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Published in:
2015 | OriginalPaper | Chapter
Syzygies in Polynomial Rings Over Valuation Domains
Author : Ihsen Yengui
Published in: Constructive Commutative Algebra
Publisher: Springer International Publishing
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It is folklore (see for example Theorem 7.3.3 in [68]) that if V is a valuation domain, then V[X1, . . . ,X k ] (k ∈ ℕ) is coherent: that is, syzygy modules of finitely-generated ideals of V[X1, . . . ,X k ] are finitely-generated. The proof in the above-mentioned reference relies on a profound and difficult result published in a huge paper by Gruson and Raynaud [75]. There is nevertheless no known general algorithm for this remarkable result, and it seems difficult to compute the syzygy module even for small polynomials.