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

Cited By: 1

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.