A counterexample to a conjecture of Somekawa

We construct an example of a torus T over a field K for which the Galois symbol K(K;T, T )/nK(K;T, T ) → H(K,T [n] ⊗ T [n]) is not injective for some n. Here K(K;T, T ) is the Milnor K-group attached to T introduced by Somekawa. Introduction Let K be a field, m a positive integer and n an integer prime to the characteristic of K. Recently Rost and Voevodsky announced a proof of the bijectivity of the Galois symbol (1) K m (K)/nK M m (K) −→ H (K,Z/nZ(m)) from Milnor K-groups to Galois cohomology (special cases have been obtained earlier by Merkurjev-Suslin, Rost and Voevodsky). In [So], Somekawa has introduced certain generalized Milnor K-groups K(K;A1, . . . , Am) attached to semi-abelian varieties A1, . . . , Am. If A1 = . . . = Am = Gm is the one-dimensional split torus they agree with the usual KM m (K). If m = 2, A1 = JacX and A2 = JacY are the Jacobians of smooth, projective and connected curves X and Y over K having a K-rational point, then K(K;A1, A2) is the kernel of the Albanese map CH0(X × Y )deg=0 → AlbX×Y (K). In general, Somekawa has defined a Galois symbol K(K;A1, . . . , Am)/nK(K;A1, . . . , Am) −→ H (K,A1[n] ⊗ . . . ⊗Am[n]) and conjectured that it is always injective. In this note we present a counterexample to the injectivity of the Galois symbol. Let us describe it briefly. Let L/K be a cyclic extension of degree n and σ a generator of the Galois group Gal(L/K). Let T be the kernel of the norm map ResL/K Gm → Gm. We show that the norm K(L;T, T ) → K(K;T, T ) induces an isomorphism K2(L;T, T )/(1 − σ) → K2(K;T, T ). On the other hand, the corresponding map of Galois cohomology groups ∗This work was done during the second author stayed at Universität Bielefeld supported by SFB 701. He is grateful to the members there.