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Year 2024, Volume: 35 Issue: 35, 61 - 81, 09.01.2024

David Ssevvıırı

https://doi.org/10.24330/ieja.1299587

Abstract

References

  • L. Alonso Tarrio, A. Jeremias Lopez and J. Lipman, Local homology and cohomology on schemes, Ann. Sci. École Norm. Sup., 30(1) (1997), 1-39.
  • I. Assem, D. Simson and A. Skowronski, Elements of the Representation Theory of Associative Algebras, Volume 1: Techniques of Representation Theory, Cambridge University Press, 2006.
  • M. Auslander, M. I. Platzeck and G. Todorov, Homological theory of idempotent ideals, Trans. Amer. Math. Soc., 332(2) (1992), 667-692.
  • T. Barthel, D. Heard and G. Valenzuela, Local duality in algebra and topology, Adv. Math., 335 (2018), 563-663.
  • T. Barthel, D. Heard and G. Valenzuela, Derived completion for comodules, Manuscripta Math., 161 (2020), 409-438.
  • M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Mathematics, 136, Cambridge University Press, Cambridge, Second Edition, 2013.
  • K. Divaani-Aazar, H. Faridian and M. Tousi, Local homology, finiteness of Tor modules and cofiniteness, J. Algebra Appl., 16(12) (2017), 1750240 (10 pp).
  • T. H. Freitas, V. H. Jorge-Perez, C. B. Miranda-Neto and P. Schenzel, Generalized local duality, canonical modules, and prescribed bound on projective dimension, J. Pure Appl. Algebra, 227(2) (2023), 107188, (17 pp).
  • J. P. C. Greenless and J. P. May, Derived functors of $I$-adic completion and local homology, J. Algebra, 149(2) (1992), 438-453.
  • R. Hartshorne, Local Cohomology, a seminar given by A. Grothendieck, Harvard University, Fall, 1961, Lecture Notes in Mathematics, 41, Springer-Verlag, Berlin-New York 1967.
  • G. Kempf, The Grothendieck-Cousin complex of an induced representation, Adv. in Math., 29(3) (1978), 310-396.
  • A. Kyomuhangi and D. Ssevviiri, The locally nilradical for modules over commutative rings, Beitr. Algebra Geom., 61(4) (2020), 759-769.
  • A. Kyomuhangi and D. Ssevviiri, Generalised reduced modules, Rend. Circ. Mat. Palermo (2), 72(1) (2023), 421-431.
  • T. K. Lee and Y. Zhou, Reduced modules, rings, modules, algebras and abelian groups, Lecture Notes in Pure and Applied Math, 236, 365-377, Marcel Dekker, New York, 2004.
  • S. MacLane, Categories for the Working Mathematician, Berlin, Heidelberg, New York, Springer-Verlag, 1971.
  • M. Porta, L. Shaul and A. Yekutieli, On the homology of completion and torsion, Algebr. Represent. Theory, 17(1) (2014), 31-67.
  • M. B. Rege and A. M. Buhphang, On reduced modules and rings, Int. Electron. J. Algebra, 3 (2008), 58-74.
  • P. Schenzel and A. M. Simon, Completion, Cech and Local Homology and Cohomology, Springer Monographs in Mathematics, 2018.
  • R. Y. Sharp, Local cohomology theory in commutative algebra, Quart. J. Math. Oxford Ser. (2), 21 (1970), 425-434.
  • D. Ssevviiri, Nilpotent elements control the structure of a module, arXiv:1812.04320 [math.RA], (2018).
  • N. Suzuki, On the generalized local cohomology and its duality, J. Math. Kyoto Univ., 18 (1978), 71-85.
  • R. Vyas and A. Yekutieli, Weak proregularity, weak stability, and the noncommutative MGM equivalence, J. Algebra, 513 (2018), 265-325.
  • R. Vyas, Weakly stable torsion classes, Algebr. Represent. Theory, 22(5) (2019), 1183-1207.
  • S. Yassemi, Generalized section functors, J. Pure Appl. Algebra, 95(1) (1994), 103-119.
  • A. Yekutieli, On flatness and completion for infinitely generated modules over Noetherian rings, Comm. Algebra, 39(11) (2011), 4221-4245.

Applications of reduced and coreduced modules I

Year 2024, Volume: 35 Issue: 35, 61 - 81, 09.01.2024

David Ssevvıırı

https://doi.org/10.24330/ieja.1299587

Abstract

This is the first in a series of papers highlighting the applications of reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ be an ideal of $R$. We show that $I$-reduced $R$-modules and $I$-coreduced $R$-modules provide a setting in which the Matlis-Greenless-May (MGM) Equivalence and the Greenless-May (GM) Duality hold. These two notions have been hitherto only known to exist in the derived category setting. We realise the $I$-torsion and the $I$-adic completion functors as representable functors and under suitable conditions compute natural transformations between them and other functors.

Keywords

torsion and adic completion coreduced module reduced module

References

  • L. Alonso Tarrio, A. Jeremias Lopez and J. Lipman, Local homology and cohomology on schemes, Ann. Sci. École Norm. Sup., 30(1) (1997), 1-39.
  • I. Assem, D. Simson and A. Skowronski, Elements of the Representation Theory of Associative Algebras, Volume 1: Techniques of Representation Theory, Cambridge University Press, 2006.
  • M. Auslander, M. I. Platzeck and G. Todorov, Homological theory of idempotent ideals, Trans. Amer. Math. Soc., 332(2) (1992), 667-692.
  • T. Barthel, D. Heard and G. Valenzuela, Local duality in algebra and topology, Adv. Math., 335 (2018), 563-663.
  • T. Barthel, D. Heard and G. Valenzuela, Derived completion for comodules, Manuscripta Math., 161 (2020), 409-438.
  • M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge Studies in Advanced Mathematics, 136, Cambridge University Press, Cambridge, Second Edition, 2013.
  • K. Divaani-Aazar, H. Faridian and M. Tousi, Local homology, finiteness of Tor modules and cofiniteness, J. Algebra Appl., 16(12) (2017), 1750240 (10 pp).
  • T. H. Freitas, V. H. Jorge-Perez, C. B. Miranda-Neto and P. Schenzel, Generalized local duality, canonical modules, and prescribed bound on projective dimension, J. Pure Appl. Algebra, 227(2) (2023), 107188, (17 pp).
  • J. P. C. Greenless and J. P. May, Derived functors of $I$-adic completion and local homology, J. Algebra, 149(2) (1992), 438-453.
  • R. Hartshorne, Local Cohomology, a seminar given by A. Grothendieck, Harvard University, Fall, 1961, Lecture Notes in Mathematics, 41, Springer-Verlag, Berlin-New York 1967.
  • G. Kempf, The Grothendieck-Cousin complex of an induced representation, Adv. in Math., 29(3) (1978), 310-396.
  • A. Kyomuhangi and D. Ssevviiri, The locally nilradical for modules over commutative rings, Beitr. Algebra Geom., 61(4) (2020), 759-769.
  • A. Kyomuhangi and D. Ssevviiri, Generalised reduced modules, Rend. Circ. Mat. Palermo (2), 72(1) (2023), 421-431.
  • T. K. Lee and Y. Zhou, Reduced modules, rings, modules, algebras and abelian groups, Lecture Notes in Pure and Applied Math, 236, 365-377, Marcel Dekker, New York, 2004.
  • S. MacLane, Categories for the Working Mathematician, Berlin, Heidelberg, New York, Springer-Verlag, 1971.
  • M. Porta, L. Shaul and A. Yekutieli, On the homology of completion and torsion, Algebr. Represent. Theory, 17(1) (2014), 31-67.
  • M. B. Rege and A. M. Buhphang, On reduced modules and rings, Int. Electron. J. Algebra, 3 (2008), 58-74.
  • P. Schenzel and A. M. Simon, Completion, Cech and Local Homology and Cohomology, Springer Monographs in Mathematics, 2018.
  • R. Y. Sharp, Local cohomology theory in commutative algebra, Quart. J. Math. Oxford Ser. (2), 21 (1970), 425-434.
  • D. Ssevviiri, Nilpotent elements control the structure of a module, arXiv:1812.04320 [math.RA], (2018).
  • N. Suzuki, On the generalized local cohomology and its duality, J. Math. Kyoto Univ., 18 (1978), 71-85.
  • R. Vyas and A. Yekutieli, Weak proregularity, weak stability, and the noncommutative MGM equivalence, J. Algebra, 513 (2018), 265-325.
  • R. Vyas, Weakly stable torsion classes, Algebr. Represent. Theory, 22(5) (2019), 1183-1207.
  • S. Yassemi, Generalized section functors, J. Pure Appl. Algebra, 95(1) (1994), 103-119.
  • A. Yekutieli, On flatness and completion for infinitely generated modules over Noetherian rings, Comm. Algebra, 39(11) (2011), 4221-4245.
There are 25 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

David Ssevvıırı This is me

Early Pub Date May 24, 2023
Publication Date January 9, 2024
Published in Issue Year 2024 Volume: 35 Issue: 35

Cite

APA Ssevvıırı, D. (2024). Applications of reduced and coreduced modules I. International Electronic Journal of Algebra, 35(35), 61-81. https://doi.org/10.24330/ieja.1299587
AMA Ssevvıırı D. Applications of reduced and coreduced modules I. IEJA. January 2024;35(35):61-81. doi:10.24330/ieja.1299587
Chicago Ssevvıırı, David. “Applications of Reduced and Coreduced Modules I”. International Electronic Journal of Algebra 35, no. 35 (January 2024): 61-81. https://doi.org/10.24330/ieja.1299587.
EndNote Ssevvıırı D (January 1, 2024) Applications of reduced and coreduced modules I. International Electronic Journal of Algebra 35 35 61–81.
IEEE D. Ssevvıırı, “Applications of reduced and coreduced modules I”, IEJA, vol. 35, no. 35, pp. 61–81, 2024, doi: 10.24330/ieja.1299587.
ISNAD Ssevvıırı, David. “Applications of Reduced and Coreduced Modules I”. International Electronic Journal of Algebra 35/35 (January 2024), 61-81. https://doi.org/10.24330/ieja.1299587.
JAMA Ssevvıırı D. Applications of reduced and coreduced modules I. IEJA. 2024;35:61–81.
MLA Ssevvıırı, David. “Applications of Reduced and Coreduced Modules I”. International Electronic Journal of Algebra, vol. 35, no. 35, 2024, pp. 61-81, doi:10.24330/ieja.1299587.
Vancouver Ssevvıırı D. Applications of reduced and coreduced modules I. IEJA. 2024;35(35):61-8.