On the endomorphism ring and Cohen-Macaulayness of local cohomology defined by a pair of ideals

Thiago H. Freitas; Victor H. Jorge Pérez

Czechoslovak Mathematical Journal (2019)

  • Volume: 69, Issue: 2, page 453-470
  • ISSN: 0011-4642

Abstract

top
Let 𝔞 , I , J be ideals of a Noetherian local ring ( R , 𝔪 , k ) . Let M and N be finitely generated R -modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of H I , J t ( M ) and D ( H I , J t ( M ) ) , where t is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and D ( - ) : = Hom R ( - , E R ( k ) ) is the Matlis dual functor. We show that if R is a d -dimensional complete Cohen-Macaulay ring and H I , J i ( R ) = 0 for all i ≠ t , the natural homomorphism R → Hom R ( H I , J t ( K R ) , H I , J t ( K R ) ) is an isomorphism, where K R denotes the canonical module of R . Also, we discuss the depth and Cohen-Macaulayness of the Matlis dual of the top local cohomology modules with respect to a pair of ideals.

How to cite

top

Freitas, Thiago H., and Jorge Pérez, Victor H.. "On the endomorphism ring and Cohen-Macaulayness of local cohomology defined by a pair of ideals." Czechoslovak Mathematical Journal 69.2 (2019): 453-470. <http://eudml.org/doc/294425>.

@article{Freitas2019,
abstract = {Let $\mathfrak \{a\}$, $I$, $J$ be ideals of a Noetherian local ring $(R,\mathfrak \{m\},k)$. Let $M$ and $N$ be finitely generated $R$-modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of $H^t_\{I,J\}(M)$ and $D(H^t_\{I,J\}(M))$, where $t$ is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and $D(-):= \{\rm Hom\}_R(-,E_R(k))$ is the Matlis dual functor. We show that if $R$ is a $d$-dimensional complete Cohen-Macaulay ring and $H^i_\{I,J\}(R)=0$ for all $i\ne t$, the natural homomorphism $R\rightarrow \{\rm Hom\}_R(H^t_\{I,J\}(K_R), H^t_\{I,J\}(K_R))$ is an isomorphism, where $K_R$ denotes the canonical module of $R$. Also, we discuss the depth and Cohen-Macaulayness of the Matlis dual of the top local cohomology modules with respect to a pair of ideals.},
author = {Freitas, Thiago H., Jorge Pérez, Victor H.},
journal = {Czechoslovak Mathematical Journal},
keywords = {local cohomology; Matlis duality; endomorphism ring},
language = {eng},
number = {2},
pages = {453-470},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the endomorphism ring and Cohen-Macaulayness of local cohomology defined by a pair of ideals},
url = {http://eudml.org/doc/294425},
volume = {69},
year = {2019},
}

TY - JOUR
AU - Freitas, Thiago H.
AU - Jorge Pérez, Victor H.
TI - On the endomorphism ring and Cohen-Macaulayness of local cohomology defined by a pair of ideals
JO - Czechoslovak Mathematical Journal
PY - 2019
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 69
IS - 2
SP - 453
EP - 470
AB - Let $\mathfrak {a}$, $I$, $J$ be ideals of a Noetherian local ring $(R,\mathfrak {m},k)$. Let $M$ and $N$ be finitely generated $R$-modules. We give a generalized version of the Duality Theorem for Cohen-Macaulay rings using local cohomology defined by a pair of ideals. We study the behavior of the endomorphism rings of $H^t_{I,J}(M)$ and $D(H^t_{I,J}(M))$, where $t$ is the smallest integer such that the local cohomology with respect to a pair of ideals is nonzero and $D(-):= {\rm Hom}_R(-,E_R(k))$ is the Matlis dual functor. We show that if $R$ is a $d$-dimensional complete Cohen-Macaulay ring and $H^i_{I,J}(R)=0$ for all $i\ne t$, the natural homomorphism $R\rightarrow {\rm Hom}_R(H^t_{I,J}(K_R), H^t_{I,J}(K_R))$ is an isomorphism, where $K_R$ denotes the canonical module of $R$. Also, we discuss the depth and Cohen-Macaulayness of the Matlis dual of the top local cohomology modules with respect to a pair of ideals.
LA - eng
KW - local cohomology; Matlis duality; endomorphism ring
UR - http://eudml.org/doc/294425
ER -

References

top
  1. Aghapournahr, M., Ahmadi-Amoli, K., Sadeghi, M. Y., 10.22044/JAS.2015.482, J. Algebr. Syst. 3 (2015), 1-10. (2015) MR3534204DOI10.22044/JAS.2015.482
  2. Ahmadi-Amoli, K., Sadeghi, M. Y., On the local cohomology modules defined by a pair of ideals and Serre subcategory, J. Math. Ext. 7 (2013), 47-62. (2013) Zbl1327.13056MR3248761
  3. Brodmann, M. P., Sharp, R. Y., 10.1017/CBO9780511629204, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1998). (1998) Zbl0903.13006MR1613627DOI10.1017/CBO9780511629204
  4. Bruns, W., Herzog, J., 10.1017/CBO9780511608681, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge (1998). (1998) Zbl0909.13005MR1251956DOI10.1017/CBO9780511608681
  5. Chu, L., 10.1090/S0002-9939-2010-10471-9, Proc. Am. Math. Soc. 139 (2011), 777-782. (2011) Zbl1210.13018MR2745630DOI10.1090/S0002-9939-2010-10471-9
  6. Chu, L.-Z., Wang, Q., 10.1215/kjm/1248983036, J. Math. Kyoto Univ. 49 (2009), 193-200. (2009) Zbl1174.13024MR2531134DOI10.1215/kjm/1248983036
  7. Eghbali, M., Schenzel, P., 10.1080/00927872.2011.588982, Commun. Algebra 40 (2012), 4295-4305. (2012) Zbl1273.13027MR2982939DOI10.1080/00927872.2011.588982
  8. Freitas, T. H., Pérez, V. H. Jorge, 10.1216/JCA-2016-8-3-337, J. Commut. Algebra 8 (2016), 337-366. (2016) Zbl1348.13026MR3546002DOI10.1216/JCA-2016-8-3-337
  9. Freitas, T. H., Pérez, V. H. Jorge, 10.1007/s13366-016-0322-6, Beitr. Algebra Geom. 58 (2017), 319-340. (2017) Zbl1387.13040MR3651656DOI10.1007/s13366-016-0322-6
  10. Hartshorne, R., 10.1007/BFb0073971, Lecture Notes in Mathematics 41, Springer, Berlin (1967). (1967) Zbl0185.49202MR224620DOI10.1007/BFb0073971
  11. Hellus, M., Schenzel, P., 10.1016/j.jalgebra.2008.09.006, J. Algebra 320 (2008), 3733-3748. (2008) Zbl1157.13012MR2457720DOI10.1016/j.jalgebra.2008.09.006
  12. Hellus, M., Stückrad, J., 10.1090/S0002-9939-08-09240-X, Proc. Am. Math. Soc. 136 (2008), 2333-2341. (2008) Zbl1152.13011MR2390499DOI10.1090/S0002-9939-08-09240-X
  13. Hochster, M., Huneke, C., 10.1090/conm/159, Commutative Algebra: Syzygies, Multiplicities, and Birational Algebra W. J. Heinzer et al. Contemp. Math. 159, American Mathematical Society, Providence (1994), 197-208. (1994) Zbl0809.13003MR1266184DOI10.1090/conm/159
  14. Iyengar, S. B., Leuschke, G. J., Leykin, A., Miller, C., Miller, E., Singh, A. K., Walther, U., 10.1090/gsm/087, Graduate Studies in Mathematics 87, American Mathematical Society, Providence (2007). (2007) Zbl1129.13001MR2355715DOI10.1090/gsm/087
  15. Khashyarmanesh, K., 10.4153/CMB-2010-072-1, Can. Math. Bull. 53 (2010), 667-673. (2010) Zbl1203.13019MR2761689DOI10.4153/CMB-2010-072-1
  16. Mafi, A., 10.1007/s10114-009-7418-y, Acta Math. Sin., Engl. Ser. 25 (2009), 917-922. (2009) Zbl1183.13025MR2511535DOI10.1007/s10114-009-7418-y
  17. Mahmood, W., On endomorphism ring of local cohomology modules, Available at https://arxiv.org/abs/1308.2584. 
  18. Payrovi, S., Parsa, M. Lotfi, Artinianness of local cohomology modules defined by a pair of ideals, Bull. Malays. Math. Sci. Soc. (2) 35 (2012), 877-883. (2012) Zbl1253.13018MR2960891
  19. Payrovi, S., Parsa, M. Lotfi, 10.1080/00927872.2011.631206, Commun. Algebra 41 (2013), 627-637. (2013) Zbl1263.13016MR3011786DOI10.1080/00927872.2011.631206
  20. Talemi, A. Pour Eshmanan, Tehranian, A., 10.7508/ijmsi.2014.02.002, Iran. J. Math. Sci. Inform. 9 (2014), 7-13. (2014) Zbl1314.13035MR3330480DOI10.7508/ijmsi.2014.02.002
  21. Sadeghi, M. Y., Eghbali, M., Ahmadi-Amoli, K., 10.1142/S0219498819501792, (to appear) in J. Algebra Appl. DOI10.1142/S0219498819501792
  22. Schenzel, P., 10.1016/j.jalgebra.2003.12.016, J. Algebra 275 (2004), 751-770. (2004) Zbl1103.13014MR2052635DOI10.1016/j.jalgebra.2003.12.016
  23. Schenzel, P., 10.1090/S0002-9939-08-09676-7, Proc. Am. Math. Soc. 137 (2009), 1315-1322. (2009) Zbl1163.13010MR2465654DOI10.1090/S0002-9939-08-09676-7
  24. Schenzel, P., 10.1007/s00013-010-0149-6, Arch. Math. 95 (2010), 115-123. (2010) Zbl1200.13028MR2674247DOI10.1007/s00013-010-0149-6
  25. Schenzel, P., 10.1016/j.jalgebra.2011.07.014, J. Algebra 344 (2011), 229-245. (2011) Zbl1236.13016MR2831938DOI10.1016/j.jalgebra.2011.07.014
  26. Strooker, J. R., 10.1017/CBO9780511629242, London Mathematical Society Lecture Note Series 145, Cambridge University Press, Cambridge (1990). (1990) Zbl0786.13008MR1074178DOI10.1017/CBO9780511629242
  27. Takahashi, R., Yoshino, Y., Yoshizawa, T., 10.1016/j.jpaa.2008.09.008, J. Pure Appl. Algebra 213 (2009), 582-600. (2009) Zbl1160.13013MR2483839DOI10.1016/j.jpaa.2008.09.008
  28. Tehranian, A., Talemi, A. Pour Eshmanan, 10.1142/S1005386713000606, Algebra Colloq. 20 (2013), 637-642. (2013) Zbl1282.13038MR3116792DOI10.1142/S1005386713000606
  29. Tehranian, A., Talemi, A. Pour Eshmanan, 10.1142/S1005386714000546, Algebra Colloq. 21 (2014), 597-604. (2014) Zbl1304.13037MR3266511DOI10.1142/S1005386714000546

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.