Cancellation Problems and Dimension Theory

ABSTRACT

A well-known cancellation problem of Zariski asks—for two given domains (fields, respectively) K ₁ and K ₂ over a field k—whether the k-isomorphism of K ₁[t] (K(t), respectively) and K ₂[t] (K ₂(t), respectively) implies the k-isomorphism of K ₁ and K ₂.

In this article, we address and systematically study a related problem: whether the k-embedding from K ₁[t] (K ₁(t), respectively) into K ₂[t] (K ₂(t), respectively) implies the k-embedding from K ₁ into K ₂. Our results are affirmative: if K ₁ and K ₂ are affine domains over an arbitrary field k, and K ₁[t] can be k-embedded into K ₂[t], then K ₁ can be k-embedded into K ₂; if K ₁ and K ₂ are affine fields over an arbitrary field k, and K ₁(t) can be k-embedded into K ₂(t), then K ₁ can be k-embedded into K ₂. Similar results are obtained for some general nonaffine domains and nonaffine fields.

These results were obtained in Belov et al. (preprint) together with L. Makar-Limanov. In this article we give an alternative proof, show connection with dimension theory, consider the case of infinite transcendental degree, and present some applications and surroundings.

2000 Mathematics Subject Classification:

• Primary 14E09, 14E25
• Secondary 14A10, 13B25

ACKNOWLEDGMENTS

Alexi Belov is grateful to the Department of Mathematics of the University of Hong Kong for its warm hospitality and stimulating atmosphere during his visits when part of this work was done.

Jie-Tai Yu was partially supported by Hong Kong RGC-CERG Grants 10203186 and 10203669.

Notes

Communicated by V. Artamonov.