# Reciprocity Map and Cycle Class Map

This might be a very naive question but here it goes. Let X be a smooth variety of dimension d over a p-adic field. We have the n part of the rerciprocity map:

$rec/n: SK_1(X)/n \to \pi^{ab}_1(X)/n$

Right hand side can be identified with $H^{2d+1}_{et}(X, \mathbb{Z}/n(d+1))$ by Poincare duality. Left hand side can be identified with $CH^{d+1}(X,1,\mathbb{Z}/n)$ (cf. Lemma 2.8, S. Landsburg, Relative Chow Groups)

So I can see this as a map

$rec/n: CH^{d+1}(X,1,\mathbb{Z}/n) \to H^{2d+1}_{et}(X, \mathbb{Z}/n(d+1))$

What I wonder is that if this map agrees with the higher cycle class map i.e. whether their kernels (cokernels) are isomorphic.

-
Can you say briefly what $SK_1(X)/n$ is? – LMN Dec 19 '12 at 21:43
$SK_1(X)$ is defined as $coker(\bigoplus_{y \in X_{(1)}}K_2(k(y)) \to \bigoplus_{x \in X_{(0)}}k(x)^*)$ where this map arises from localization theory of $K$ groups. Here $X_{(a)}$ denotes the points of dimension $a$. – Grilo Dec 20 '12 at 1:17