The Nilpotence Conjecture in K-theory of Toric Varieties

It is shown that all nontrivial elements in higher K-groups of toric varieties over a class of regular rings are annihilated by iterations of the natural Frobenius type endomorphisms. This is a higher analog of the triviality of vector bundles on affine toric varieties. 1. Statement of the main result The nilpotence conjecture in K-theory of toric varieties, treated in our previous works, asserts the following: Conjecture 1.1. 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. Then for every sequence c = (c1, c2, . . . ) of natural numbers ≥ 2 and every 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). Speaking loosely, this conjecture says that the multiplicative monoid of natural numbers acts nilpotently on Ki(R[M ]). The following is a reformulation of Conjecture 1.1 in a typical case: Conjecture. Let R and i be as above and c be a natural number ≥ 2. Assume C ⊂ R (n ∈ N) is a convex cone, containing no affine lines. Then Ki(R) = Ki(R[C ∩ (c Z)]). (Here cZ is the additive group of the localization of Z at c.) Although the conjecture is stated for affine (not necessarily normal) toric varieties it yields a similar result for all quasi-projective toric varieties. In fact, the motivation behind the nilpotence conjecture can be described by the diagram of relationships: 1991 Mathematics Subject Classification. Primary 14M25, 19D55; Secondary 19D25, 19E08. Supported by MSRI, INTAS grant 99-00817 and TMR grant ERB FMRX CT-97-0107. 1