The Coefficient Field in the Nilpotence Conjecture for Toric Varieties

The main result of the work “The nilpotence conjecture in K-theory of toric varieties” is extended to all coefficient fields of characteristic 0, thus covering the class of genuine toric varieties. 1. The statement Let R be a (commutative) regular ring, M be arbitrary commutative, cancellative, torsion free monoid without nontrivial units, and i be a nonnegative integral number. The nilpotence conjecture asserts that for any sequence c = (c1, c2, . . . ) of natural numbers ≥ 2 and any element x ∈ Ki(R[M ]) there exists an index jx ∈ N such that (c1 · · · cj)∗(x) ∈ Ki(R) for all j > jx. Here R[M ] is the monoid R-algebra of M and for a natural number c the endomorphism of Ki(R[M ]), induced by the R-algebra endomorphism R[M ] → R[M ], m 7→ m, m ∈ M , is denoted by c∗ (writing the monoid operation multiplicatively). We call this action the multiplicative action of N on Ki(R[M ]). Theorem 1.2 in [G2] verifies the conjecture for the coefficient rings R of type Sk[T1, . . . , Td] where k is a number field and S ⊂ k[T1, . . . , Td] is arbitrary multiplicative subset of nonzero polynomials (d ∈ Z+). In particular, the nilpotence conjecture is valid for purely transcendental extensions of Q. On the other hand K-groups commute with filtered colimits. Therefore, the following induction proposition, together with [G2, Theorem 2.1], proves the conjecture for all characteristic 0 fields. Proposition 1. The validity of the nilpotence conjecture for a field k1 of characteristic 0 transfers to any finite field extension k1 ⊂ k2. This note should be viewed as an addendum to the main paper [G2]. In Sections 2 and 3 for the reader’s convenience we recall in a convenient form the needed results on Galois descent and Bloch-Stienstra operations in K-theory. 2. Galois descent for rational K-theory We need the fact that rational K-theory of rings (essentially) satisfies Galois descent – a very special case of Thomason’s étale descent for localized versions of K-theory, first proved in [T] in the smooth case and then extended to the singular case as an application of the new local-to-global technique [TT]. The argument 2000 Mathematics Subject Classification. Primary 14M25, 19D55; Secondary 19D25, 19E08. Supported by MSRI, INTAS grant 99-00817 and TMR grant ERB FMRX CT-97-0107.