A generalization of Hartshorne's connectedness theorem
DOI:
https://doi.org/10.5269/bspm.44323Abstract
In this paper, we use local cohomology theory to present some results about connectedness property of prime spectrum of modules. In particular, we generalize the Hartshorne's connectedness theorem.
References
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9. M. Eghbali and P. Schenzel, On an endomorphism ring of local cohomology, Comm. Algebra 40 (2012), 4295-4305. https://doi.org/10.1080/00927872.2011.588982
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12. R. Hartshorne, Complete intersection and connectedness, Amer. J. Math. 84 (1962), 497-508. https://doi.org/10.2307/2372986
13. D. Hassanzadeh-Lelekaami and H. Roshan-Shekalgourabi, Prime submodules and a sheaf on the prime spectra of modules, Comm. Algebra 42 (2014), no. 7, 3063-3077. https://doi.org/10.1080/00927872.2013.780063
14. Chin-Pi Lu, Prime submodules of modules, Comment. Math. Univ. St. Pauli 33 (1984), no. 1, 61-69.
15. Chin-Pi Lu, Spectra of modules, Comm. Algebra 23 (1995), no. 10, 3741-3752. https://doi.org/10.1080/00927879508825430
16. Chin-Pi Lu, The Zariski topology on the prime spectrum of a module, Houston J. Math. 25 (1999), no. 3, 417-432.
17. Chin-Pi Lu, Modules with Noetherian spectrum, Comm. Algebra 38 (2010), no. 3, 807-828. https://doi.org/10.1080/00927870802578050
18. R. L. McCasland, M. E. Moore, and P. F. Smith, On the spectrum of a module over a commutative ring, Comm. Algebra 25 (1997), no. 1, 79-103. https://doi.org/10.1080/00927879708825840
19. P. Schenzel, On formal local cohomology and connectedness, J. Algebra 315 (2007), 894-923. https://doi.org/10.1016/j.jalgebra.2007.06.015
20. M. Tousi and S. Yassemi, The Lichtenbaum-Hartshorne theorem for modules which are finite over a ring homomorphism, J. Pure Appl. Algebra 212 (2008), 1222-1228. https://doi.org/10.1016/j.jpaa.2007.09.003
2. M. Behboodi, A generalization of the classical Krull dimension for modules, J. Algebra 305 (2006), 1128-1148. https://doi.org/10.1016/j.jalgebra.2006.04.010
3. M. Behboodi and M. R. Haddadi, Classical Zariski topology of modules and spectral spaces I, Int. Electron. J. Algebra 4 (2008), 104-130.
4. M. P. Brodmann and R. Y. Sharp, Local cohomology: An algebraic introduction with geometric applications, Cambridge University Press, 1998. https://doi.org/10.1017/CBO9780511629204
5. W. Bruns and J. Herzog, Cohen-Macaulay rings, revised ed., Cambridge University Press, 1998. https://doi.org/10.1017/CBO9780511608681
6. J. Dauns, Prime modules, J. Reine Angew. Math. 298 (1978), 156-181. https://doi.org/10.1515/crll.1978.298.156
7. K. Divaani-Aazar, R. Naghipour, and M. Tousi, The Lichtenbaum-Hartshorne theorem for generalized local cohomology and connectedness, Comm. Algebra 30 (2002), no. 8, 3687-3702. https://doi.org/10.1081/AGB-120005813
8. K. Divaani-Aazar and P. Schenzel, Ideal topologies, local cohomology and connectedness, Math. Proc. Cambridge Philos. Soc. 131 (2001), 211-226. https://doi.org/10.1017/S0305004101005229
9. M. Eghbali and P. Schenzel, On an endomorphism ring of local cohomology, Comm. Algebra 40 (2012), 4295-4305. https://doi.org/10.1080/00927872.2011.588982
10. G. Faltings, Algebraisation of some formal vector bundles, Ann. of Math. 110 (1979), 501-514. https://doi.org/10.2307/1971235
11. E. H. Feller and E. W. Swokowski, Prime modules, Canad. J. Math. 17 (1965), 1041-1052. https://doi.org/10.4153/CJM-1965-099-5
12. R. Hartshorne, Complete intersection and connectedness, Amer. J. Math. 84 (1962), 497-508. https://doi.org/10.2307/2372986
13. D. Hassanzadeh-Lelekaami and H. Roshan-Shekalgourabi, Prime submodules and a sheaf on the prime spectra of modules, Comm. Algebra 42 (2014), no. 7, 3063-3077. https://doi.org/10.1080/00927872.2013.780063
14. Chin-Pi Lu, Prime submodules of modules, Comment. Math. Univ. St. Pauli 33 (1984), no. 1, 61-69.
15. Chin-Pi Lu, Spectra of modules, Comm. Algebra 23 (1995), no. 10, 3741-3752. https://doi.org/10.1080/00927879508825430
16. Chin-Pi Lu, The Zariski topology on the prime spectrum of a module, Houston J. Math. 25 (1999), no. 3, 417-432.
17. Chin-Pi Lu, Modules with Noetherian spectrum, Comm. Algebra 38 (2010), no. 3, 807-828. https://doi.org/10.1080/00927870802578050
18. R. L. McCasland, M. E. Moore, and P. F. Smith, On the spectrum of a module over a commutative ring, Comm. Algebra 25 (1997), no. 1, 79-103. https://doi.org/10.1080/00927879708825840
19. P. Schenzel, On formal local cohomology and connectedness, J. Algebra 315 (2007), 894-923. https://doi.org/10.1016/j.jalgebra.2007.06.015
20. M. Tousi and S. Yassemi, The Lichtenbaum-Hartshorne theorem for modules which are finite over a ring homomorphism, J. Pure Appl. Algebra 212 (2008), 1222-1228. https://doi.org/10.1016/j.jpaa.2007.09.003
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2021-12-18
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