A note on the strong direct summand conjecture

McCullough, Jason

doi:10.1090/S0002-9939-09-09853-0

A note on the strong direct summand conjecture
HTML articles powered by AMS MathViewer

by Jason McCullough PDF

Proc. Amer. Math. Soc. 137 (2009), 2857-2864 Request permission

Abstract:

We prove special cases of the strong direct summand conjecture and the related strong monomial conjecture.

References

S. P. Dutta, Dualizing complex and the canonical element conjecture. II, J. London Math. Soc. (2) 56 (1997), no. 1, 49–63. MR 1462825, DOI 10.1112/S0024610797005292
S. P. Dutta, A note on the monomial conjecture, Trans. Amer. Math. Soc. 350 (1998), no. 7, 2871–2878. MR 1466948, DOI 10.1090/S0002-9947-98-02158-8
S. P. Dutta, Splitting of local cohomology of syzygies of the residue field, J. Algebra 244 (2001), no. 1, 168–185. MR 1856533, DOI 10.1006/jabr.2001.8862
E. Graham Evans and Phillip Griffith, The syzygy problem, Ann. of Math. (2) 114 (1981), no. 2, 323–333. MR 632842, DOI 10.2307/1971296
Shiro Goto, On the associated graded rings of parameter ideals in Buchsbaum rings, J. Algebra 85 (1983), no. 2, 490–534. MR 725097, DOI 10.1016/0021-8693(83)90109-6
Raymond C. Heitmann, The direct summand conjecture in dimension three, Ann. of Math. (2) 156 (2002), no. 2, 695–712. MR 1933722, DOI 10.2307/3597204
Melvin Hochster and Craig Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc. 3 (1990), no. 1, 31–116. MR 1017784, DOI 10.1090/S0894-0347-1990-1017784-6
Melvin Hochster and Craig Huneke, Phantom homology, Mem. Amer. Math. Soc. 103 (1993), no. 490, vi+91. MR 1144758, DOI 10.1090/memo/0490
Melvin Hochster and Craig Huneke, Applications of the existence of big Cohen-Macaulay algebras, Adv. Math. 113 (1995), no. 1, 45–117. MR 1332808, DOI 10.1006/aima.1995.1035
M. Hochster, Contracted ideals from integral extensions of regular rings, Nagoya Math. J. 51 (1973), 25–43. MR 349656
Melvin Hochster, Canonical elements in local cohomology modules and the direct summand conjecture, J. Algebra 84 (1983), no. 2, 503–553. MR 723406, DOI 10.1016/0021-8693(83)90092-3
Melvin Hochster, Big Cohen-Macaulay algebras in dimension three via Heitmann’s theorem, J. Algebra 254 (2002), no. 2, 395–408. MR 1933876, DOI 10.1016/S0021-8693(02)00086-8
Melvin Hochster, Homological conjectures, old and new, Illinois J. Math. 51 (2007), no. 1, 151–169. MR 2346192
Jee Koh, Degree $p$ extensions of an unramified regular local ring of mixed characteristic $p$, J. Algebra 99 (1986), no. 2, 310–323. MR 837546, DOI 10.1016/0021-8693(86)90029-3
Nandini Ranganathan, Splitting in module-finite extension rings and the vanishing conjecture for maps of tor, Ph.D. thesis, University of Michigan, 2000.
Paul Roberts, Two applications of dualizing complexes over local rings, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 1, 103–106. MR 399075
Jan R. Strooker and Jürgen Stückrad, Monomial conjecture and complete intersections, Manuscripta Math. 79 (1993), no. 2, 153–159. MR 1216771, DOI 10.1007/BF02568334
Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, Graduate Texts in Mathematics, Vol. 29, Springer-Verlag, New York-Heidelberg, 1975. Reprint of the 1960 edition. MR 0389876