Cofiniteness and finiteness of local cohomology modules over regular local rings

Jafar A'zami; Naser Pourreza

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 3, page 733-740
  • ISSN: 0011-4642

Abstract

top
Let ( R , 𝔪 ) be a commutative Noetherian regular local ring of dimension d and I be a proper ideal of R such that mAss R ( R / I ) = Assh R ( I ) . It is shown that the R -module H I ht ( I ) ( R ) is I -cofinite if and only if cd ( I , R ) = ht ( I ) . Also we present a sufficient condition under which this condition the R -module H I i ( R ) is finitely generated if and only if it vanishes.

How to cite

top

A'zami, Jafar, and Pourreza, Naser. "Cofiniteness and finiteness of local cohomology modules over regular local rings." Czechoslovak Mathematical Journal 67.3 (2017): 733-740. <http://eudml.org/doc/294087>.

@article{Azami2017,
abstract = {Let $(R,\mathfrak \{m\})$ be a commutative Noetherian regular local ring of dimension $d$ and $I$ be a proper ideal of $R$ such that $\{\rm mAss\}_R(R/I)=\{\rm Assh\}_R(I)$. It is shown that the $R$-module $H^\{\{\rm ht\}(I)\}_I(R)$ is $I$-cofinite if and only if $\{\rm cd\}(I,R)=\{\rm ht\}(I)$. Also we present a sufficient condition under which this condition the $R$-module $H^i_I(R)$ is finitely generated if and only if it vanishes.},
author = {A'zami, Jafar, Pourreza, Naser},
journal = {Czechoslovak Mathematical Journal},
keywords = {cofinite module; Cohen-Macaulay ring; Krull dimension; local cohomology; regular ring},
language = {eng},
number = {3},
pages = {733-740},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Cofiniteness and finiteness of local cohomology modules over regular local rings},
url = {http://eudml.org/doc/294087},
volume = {67},
year = {2017},
}

TY - JOUR
AU - A'zami, Jafar
AU - Pourreza, Naser
TI - Cofiniteness and finiteness of local cohomology modules over regular local rings
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 3
SP - 733
EP - 740
AB - Let $(R,\mathfrak {m})$ be a commutative Noetherian regular local ring of dimension $d$ and $I$ be a proper ideal of $R$ such that ${\rm mAss}_R(R/I)={\rm Assh}_R(I)$. It is shown that the $R$-module $H^{{\rm ht}(I)}_I(R)$ is $I$-cofinite if and only if ${\rm cd}(I,R)={\rm ht}(I)$. Also we present a sufficient condition under which this condition the $R$-module $H^i_I(R)$ is finitely generated if and only if it vanishes.
LA - eng
KW - cofinite module; Cohen-Macaulay ring; Krull dimension; local cohomology; regular ring
UR - http://eudml.org/doc/294087
ER -

References

top
  1. Bagheriyeh, I., Bahmanpour, K., A'zami, J., 10.1216/JCA-2014-6-3-305, J. Commut. Algebra 6 (2014), 305-321. (2014) Zbl1299.13019MR3278806DOI10.1216/JCA-2014-6-3-305
  2. Bahmanpour, K., 10.1080/00927872.2014.900687, Commun. Algebra 43 (2015), 2509-2515. (2015) Zbl1323.13003MR3344203DOI10.1080/00927872.2014.900687
  3. Bahmanpour, K., A'zami, J., Ghasemi, G., 10.1016/j.jalgebra.2012.03.026, J. Algebra 363 (2012), 8-13. (2012) Zbl1262.13027MR2925842DOI10.1016/j.jalgebra.2012.03.026
  4. Bahmanpour, K., Naghipour, R., 10.1016/j.jalgebra.2008.05.014, J. Algebra 320 (2008), 2632-2641. (2008) Zbl1149.13008MR2441778DOI10.1016/j.jalgebra.2008.05.014
  5. Bahmanpour, K., Naghipour, R., 10.1016/j.jalgebra.2008.12.020, J. Algebra 321 (2009), 1997-2011. (2009) Zbl1168.13016MR2494753DOI10.1016/j.jalgebra.2008.12.020
  6. Brodmann, M. P., Sharp, R. Y., 10.1017/CBO9780511629204, Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, Cambridge (1998). (1998) Zbl0903.13006MR1613627DOI10.1017/CBO9780511629204
  7. Grothendieck, A., Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz loceaux et globeaux (SGA 2), Séminaire de Géométrie Algébrique du Bois-Marie, 1962, Advanced Studies in Pure Mathematics 2, North-Holland Publishing Company, Amsterdam; Masson & Cie, Éditeur, Paris (1968), French. (1968) Zbl0197.47202MR0476737
  8. Hartshorne, R., 10.1007/BF01404554, Invent. Math. 9 (1970), 145-164. (1970) Zbl0196.24301MR0257096DOI10.1007/BF01404554
  9. Hellus, M., 10.1006/jabr.2000.8580, J. Algebra 237 (2001), 406-419. (2001) Zbl1027.13009MR1813886DOI10.1006/jabr.2000.8580
  10. Huneke, C., Koh, J., 10.1017/S0305004100070493, Math. Proc. Camb. Philos. Soc. 110 (1991), 421-429. (1991) Zbl0749.13007MR1120477DOI10.1017/S0305004100070493
  11. Khashyarmanesh, K., 10.1090/S0002-9939-06-08664-3, Proc. Am. Math. Soc. 135 (2007), 1319-1327. (2007) Zbl1111.13016MR2276640DOI10.1090/S0002-9939-06-08664-3
  12. Khashyarmanesh, K., Salarian, Sh., 10.1080/00927879808826293, Commun. Algebra 26 (1998), 2483-2490. (1998) Zbl0909.13007MR1627876DOI10.1080/00927879808826293
  13. Melkersson, L., 10.1016/j.jalgebra.2004.08.037, J. Algebra 285 (2005), 649-668. (2005) Zbl1093.13012MR2125457DOI10.1016/j.jalgebra.2004.08.037
  14. Schenzel, P., 10.7146/math.scand.a-14399, Math. Scand. 92 (2003), 161-180. (2003) Zbl1023.13011MR1973941DOI10.7146/math.scand.a-14399

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.