Comptes Rendus
Homological Algebra
Mixed characteristic homological theorems in low degrees
Comptes Rendus. Mathématique, Volume 336 (2003) no. 6, pp. 463-466.

Let R be a locally finitely generated algebra over a discrete valuation ring V of mixed characteristic. For any of the homological properties, the Direct Summand Theorem, the Monomial Theorem, the Improved New Intersection Theorem, the Vanishing of Maps of Tors and the Hochster–Roberts Theorem, we show that it holds for R and possibly some other data defined over R, provided the residual characteristic of V is sufficiently large in terms of the complexity of the data, where the complexity is primarily given in terms of the degrees of the polynomials over V that define the data, but possibly also by some additional invariants.

Soit ϖ un ensemble fini d'objets algébriques (comme des anneaux, des modules, etc.) de type fini sur un anneau de valuation discrète V en caractéristique mixte. Pour plusieurs propriétés homologiques, nous montrons que la propriété est satisfaite par ϖ, pourvu que la caractéristique résiduelle de V soit supérieure à une borne qui ne dépend que de la complexité de ϖ, où la complexité est déterminée notamment par les degrés des polynômes définissant ϖ.

Received:
Revised:
Published online:
DOI: 10.1016/S1631-073X(03)00114-6
Hans Schoutens 1

1 Department of Mathematics, Ohio State University, Columbus, OH 43210, USA
@article{CRMATH_2003__336_6_463_0,
     author = {Hans Schoutens},
     title = {Mixed characteristic homological theorems in low degrees},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {463--466},
     publisher = {Elsevier},
     volume = {336},
     number = {6},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00114-6},
     language = {en},
}
TY  - JOUR
AU  - Hans Schoutens
TI  - Mixed characteristic homological theorems in low degrees
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 463
EP  - 466
VL  - 336
IS  - 6
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00114-6
LA  - en
ID  - CRMATH_2003__336_6_463_0
ER  - 
%0 Journal Article
%A Hans Schoutens
%T Mixed characteristic homological theorems in low degrees
%J Comptes Rendus. Mathématique
%D 2003
%P 463-466
%V 336
%N 6
%I Elsevier
%R 10.1016/S1631-073X(03)00114-6
%G en
%F CRMATH_2003__336_6_463_0
Hans Schoutens. Mixed characteristic homological theorems in low degrees. Comptes Rendus. Mathématique, Volume 336 (2003) no. 6, pp. 463-466. doi : 10.1016/S1631-073X(03)00114-6. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00114-6/

[1] M. Aschenbrenner, Ideal membership in polynomial rings over the integers, Ph.D. thesis, University of Illinois, Urbana-Champaign, 2001

[2] J. Ax; S. Kochen Diophantine problems over local fields I, II, Amer. J. Math., Volume 87 (1965), pp. 605-630 (631–648)

[3] W. Bruns; J. Herzog Cohen–Macaulay Rings, Cambridge University Press, Cambridge, 1993

[4] Y. Eršhov On the elementary theory of maximal normed fields I, Algebra i Logica, Volume 4 (1965), pp. 31-69

[5] Y. Eršhov On the elementary theory of maximal normed fields II, Algebra i Logica, Volume 5 (1966), pp. 8-40

[6] M. Hochster Grade-sensitive modules and perfect modules, Proc. London Math. Soc., Volume 29 (1974), pp. 55-76

[7] M. Hochster Topics in the homological theory of modules over commutative rings, CBMS Regional Conf. Ser. in Math., 24, American Mathematical Society, Providence, RI, 1975

[8] M. Hochster Canonical elements in local cohomology modules and the direct summand conjecture, J. Algebra, Volume 84 (1983), pp. 503-553

[9] M. Hochster; C. Huneke Infinite integral extensions and big Cohen–Macaulay algebras, Ann. of Math., Volume 135 (1992), pp. 53-89

[10] M. Hochster; C. Huneke Applications of the existence of big Cohen–Macaulay algebras, Adv. Math., Volume 113 (1995), pp. 45-117

[11] W. Hodges Model Theory, Cambridge University Press, Cambridge, 1993

[12] K. Schmidt; L. van den Dries Bounds in the theory of polynomial rings over fields. A non-standard approach, Invent. Math., Volume 76 (1984), pp. 77-91

[13] H. Schoutens Bounds in cohomology, Israel J. Math., Volume 116 (2000), pp. 125-169

[14] H. Schoutens Uniform bounds in algebraic geometry and commutative algebra (A. Macintyre, ed.), Connections between Model Theory and Algebraic and Analytic Geometry, Quad. Math., 6, 2000, pp. 43-93

[15] H. Schoutens, Absolute bounds on the number of generators of Cohen–Macaulay ideals of height at most 2, Preprint, http://www.math.ohio-state.edu/~schoutens, 2001

[16] H. Schoutens, Non-standard tight closure for affine -algebras, Manuscripta Mathematica, to appear

[17] H. Schoutens, Asymptotic homological conjectures in mixed characteristic, Manuscript, 2002

[18] H. Schoutens, Canonical big Cohen–Macaulay algebras and rational singularities, Preprint, http://www.math.ohio-state.edu/~schoutens, 2002

[19] J. Strooker Homological Questions in Local Algebra, London Math. Soc. Lecture Note Ser., 145, Cambridge University Press, Cambridge, 1990

Cited by Sources:

Comments - Policy


Articles of potential interest

A criterion for regularity of local rings

Tom Bridgeland; Srikanth Iyengar

C. R. Math (2006)


Metric ultraproducts of finite simple groups

Andreas Thom; John S. Wilson

C. R. Math (2014)


Equivalent condition for approximately Cohen–Macaulay complexes

Michał Lasoń

C. R. Math (2012)