we prove that $\inf\{i: H^I_i(M,N) \text{ is not minimax}\}\!=\!\inf\{r^{IR_{\fp}}(M_{\fp},_{\fp}N) : \fp\in \Spec(R)\setminus\Max(R)\}.$
참고문헌 (20)
- N. T. Cuong, "On the co-localization, co-support and co-associated primes of local homology modules", Vietnam J. Math, 29(4), pp.359~368 (2001)
- N. T. Cuong, "A local homology theory for linearly compact modules", J. Algebra, 319(11), pp.4712~4737 (2008)[DOI]
- Yen Ngoc Do, "Some finiteness results for co-associated primes of generalized local homology modules and applications", 대한수학회지, 57(5), pp.1061~1078 (2020), 대한수학회[DOI]
- G. Faltings, "Der Endlichkeitssatz in der lokalen Kohomologie", Math. Ann, 255(1), pp.45~56 (1981)[DOI]
- J. Herzog, "Komplexe, Auosungen und dualitat in der localen Algebra", pp.1970
- C. U. Jensen, "Les foncteurs d´eriv´es de lim←−et leurs applications en th´eorie des modules, Lecture Notes in Mathematics, Vol. 254" (1972), Springer-Verlag
- I. G. Macdonald, "Duality over complete local rings", Topology, 1, pp.213~235 (1962)[DOI]
- I. G. Macdonald, "Secondary representation of modules over a commutative ring, Symposia Mathematica, Vol. XI" (1973), Academic Press
- Leif Melkersson, "The co-localization of an Artinian module", Proceedings of the Edinburgh Mathematical Society, 38(1), pp.121~131 (1995), Cambridge University Press (CUP)[DOI]
- T. T. Nam, "Co-support and coartinian modules", Algebra Colloq, 15(1), pp.83~96 (2008)[DOI]
- T. T. Nam, "A finiteness result for co-associated and associated primes of generalized local homology and cohomology modules", Comm. Algebra, 37(5), pp.1748~1757 (2009)[DOI]
- T. T. Nam, "Left-derived functors of the generalized I-adic completion and generalized local homology", Comm. Algebra, 38(2), pp.440~453 (2010)[DOI]
- Tran Tuan Nam, "Some properties of generalized local homology and cohomology modules", 대한수학회보, 50(6), pp.2013~2020 (2013), 대한수학회
- Shahram Rezaei, "Colocalization of local homology modules", 대한수학회보, 57(1), pp.167~177 (2020), 대한수학회[DOI]
- R. N. ROBERTS, "KRULL DIMENSION FOR ARTINIAN MODULES OVER QUASI LOCAL COMMUTATIVE RINGS", The Quarterly Journal of Mathematics, 26(1), pp.269~273 (1975), Oxford University Press (OUP)[DOI]
- S. Yassemi, "Coassociated primes", Comm. Algebra, 23(4), pp.1473~1498 (1995)[DOI]
- D. N. Yen, "Generalized local homology and duality", Internat. J. Algebra Comput, 29(3), pp.581~601 (2019)[DOI]
- H. Zoschinger, "Koatomare Moduln", Math. Z, 170(3), pp.221~232 (1980)[DOI]
- H. Zoschinger, "Minimax-moduln", J. Algebra, 102(1), pp.1~32 (1986)[DOI]
- H. Zoschinger, "Uber koassoziierte Primideale ¨", Math. Scand, 63(2), pp.196~211 (1988)[DOI]