The nilpotence conjecture in \(K\)-theory of toric varieties. (English) Zbl 1075.14051
In the paper the author proves the nilpotence conjecture in \(K\)-theory of toric varieties for all fields of characteristic \(0.\) The main result is the following.
Theorem 1.2. Let \({\mathbf k}\) be a field of characteristic \(0,\) \(M\) be an arbitrary commutative, cancellative, torsion free monoid without nontrivial units, \(p\) be a natural number, and \({\mathbf c} =(c_1,c_2,\dots )\) be a sequence of natural numbers \(\geq 2.\) Then for every element \(x\in K_p({\mathbf k}[M])\) there exists an index \(j_0 \in \mathbb N\) such that \((c_1\cdot \dots \cdot c_j)_{*}(x) \in K_p({\mathbf k})\) whenever \(j>j_0 .\)
Here \({\mathbf k}[M]\) denotes the monoid \({\mathbf k}\)-algebra of \(M\) and for a natural number \(c\), \(c_*\) is the endomorphism of \(K_p({\mathbf k}[M])\) induced by the \({\mathbf k}\)-algebra endomorphism \({\mathbf k}[M]\rightarrow {\mathbf k}[M]\) given by the assignment \(m \rightarrow m^c,\) for \(m\in M\). In the introduction the author describes what is known if \({\mathbf k}\) is replaced by a ring \(R\) as well as possible directions of generalization of Theorem 1.2.
Theorem 1.2. Let \({\mathbf k}\) be a field of characteristic \(0,\) \(M\) be an arbitrary commutative, cancellative, torsion free monoid without nontrivial units, \(p\) be a natural number, and \({\mathbf c} =(c_1,c_2,\dots )\) be a sequence of natural numbers \(\geq 2.\) Then for every element \(x\in K_p({\mathbf k}[M])\) there exists an index \(j_0 \in \mathbb N\) such that \((c_1\cdot \dots \cdot c_j)_{*}(x) \in K_p({\mathbf k})\) whenever \(j>j_0 .\)
Here \({\mathbf k}[M]\) denotes the monoid \({\mathbf k}\)-algebra of \(M\) and for a natural number \(c\), \(c_*\) is the endomorphism of \(K_p({\mathbf k}[M])\) induced by the \({\mathbf k}\)-algebra endomorphism \({\mathbf k}[M]\rightarrow {\mathbf k}[M]\) given by the assignment \(m \rightarrow m^c,\) for \(m\in M\). In the introduction the author describes what is known if \({\mathbf k}\) is replaced by a ring \(R\) as well as possible directions of generalization of Theorem 1.2.
Reviewer: Piotr Krasoń (Szczecin)
MSC:
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 19D55 | \(K\)-theory and homology; cyclic homology and cohomology |
| 19E08 | \(K\)-theory of schemes |
| 19D25 | Karoubi-Villamayor-Gersten \(K\)-theory |
| 14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |
Keywords:
higher algebraic \(K\)-theoryReferences:
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