[Théorèmes homologiques en petits degrés dans le cas de caractéristique mixte]
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.
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.
Révisé le :
Publié le :
@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},
}
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] Diophantine problems over local fields I, II, Amer. J. Math., Volume 87 (1965), pp. 605-630 (631–648)
[3] Cohen–Macaulay Rings, Cambridge University Press, Cambridge, 1993
[4] On the elementary theory of maximal normed fields I, Algebra i Logica, Volume 4 (1965), pp. 31-69
[5] On the elementary theory of maximal normed fields II, Algebra i Logica, Volume 5 (1966), pp. 8-40
[6] Grade-sensitive modules and perfect modules, Proc. London Math. Soc., Volume 29 (1974), pp. 55-76
[7] Topics in the homological theory of modules over commutative rings, CBMS Regional Conf. Ser. in Math., 24, American Mathematical Society, Providence, RI, 1975
[8] Canonical elements in local cohomology modules and the direct summand conjecture, J. Algebra, Volume 84 (1983), pp. 503-553
[9] Infinite integral extensions and big Cohen–Macaulay algebras, Ann. of Math., Volume 135 (1992), pp. 53-89
[10] Applications of the existence of big Cohen–Macaulay algebras, Adv. Math., Volume 113 (1995), pp. 45-117
[11] Model Theory, Cambridge University Press, Cambridge, 1993
[12] Bounds in the theory of polynomial rings over fields. A non-standard approach, Invent. Math., Volume 76 (1984), pp. 77-91
[13] Bounds in cohomology, Israel J. Math., Volume 116 (2000), pp. 125-169
[14] 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] Homological Questions in Local Algebra, London Math. Soc. Lecture Note Ser., 145, Cambridge University Press, Cambridge, 1990
Cité par Sources :
Commentaires - Politique