Title: | Regularity and intersections of bracket powers (English) |
Author: | Epstein, Neil |
Language: | English |
Journal: | Czechoslovak Mathematical Journal |
ISSN: | 0011-4642 (print) |
ISSN: | 1572-9141 (online) |
Volume: | 72 |
Issue: | 2 |
Year: | 2022 |
Pages: | 593-599 |
Summary lang: | English |
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Category: | math |
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Summary: | Among reduced Noetherian prime characteristic commutative rings, we prove that a regular ring is precisely that where the finite intersection of ideals commutes with taking bracket powers. However, reducedness is essential for this equivalence. Connections are made with Ohm-Rush content theory, intersection-flatness of the Frobenius map, and various flatness criteria. (English) |
Keyword: | regular ring |
Keyword: | Ohm-Rush content theory |
Keyword: | intersection flat |
Keyword: | bracket power |
Keyword: | Frobenius endomorphism |
MSC: | 13A35 |
MSC: | 13B40 |
MSC: | 13H05 |
idZBL: | Zbl 07547221 |
idMR: | MR4412776 |
DOI: | 10.21136/CMJ.2022.0066-21 |
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Date available: | 2022-04-21T19:06:03Z |
Last updated: | 2022-09-08 |
Stable URL: | http://hdl.handle.net/10338.dmlcz/150418 |
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Reference: | [1] Bourbaki, N.: Elements of Mathematics. Commutative Algebra.Hermann, Paris (1972). Zbl 0279.13001, MR 0360549 |
Reference: | [2] Chevalley, C.: On the theory of local rings.Ann. Math. (2) 44 (1943), 690-708. Zbl 0060.06908, MR 0009603, 10.2307/1969105 |
Reference: | [3] Epstein, N., Shapiro, J.: The Ohm-Rush content function.J. Algebra Appl. 15 (2016), Article ID 1650009, 14 pages. Zbl 1333.13009, MR 3393938, 10.1142/S0219498816500092 |
Reference: | [4] Epstein, N., Yao, Y.: Criteria for flatness and injectivity.Math. Z. 271 (2012), 1193-1210. Zbl 1245.13009, MR 2945604, 10.1007/s00209-011-0910-y |
Reference: | [5] Hochster, M., Huneke, C.: $F$-regularity, test elements, and smooth base change.Trans. Am. Math. Soc. 346 (1994), 1-62. Zbl 0844.13002, MR 1273534, 10.2307/2154942 |
Reference: | [6] Hochster, M., Jeffries, J.: Extensions of primes, flatness, and intersection flatness 150 Years with Roger and Sylvia Wiegand.Commutative Algebra - 150 Years with Roger and Sylvia Wiegand Contemporary Mathematics 773. AMS, Providence (2021). MR 4321391, 10.1090/conm/773 |
Reference: | [7] Katzman, M.: Parameter-test-ideals of Cohen-Macaulay rings.Compos. Math. 144 (2008), 933-948. Zbl 1152.13005, MR 2441251, 10.1112/S0010437X07003417 |
Reference: | [8] Kunz, E.: Characterizations of regular local rings of characteristic $p$.Am. J. Math. 91 (1969), 772-784. Zbl 0188.33702, MR 0252389, 10.2307/2373351 |
Reference: | [9] Miller, C.: The Frobenius endomorphism and homological dimensions.Commutative Algebra: Interactions with Algebraic Geometry Contemporary Mathematics 331. AMS, Providence (2003), 207-234. Zbl 1085.13502, MR 2013168, 10.1090/conm/331 |
Reference: | [10] Ohm, J., Rush, D. E.: Content modules and algebras.Math. Scand. 31 (1972), 49-68. Zbl 0248.13013, MR 0344289, 10.7146/math.scand.a-11411 |
Reference: | [11] Sharp, R. Y.: Big tight closure test elements for some non-reduced excellent rings.J. Algebra 349 (2012), 284-316. Zbl 1256.13004, MR 2853638, 10.1016/j.jalgebra.2011.08.009 |
Reference: | [12] Zhang, W.: On the Frobenius power and colon ideals.Commun. Algebra 37 (2009), 2391-2395. Zbl 1185.13011, MR 2536926, 10.1080/00927870802216438 |
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