The \(K\)-theory of toric varieties in positive characteristic (Q2879010)

Let \(X\) be a toric variety. Given an integer \(c\geq 2\), the associated endomorphism \(\theta_c\) of \(X\) induced by a multiplication by \(c\) on the lattice defining \(X\) is called a dilation. Any sequence \((c_1,c_2, \dots )\) yields the sequence of endomorphisms \(\theta_{c_i}\) of the \(K\)-theory \(K_*(X)\) and the homotopy \(K\)-theory \(KH_*(X).\) The Dilation theorem states that on toric varieties one has an isomorphism NEWLINE\[NEWLINE\varinjlim_{\theta_c}K_*(X) \cong \varinjlim_{\theta_c}KH_*(X).NEWLINE\]NEWLINE For \(k\) of characteristic zero this was proved by \textit{J. Gubeladze} for affine toric varieties (cf. [Invent. Math. 160, No. 1, 173--216 (2005; Zbl 1075.14051)]) and the general case was proved by the authors in [Trans. Am. Math. Soc. 361, No. 6, 3325--3341 (2009; Zbl 1170.19001)]. Let \(X_{R}=X\times \mathrm{Spec}(R).\) The main result of the paper is the Dilation theorem for \(X_{R}.\) As a consequence the authors obtain a conjecture of Gubeladze concerning the monoid algebras \(k[A]\) when \(k\) is any regular ring. This states that if \(A\) is a cancellative, torsion-free commutative monoid with no non-trivial units then for every sequence \((c_1,c_2,\dots )\) of integers \(\geq 2\) and every regular ring \(k\) containing a field, there is an isomorphism NEWLINE\[NEWLINEK_*(k) \cong\varinjlim_{\theta_c}K_*(k[A]).NEWLINE\]