A Purity Theorem for Functors with Transfers, by Kirill Zainoulline

Let R be a local regular ring got by localizing a smooth k-algebra A and let K be it's field of fractions. Let F be a covariant functor from the category of A-algebras to abelian groups that satisfies some additional properties (continuity, existence of well behaved transfer map). In this paper we show that the subgroup im{F(R) --> F(K)} of the group F(K) coincides with the intersection of the subgroups im{F(Rp) --> F(K)}, where all maps are induced by the canonical inclusions and Rp runs through localizations of the local ring R at all prime ideals p of height 1.

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