Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
regular ring; Ohm-Rush content theory; intersection flat; bracket power; Frobenius endomorphism
regular ring; Ohm-Rush content theory; intersection flat; bracket power; Frobenius endomorphism
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.
References:
[1] Bourbaki, N.: Elements of Mathematics. Commutative Algebra. Hermann, Paris (1972). MR 0360549 | Zbl 0279.13001
[2] Chevalley, C.: On the theory of local rings. Ann. Math. (2) 44 (1943), 690-708. DOI 10.2307/1969105 | MR 0009603 | Zbl 0060.06908
[3] Epstein, N., Shapiro, J.: The Ohm-Rush content function. J. Algebra Appl. 15 (2016), Article ID 1650009, 14 pages. DOI 10.1142/S0219498816500092 | MR 3393938 | Zbl 1333.13009
[4] Epstein, N., Yao, Y.: Criteria for flatness and injectivity. Math. Z. 271 (2012), 1193-1210. DOI 10.1007/s00209-011-0910-y | MR 2945604 | Zbl 1245.13009
[5] Hochster, M., Huneke, C.: $F$-regularity, test elements, and smooth base change. Trans. Am. Math. Soc. 346 (1994), 1-62. DOI 10.2307/2154942 | MR 1273534 | Zbl 0844.13002
[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). DOI 10.1090/conm/773 | MR 4321391
[7] Katzman, M.: Parameter-test-ideals of Cohen-Macaulay rings. Compos. Math. 144 (2008), 933-948. DOI 10.1112/S0010437X07003417 | MR 2441251 | Zbl 1152.13005
[8] Kunz, E.: Characterizations of regular local rings of characteristic $p$. Am. J. Math. 91 (1969), 772-784. DOI 10.2307/2373351 | MR 0252389 | Zbl 0188.33702
[9] Miller, C.: The Frobenius endomorphism and homological dimensions. Commutative Algebra: Interactions with Algebraic Geometry Contemporary Mathematics 331. AMS, Providence (2003), 207-234. DOI 10.1090/conm/331 | MR 2013168 | Zbl 1085.13502
[10] Ohm, J., Rush, D. E.: Content modules and algebras. Math. Scand. 31 (1972), 49-68. DOI 10.7146/math.scand.a-11411 | MR 0344289 | Zbl 0248.13013
[11] Sharp, R. Y.: Big tight closure test elements for some non-reduced excellent rings. J. Algebra 349 (2012), 284-316. DOI 10.1016/j.jalgebra.2011.08.009 | MR 2853638 | Zbl 1256.13004
[12] Zhang, W.: On the Frobenius power and colon ideals. Commun. Algebra 37 (2009), 2391-2395. DOI 10.1080/00927870802216438 | MR 2536926 | Zbl 1185.13011
Search
Partner of
