Mixed characteristic homological theorems in low degrees

Schoutens, Hans

Comptes Rendus. Mathématique, Tome 336 (2003) no. 6, pp. 463-466.

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.

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