Big Cohen-Macaulay algebras and seeds

Dietz, Geoffrey

doi:10.1090/S0002-9947-07-04252-3

Big Cohen-Macaulay algebras and seeds
HTML articles powered by AMS MathViewer

by Geoffrey D. Dietz PDF

Trans. Amer. Math. Soc. 359 (2007), 5959-5989 Request permission

Abstract:

In this article, we delve into the properties possessed by algebras, which we have termed seeds, that map to big Cohen-Macaulay algebras. We will show that over a complete local domain of positive characteristic any two big Cohen-Macaulay algebras map to a common big Cohen-Macaulay algebra. We will also strengthen Hochster and Huneke’s “weakly functorial" existence result for big Cohen-Macaulay algebras by showing that the seed property is stable under base change between complete local domains of positive characteristic. We also show that every seed over a positive characteristic ring $(R,m)$ maps to a balanced big Cohen-Macaulay $R$-algebra that is an absolutely integrally closed, $m$-adically separated, quasilocal domain.

References

Luchezar L. Avramov, Hans-Bjørn Foxby, and Bernd Herzog, Structure of local homomorphisms, J. Algebra 164 (1994), no. 1, 124–145. MR 1268330, DOI 10.1006/jabr.1994.1057
Jaap Bartijn and Jan R. Strooker, Modifications monomiales, Paul Dubreil and Marie-Paule Malliavin algebra seminar, 35th year (Paris, 1982) Lecture Notes in Math., vol. 1029, Springer, Berlin, 1983, pp. 192–217 (French). MR 732476, DOI 10.1007/BFb0098932
Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
G. Dietz, Closure Operations in Positive Characteristic and Big Cohen-Macaulay Algebras, Thesis, Univ. of Michigan, 2005.
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
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, Topics in the homological theory of modules over commutative rings, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 24, Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, Providence, R.I., 1975. Expository lectures from the CBMS Regional Conference held at the University of Nebraska, Lincoln, Neb., June 24–28, 1974. MR 0371879, DOI 10.1090/cbms/024
Melvin Hochster, Solid closure, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 103–172. MR 1266182, DOI 10.1090/conm/159/01508
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 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, Infinite integral extensions and big Cohen-Macaulay algebras, Ann. of Math. (2) 135 (1992), no. 1, 53–89. MR 1147957, DOI 10.2307/2946563
Melvin Hochster and Craig Huneke, Tight closure of parameter ideals and splitting in module-finite extensions, J. Algebraic Geom. 3 (1994), no. 4, 599–670. MR 1297848
Melvin Hochster and Craig Huneke, $F$-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), no. 1, 1–62. MR 1273534, DOI 10.1090/S0002-9947-1994-1273534-X
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
Karel Hrbáček and Thomas Jech, Introduction to set theory, 2nd ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 85, Marcel Dekker, Inc., New York, 1984. MR 758796
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
M. Nagata, Lectures on the fourteenth problem of Hilbert, Tata Institute of Fundamental Research, Bombay, 1965. MR 0215828
Masayoshi Nagata, Local rings, Robert E. Krieger Publishing Co., Huntington, N.Y., 1975. Corrected reprint. MR 0460307