\(K\)-theory of affine toric varieties.

*(English)*Zbl 0920.19001This 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)].

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)].

Reviewer: L.G.Roberts (Kingston/Ontario)