Finiteness of cousin cohomologies

Kawasaki, Takesi

doi:10.1090/S0002-9947-07-04418-2

Finiteness of cousin cohomologies
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Trans. Amer. Math. Soc. 360 (2008), 2709-2739 Request permission

Abstract:

The notion of the Cousin complex of a module was given by Sharp in 1969. It wasn’t known whether its cohomologies are finitely generated until recently. In 2001, Dibaei and Tousi showed that the Cousin cohomologies of a finitely generated $A$-module $M$ are finitely generated if the base ring $A$ is local, has a dualizing complex, $M$ satisfies Serre’s $(S_2)$-condition and is equidimensional. In the present article, the author improves their result. He shows that the Cousin cohomologies of $M$ are finitely generated if $A$ is universally catenary, all the formal fibers of all the localizations of $A$ are Cohen-Macaulay, the Cohen-Macaulay locus of each finitely generated $A$-algebra is open and all the localizations of $M$ are equidimensional. As a consequence of this, he gives a necessary and sufficient condition for a Noetherian ring to have an arithmetic Macaulayfication.

References

Yôichi Aoyama, Some basic results on canonical modules, J. Math. Kyoto Univ. 23 (1983), no. 1, 85–94. MR 692731, DOI 10.1215/kjm/1250521612
Nguyễn Tụ’ Cu’ò’ng, On the length of the powers of systems of parameters in local ring, Nagoya Math. J. 120 (1990), 77–88. MR 1086571, DOI 10.1017/s0027763000003263
Nguyễn Tụ’ Cu’ò’ng, On the dimension of the non-Cohen-Macaulay locus of local rings admitting dualizing complexes, Math. Proc. Cambridge Philos. Soc. 109 (1991), no. 3, 479–488. MR 1094747, DOI 10.1017/S0305004100069929
M. T. Dibaei and M. Tousi, The structure of dualizing complex for a ring which is $(\textrm {S}_2)$, J. Math. Kyoto Univ. 38 (1998), no. 3, 503–516. MR 1661220, DOI 10.1215/kjm/1250518063
M. T. Dibaei and M. Tousi, A generalization of the dualizing complex structure and its applications, J. Pure Appl. Algebra 155 (2001), no. 1, 17–28. MR 1804325, DOI 10.1016/S0022-4049(99)00160-7
Gerd Faltings, Der Endlichkeitssatz in der lokalen Kohomologie, Math. Ann. 255 (1981), no. 1, 45–56 (German). MR 611272, DOI 10.1007/BF01450555
Robert Fossum, Hans-Bjørn Foxby, Phillip Griffith, and Idun Reiten, Minimal injective resolutions with applications to dualizing modules and Gorenstein modules, Inst. Hautes Études Sci. Publ. Math. 45 (1975), 193–215. MR 396529, DOI 10.1007/BF02684302
Shiro Goto, Approximately Cohen-Macaulay rings, J. Algebra 76 (1982), no. 1, 214–225. MR 659220, DOI 10.1016/0021-8693(82)90248-4
Shiro Goto and Kikumichi Yamagishi, The theory of unconditioned strong d-sequences and modules of finite local cohomology, preprint.
Janet E. Hall, Fundamental dualizing complexes for commutative Noetherian rings, Quart. J. Math. Oxford Ser. (2) 30 (1979), no. 117, 21–32. MR 528888, DOI 10.1093/qmath/30.1.21
Robin Hartshorne, Residues and duality, Lecture Notes in Mathematics, No. 20, Springer-Verlag, Berlin-New York, 1966. Lecture notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64; With an appendix by P. Deligne. MR 0222093, DOI 10.1007/BFb0080482
Craig Huneke, The theory of $d$-sequences and powers of ideals, Adv. in Math. 46 (1982), no. 3, 249–279. MR 683201, DOI 10.1016/0001-8708(82)90045-7
Eero Hyry, The diagonal subring and the Cohen-Macaulay property of a multigraded ring, Trans. Amer. Math. Soc. 351 (1999), no. 6, 2213–2232. MR 1467469, DOI 10.1090/S0002-9947-99-02143-1
Takesi Kawasaki, On Macaulayfication of Noetherian schemes, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2517–2552. MR 1707481, DOI 10.1090/S0002-9947-00-02603-9
Takesi Kawasaki, On arithmetic Macaulayfication of Noetherian rings, Trans. Amer. Math. Soc. 354 (2002), no. 1, 123–149. MR 1859029, DOI 10.1090/S0002-9947-01-02817-3
Tetsushi Ogoma, Fibre products of Noetherian rings and their applications, Math. Proc. Cambridge Philos. Soc. 97 (1985), no. 2, 231–241. MR 771818, DOI 10.1017/S0305004100062794
Henrike Petzl, Cousin complexes and flat ring extensions, Comm. Algebra 25 (1997), no. 1, 311–339. MR 1429765, DOI 10.1080/00927879708825856
L. J. Ratliff Jr., Characterizations of catenary rings, Amer. J. Math. 93 (1971), 1070–1108. MR 297752, DOI 10.2307/2373746
Peter Schenzel, Cohomological annihilators, Math. Proc. Cambridge Philos. Soc. 91 (1982), no. 3, 345–350. MR 654081, DOI 10.1017/S0305004100059417
Peter Schenzel, Standard systems of parameters and their blowing-up rings, J. Reine Angew. Math. 344 (1983), 201–220. MR 716256, DOI 10.1515/crll.1983.344.201
R. Y. Sharp, Cousin complex characterizations of two classes of commutative Noetherian rings, J. London Math. Soc. (2) 3 (1971), 621–624. MR 294323, DOI 10.1112/jlms/s2-3.4.621
Rodney Y. Sharp, The Cousin complex for a module over a commutative Noetherian ring, Math. Z. 112 (1969), 340–356. MR 263800, DOI 10.1007/BF01110229
Rodney Y. Sharp, Acceptable rings and homomorphic images of Gorenstein rings, J. Algebra 44 (1977), no. 1, 246–261. MR 441957, DOI 10.1016/0021-8693(77)90180-6
Rodney Y. Sharp, Local cohomology and the Cousin complex for a commutative Noetherian ring, Math. Z. 153 (1977), no. 1, 19–22. MR 442062, DOI 10.1007/BF01214729
Ngô Việt Trung, Toward a theory of generalized Cohen-Macaulay modules, Nagoya Math. J. 102 (1986), 1–49. MR 846128, DOI 10.1017/s0027763000000416