A note on Non-Noetherian Cohen-Macaulay rings

Kim, Youngsu; Walker, Andrew

doi:10.1090/proc/14836

A note on Non-Noetherian Cohen-Macaulay rings
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by Youngsu Kim and Andrew Walker

Proc. Amer. Math. Soc. 148 (2020), 1031-1042

DOI: https://doi.org/10.1090/proc/14836

Published electronically: December 6, 2019

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Abstract:

In this note, we study the Cohen-Macaulayness of non-Noetherian rings. We show that Hochster’s celebrated theorem that a finitely generated normal semigroup ring is Cohen-Macaulay does not extend to non-Noetherian rings. We also show that for any valuation domain $V$ of finite Krull dimension, $V[x]$ is Cohen-Macaulay in the sense of Hamilton-Marley.

References

Mohsen Asgharzadeh, Mehdi Dorreh, and Massoud Tousi, Direct limits of Cohen-Macaulay rings, J. Pure Appl. Algebra 218 (2014), no. 9, 1730–1744. MR 3188868, DOI 10.1016/j.jpaa.2014.01.010
Bernard Alfonsi, Grade non-noethérien, Comm. Algebra 9 (1981), no. 8, 811–840 (French). MR 611560, DOI 10.1080/00927878108822620
David F. Anderson, Robert Gilmer’s work on semigroup rings, Multiplicative ideal theory in commutative algebra, Springer, New York, 2006, pp. 21–37. MR 2265799, DOI 10.1007/978-0-387-36717-0_{2}
Mohsen Asgharzadeh and Massoud Tousi, On the notion of Cohen-Macaulayness for non-Noetherian rings, J. Algebra 322 (2009), no. 7, 2297–2320. MR 2553201, DOI 10.1016/j.jalgebra.2009.06.017
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
M. P. Brodmann and R. Y. Sharp, Local cohomology, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 136, Cambridge University Press, Cambridge, 2013. An algebraic introduction with geometric applications. MR 3014449
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
Marco D’Anna, Carmelo A. Finocchiaro, and Marco Fontana, Properties of chains of prime ideals in an amalgamated algebra along an ideal, J. Pure Appl. Algebra 214 (2010), no. 9, 1633–1641. MR 2593689, DOI 10.1016/j.jpaa.2009.12.008
Sarah Glaz, On the weak dimension of coherent group rings, Comm. Algebra 15 (1987), no. 9, 1841–1858. MR 898295, DOI 10.1080/00927878708823507
Sarah Glaz, Commutative coherent rings, Lecture Notes in Mathematics, vol. 1371, Springer-Verlag, Berlin, 1989. MR 999133, DOI 10.1007/BFb0084570
Sarah Glaz, On the coherence and weak dimension of the rings $R\langle x\rangle$ and $R(x)$, Proc. Amer. Math. Soc. 106 (1989), no. 3, 579–587. MR 961405, DOI 10.1090/S0002-9939-1989-0961405-9
Sarah Glaz, Coherence, regularity and homological dimensions of commutative fixed rings, Commutative algebra (Trieste, 1992) World Sci. Publ., River Edge, NJ, 1994, pp. 89–106. MR 1421079
Sarah Glaz, Homological dimensions of localizations of polynomial rings, Zero-dimensional commutative rings (Knoxville, TN, 1994) Lecture Notes in Pure and Appl. Math., vol. 171, Dekker, New York, 1995, pp. 209–222. MR 1335716
Tracy Dawn Hamilton, Weak Bourbaki unmixed rings: a step towards non-Noetherian Cohen-Macaulayness, Rocky Mountain J. Math. 34 (2004), no. 3, 963–977. MR 2087441, DOI 10.1216/rmjm/1181069837
Robin Hartshorne, Local cohomology, Lecture Notes in Mathematics, No. 41, Springer-Verlag, Berlin-New York, 1967. A seminar given by A. Grothendieck, Harvard University, Fall, 1961. MR 0224620, DOI 10.1007/BFb0073971
Tracy Dawn Hamilton and Thomas Marley, Non-Noetherian Cohen-Macaulay rings, J. Algebra 307 (2007), no. 1, 343–360. MR 2278059, DOI 10.1016/j.jalgebra.2006.08.003
Livia Hummel and Thomas Marley, The Auslander-Bridger formula and the Gorenstein property for coherent rings, J. Commut. Algebra 1 (2009), no. 2, 283–314. MR 2504937, DOI 10.1216/JCA-2009-1-2-283
M. Hochster, Rings of invariants of tori, Cohen-Macaulay rings generated by monomials, and polytopes, Ann. of Math. (2) 96 (1972), 318–337. MR 304376, DOI 10.2307/1970791
M. Hochster, Grade-sensitive modules and perfect modules, Proc. London Math. Soc. (3) 29 (1974), 55–76. MR 374118, DOI 10.1112/plms/s3-29.1.55
Srikanth B. Iyengar, Graham J. Leuschke, Anton Leykin, Claudia Miller, Ezra Miller, Anurag K. Singh, and Uli Walther, Twenty-four hours of local cohomology, Graduate Studies in Mathematics, vol. 87, American Mathematical Society, Providence, RI, 2007. MR 2355715, DOI 10.1090/gsm/087
Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
D. G. Northcott, Finite free resolutions, Cambridge Tracts in Mathematics, No. 71, Cambridge University Press, Cambridge-New York-Melbourne, 1976. MR 0460383, DOI 10.1017/CBO9780511565892
Gabriel Sabbagh, Coherence of polynomial rings and bounds in polynomial ideals, J. Algebra 31 (1974), 499–507. MR 347796, DOI 10.1016/0021-8693(74)90128-8
Peter Schenzel, Proregular sequences, local cohomology, and completion, Math. Scand. 92 (2003), no. 2, 161–180. MR 1973941, DOI 10.7146/math.scand.a-14399
Jean-Pierre Soublin, Anneaux et modules cohérents, J. Algebra 15 (1970), 455–472 (French). MR 260799, DOI 10.1016/0021-8693(70)90050-5
Wolmer V. Vasconcelos, The rings of dimension two, Lecture Notes in Pure and Applied Mathematics, Vol. 22, Marcel Dekker, Inc., New York-Basel, 1976. MR 0427290