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Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Let $\pi\colon X\rightarrow \text{Spec}(R)$ be a smooth projective morphism with geometrically integral fibers.

In Fulton's intersection theory book, Section 20.3, he defines a specialization morphism $\text{sp}\colon \text{CH}^p(X_K) \rightarrow \text{CH}^p(X_k)$. The fact that it is well-defined means that $\text{sp}$ preserves rational equivalence.

Question: Doe the specialization map $\text{sp}$ also preserve algebraic equivalence?

I couldn't find an answer to this question in Fulton's book or the literature (but the literature on this topic is huge, so I probably missed it). Any references or pointers would be appreciated!

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  • $\begingroup$ Doesn't the same proof work as for rational equivalence? $\endgroup$
    – anon
    Commented May 2 at 18:37
  • $\begingroup$ This would require knowing that the `generic fiber' homomorphism $\text{CH}^p(X)\rightarrow \text{CH}^p(X_K)$ is surjective when restricted to the subgroups of algebraically trivial cycle classes. I don't see an obvious reason why that's true, nor what the definition of such a subgroup of $\text{CH}^p(X)$ would actually be. (Fulton only considers algebraic triviality for varieties over a field.) $\endgroup$
    – Jef
    Commented May 3 at 10:08

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