Zero-cycles on varieties over finite fields
نویسنده
چکیده
For any field k, Milnor [Mi] defined a sequence of groups K 0 (k), K M 1 (k), K M 2 (k), . . . which later came to be known as Milnor K-groups. These were studied extensively by Bass and Tate [BT], Suslin [Su], Kato [Ka1], [Ka2] and others. In [Som], Somekawa investigates a generalization of this definition proposed by Kato: given semi-abelian varieties G1, . . . , Gs over a field k, there is a group K(k;G1, . . . , Gs) which is isomorphic to K s (k) in the case that G1 = . . . = Gs = Gm, the multiplicative group scheme. Raskind and Spiess [RS] use a similar idea to define a Milnor-type group K(k; CH0(X1), . . . , CH0(Xr)) associated to a family X1, . . . , Xr of smooth projective varieties over k and prove that this group is isomorphic to the Chow group CH0(X1 ×k . . . ×k Xr) of zero-cycles. These two definitions were amalgamated in [A2] to define groups K(k; CH0(X1), . . . , CH0(Xr);G1, . . . , Gs) where the Gi are as above and this time X1, . . . , Xr are only assumed to be quasiprojective. In the case G1 = . . . = Gs = Gm, we use the abbreviated notation Ks(k; CH0(X1), . . . , CH0(Xr); Gm) for this group. The main result of [A2] is the following:
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