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$$K$$-theory of affine toric varieties. (English) Zbl 0920.19001
This paper contains two sections. The first is a survey of the results in a long series of papers by the author and others, mostly concerned with the regularity of monoid rings for the functors $$K_i$$ of algebraic $$K$$-theory. If $$F$$ is a functor from rings to abelian groups and $$R$$ is a ring then $$R$$ is $$F$$-regular if the inclusion $$R\to R[t]$$ ($$t$$ an indeterminate over $$R$$) induces an isomorphism $$F(R)\to F(R[t])$$. All monoids will be commutative, cancellative, and torsion-free. Sometimes, but not always, they are finitely generated and have no units other than units of $$R$$ (‘trivial units’). A monoid $$M$$ with quotient group $$\text{gp}(M)$$ is seminormal if $$x\in \text{gp}(M)$$, $$2x\in M$$, $$3x\in M$$ implies $$x\in M$$. A few of these results are as follows: Theorem 1.1 [J. Gubeladze, “Anderson’s conjecture and the maximal monoid class over which projective modules are free”, Math. USSR Sb. 63, 165-180 (1989; Zbl 0668.13011); translation from Mat. Sb., Nov. Ser. 135(177), No. 2, 169-185 (1988; Zbl 0654.13013)]: For any principal ideal domain $$R$$ and any monoid $$M$$ (maybe infinitely generated and with non-trivial units) the following are equivalent: (a) $$\text{Pic}(R[M])=0$$; (b) $$K_0(R[M]) = {\mathbb Z}$$ (the integers); (c) finitely generated projective $$R[M]$$-modules are all free; (d) $$M$$ is seminormal.
Theorem 1.1 is a confirmation of Anderson’s conjecture [D. F. Anderson, “Projective modules over subrings of $$k[x,y]$$ generated by monomials”, Pac. J. Math. 79, 5-17 (1979; Zbl 0372.13006)] (by applying (b) to $$M$$ and $$M\oplus {\mathbb Z}$$ we see that $$R[M]$$ is $$K_0$$-regular). There are similar $$K_0$$-results for regular rings $$R$$, and also unstable results, i.e., results about projective modules themselves rather than the Grothendieck group $$K_0$$. If $$i>0$$ then it is harder for a monoid ring to be $$K_i$$-regular. However one can salvage $$K_i$$-regularity by considering $$c$$-divisible monoids and monoids $$M$$ such that $${\mathbb Z}^n_+\subset M\subset {\mathbb Q}^n_+$$ where $${\mathbb Z}^n_+$$ denotes the non-negative integers, similarly for the rationals $${\mathbb Q}$$. Then one has theorem 1.12 [J. Gubeladze, “Geometric and algebraic representations of commutative cancellative monoids”, Proc. A. Razmadze Math. Inst. 113, 31-81 (1995; Zbl 0871.19001)]: $$K_i(R) = K_i(R[M])$$ for any regular ring $$R$$ provided $$\mathbb Z^n_+\subset M\subset \mathbb Q^n_+$$, $$i\geq 1$$, $$n\geq 1$$ (and $$M$$ has no non-trivial units and is $$c$$-divisible for some integer $$c>1$$).
The second section contains several conjectures, which in the words of the author “include the results from section 1 in a uniform way and, simultaneously, provide their final possible generalizations”. To give one example of these conjectures, define an action of the positive integers $$\mathbb N$$ on $$M$$ by $$m\to m^c$$ (writing $$M$$ multiplicatively). Then we have conjecture 2.1: For any index $$i\in {\mathbb Z_+}$$, any regular ring $$R$$ and any monoid $$M$$ without non-trivial units the multiplicative action of $$\mathbb N$$ on $$K_i(R[M])/K_i(R)$$ is nilpotent.
A consequence of this is that if $$M$$ is $$c$$-divisible without non-trivial units, then $$K_i(R)=K_i(R[M])$$, generalizing, for example, theorem 1.12. There is another conjecture involving unstable results, phrased in Volodin’s $$K$$-Theory [I. A. Volodin, “Algebraic $$K$$-theory as extraordinary homology theory on the category of associative rings with unity”, Math. USSR Izv. 5(1971), 859-887 (1972; Zbl 0252.18010); translation from Izv. Akad. Nauk SSSR, Ser. mat. 35, 844-873 (1971; Zbl 0229.18010)].

##### MSC:
 19-02 Research exposition (monographs, survey articles) pertaining to $$K$$-theory 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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