Finiteness of local homology modules

Shahram Rezaei

Archivum Mathematicum (2020)

  • Volume: 056, Issue: 1, page 31-41
  • ISSN: 0044-8753

Abstract

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Let I be an ideal of Noetherian ring R and M a finitely generated R -module. In this paper, we introduce the concept of weakly colaskerian modules and by using this concept, we give some vanishing and finiteness results for local homology modules. Let I M : = Ann R ( M / I M ) , we will prove that for any integer n

If N N is a weakly colaskerian linearly compact R R -module such that ( 0 : N I M ) ≠ 0 (0:_N {I_M})\ne 0 then

width I M ( N ) = inf { i ∣ H i I M ( N ) ≠ 0 } = inf { i ∣ H i I ( M , N ) ≠ 0 } . \operatorname{width}_{I_M}(N)= \inf \lbrace i\mid \operatorname{H}_i^{I_M}(N)\ne 0 \rbrace =\inf \lbrace i \mid \operatorname{H}_i^I(M,N)\ne 0 \rbrace \,.

If ( R , 𝔪 ) (R,\mathfrak {m}) is a Noetherian local ring and N N is an artinian R R -module then

∪ i < n Cos R ( H i I M ( N ) ) = ∪ i < n Cos R ( H i I ( M , N ) ) = ∪ i < n Cos R ( Tor i R ( M / I M , N ) ) , \cup _{i<n}\operatorname{Cos}_R\big (\operatorname{H}_i^{I_M}(N)\big )=\cup _{i<n}\operatorname{Cos}_R\big (\operatorname{H}_i^I(M,N)\big )=\\ \cup _{i<n}\operatorname{Cos}_R\big (\operatorname{Tor}_i^R(M/IM,N)\big )\,,

inf { i ∣ H i I M ( N ) is not Noetherian R -module } = inf { i ∣ H i I ( M , N ) i s n o t N o e t h e r i a n R - m o d u l e } . \inf \lbrace i \mid \operatorname{H}_i^{I_M}(N) \text{ is not Noetherian $R$-module\,} \rbrace =\\ \inf \lbrace i \mid \operatorname{H}_i^I(M,N) \mbox {\ is not Noetherian R-module\,}\rbrace \,.

How to cite

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Rezaei, Shahram. "Finiteness of local homology modules." Archivum Mathematicum 056.1 (2020): 31-41. <http://eudml.org/doc/295083>.

@article{Rezaei2020,
abstract = {Let $I$ be an ideal of Noetherian ring $R$ and $M$ a finitely generated $R$-module. In this paper, we introduce the concept of weakly colaskerian modules and by using this concept, we give some vanishing and finiteness results for local homology modules. Let $I_\{M\}:=\operatorname\{Ann\}_\{R\}(M/IM)$, we will prove that for any integer $n$},
author = {Rezaei, Shahram},
journal = {Archivum Mathematicum},
keywords = {coregular sequence; local homology; weakly colaskerian},
language = {eng},
number = {1},
pages = {31-41},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Finiteness of local homology modules},
url = {http://eudml.org/doc/295083},
volume = {056},
year = {2020},
}

TY - JOUR
AU - Rezaei, Shahram
TI - Finiteness of local homology modules
JO - Archivum Mathematicum
PY - 2020
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 056
IS - 1
SP - 31
EP - 41
AB - Let $I$ be an ideal of Noetherian ring $R$ and $M$ a finitely generated $R$-module. In this paper, we introduce the concept of weakly colaskerian modules and by using this concept, we give some vanishing and finiteness results for local homology modules. Let $I_{M}:=\operatorname{Ann}_{R}(M/IM)$, we will prove that for any integer $n$
LA - eng
KW - coregular sequence; local homology; weakly colaskerian
UR - http://eudml.org/doc/295083
ER -

References

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  1. Brodmann, M.P., Sharp, R.Y., Local cohomology – An algebric introduction with geometric applications, Cambridge University Press, 1998. (1998) MR1613627
  2. Cuong, N.T., Nam, T.T., On the co-localization, Co-support and Co-associated primes of local homology modules, Vietnam J. Math. 29 (4) (2001), 359–368. (2001) MR1934215
  3. Cuong, N.T., Nam, T.T., 10.1090/S0002-9939-01-06298-0, Proc. Amer. Math. Soc. 130 (7) (2001), 1927–1936. (2001) MR1896024DOI10.1090/S0002-9939-01-06298-0
  4. Cuong, N.T., Nam, T.T., 10.1017/S0305004101005126, Math. Proc. Cambridge Philos. Soc. 131 (2001), 61–72. (2001) MR1833074DOI10.1017/S0305004101005126
  5. Cuong, N.T., Nam, T.T., 10.1016/j.jalgebra.2007.11.030, J. Algebra 319 (2008), 4712–4737. (2008) MR2416740DOI10.1016/j.jalgebra.2007.11.030
  6. Divaani-Aazar, K., Mafi, A., 10.1090/S0002-9939-04-07728-7, Proc. Amer. Math. Soc. 133 (3) (2005), 655–660. (2005) Zbl1103.13010MR2113911DOI10.1090/S0002-9939-04-07728-7
  7. Greenlees, J.P.C., May, J.P., 10.1016/0021-8693(92)90026-I, J. Algebra 149 (1992), 438–453. (1992) MR1172439DOI10.1016/0021-8693(92)90026-I
  8. Macdonald, I.G., 10.1016/0040-9383(62)90104-0, Topology 1 (1962), 213–235. (1962) MR0151491DOI10.1016/0040-9383(62)90104-0
  9. Macdonald, I.G., Secondary representation of modules over a commutative ring, Symposia Mathematica 11 (1973), 23–43. (1973) MR0342506
  10. Melkersson, L., Schenzel, P., The co-localization of an artinian module, Proc. Edinburgh Math. Soc. 38 (1995), 121–131. (1995) MR1317331
  11. Nam, T.T., 10.1142/S1005386708000084, Algebra Colloquium 15 (1) (2008), 83–96. (2008) MR2371580DOI10.1142/S1005386708000084
  12. Nam, T.T., 10.1080/00927870802216396, Comm. Algebra 37 (2009), 1748–1757. (2009) MR2526337DOI10.1080/00927870802216396
  13. Nam, T.T., 10.1080/00927870802578043, Comm. Algebra 38 (2010), 440–453. (2010) MR2598892DOI10.1080/00927870802578043
  14. Nam, T.T., Generalized local homology for artinian modules, Algebra Colloquium 1 (2012), 11205–1212. (2012) MR3073409
  15. Ooishi, A., 10.32917/hmj/1206136213, Hiroshima Math. J. 6 (1976), 573–587. (1976) MR0422243DOI10.32917/hmj/1206136213
  16. Shar, R.Y., Steps in commutative algebra, London Mathematical Society Student Texts, vol. 19, Cambridge University Press, 1990. (1990) MR1070568
  17. Tang, Z., 10.1080/00927879408824928, Comm. Algebra 22 (1994), 1675–1684. (1994) MR1264734DOI10.1080/00927879408824928
  18. Yassemi, S., 10.1080/00927879508825288, Comm. Algebra 23 (1995), 1473–1498. (1995) MR1317409DOI10.1080/00927879508825288
  19. Yen, D.N., Nam, T.T., 10.1142/S0218196719500152, Int. J. Algebra Comput. 29 (3) (2019), 581–601. (2019) MR3955823DOI10.1142/S0218196719500152
  20. Zelinsky, D., 10.2307/2372616, Amer. J. Math. 75 (1953), 79–90. (1953) MR0051832DOI10.2307/2372616
  21. Zöschinger, H., Koatomare Moduln, Math. Z. 170 (1980), 221–232. (1980) MR0564202

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