Classical algebraic K-theory of monid algebras

Higher K-Theory of Toric Varieties

“…We also show that all of this generalizes to not necessarily affine toric varieties (Proposition 4.7). Such a positive answer for i = 1 is obtained in [Gu2] and for i = 2 in [Msh]. In [Gu3] we showed that the answer is again 'yes' for all higher K-groups when the cone R + M ⊂ R r is simplicial, i. e. spanned by linearly independent vectors.…”

“…Here we recall the relationship between elementary convex geometry and monoids as developed in [Gu1] [Gu2]. A pure algebraic alternative is found in [Sw,§4,§5].…”

“…In [Gu1] we showed that vector bundles on affine toric varieties are trivial. An easy generalization of this result to the stable situation yields the equality K 0 (R) = K 0 (R[M]) for every regular ring R and every seminormal monoid M (see [Gu2], also [Sw,Corollary 1.4]). The key case is the class of finitely generated normal monoids which, up to isomorphism, can be described as additive submonoids of Z r of the type C ∩ Z r for some finite-polyhedral convex rational cone C ⊂ R r , r ∈ N.…”

“…Its construction involves higher K-theoretic analogues of essentially all steps in [Gu1], with use of results from [Gu2]. This becomes possible in combination with Suslin-Wodzicki's solution to the excision problem and the Mayer-Vietoris sequence for singular varieties due to Thomason. If the action of the big Witt vectors on K i (A, A + ) for graded not necessarily commutative k-algebras A = A 0 ⊕ A 1 ⊕ · · · satisfies a very natural condition when char k = 0 (see Question 10.2) then one can also eliminate the distinguished monomial in Λ.…”

“…In [Gu4], using different arguments, we showed that K 1 (R) = K 1 (R [M]) for essentially all finitely generated submonoids of Z r , not isomorphic to Z r + . As a plausible substitute for the equalities K i (R) = K i (R[M]), we raised in [Gu2] the following question. Assume c is a natural number ≥ 2 and M ⊂ Z r is a submonoid without non-trivial units.…”