Algebraic K-theory of Monoid Rings

Are all finitely generated projective k[t1, . . . , td]-modules free for an arbitrary field k and arbitrary d ∈ N? This question, set in Serre’s famous paper FAC in 1955, inspired an enormous activity of algebraists worldwide. The activity culminated in two independent confirmations of the question in 1976 by Quillen and Suslin. In the meanwhile the algebraic K-theory was created, in which one of the central topics was the so called homotopy properties of algebraic K-functors. Grothendieck-Serre’s classical theorem on K0-regularity of a regular ring was a starting point here. These two mentioned results, concerning respectively unstable and stable homotopic behavior of the classical functor K0, proved to be a fruitful domain for further search and generalizations. Almost the whole of this activity, save few exceptional cases, was concentrated on the consideration of polynomial ring extensions. On the other hand polynomial (and Laurent polynomial) rings are simplest (and the only regular) representatives of monoid rings and we could ask what results, established previuosly for polynomial rings, generalize to the latter ones. Below we shall survey the progress made in this direction during last years. A brief account of this search looks as follows. All the considered monoids and rings we deal with are supposed to be commutative. In addition monoids are cancellative and, if the contrary is not explicitly stated, torsion free (that is the corresponding groups of fractions are so). A monoid M is called normal if (writing additively) nx ∈ M for n ∈ N and x ∈ K(M) (the group of fractions) imply x ∈ M ; M is called seminormal if 2x ∈ M and 3x ∈ M imply x ∈ M ; we shall say that M is c-divisible for some c ∈ N if for any x ∈ M there exists y ∈ M for which cy = x (observe that a c-divisible monoid is always seminormal). Later on Z+ will denote the additive monoid of nonnegative rational integers and Q+ that of nonnegative rationals. It is well known that a monoid domain R[M ] is normal (seminormal) if and only if the domain R and the monoid M are normal (seminormal respectively). We remark that analogous statement for a completion of a monoid domain (with respect to the natural augmentation ideal) is also valid [G5]. In order to present a complete picture we start with the following relatevely old (1986) result, which confirms Anderson’s conjecture: THEOREM A [G1]. For any principal ideal domain (PID) R and a monoid M the following conditions are equivalent (a) Pic(R[M ]) = 0, (b) K0(R[M ]) = Z, (c) Finitely generated projective R[M ]-modules are all free, (d) M is seminormal. THEOREM B [G1+G5]. For any regular ring R and a monoid M we have SK0(R) = SK0(R[M ]) and K−i(R) = K−i(R[M ]) = 0 (Bass negative K-groups); the following conditions are equivalent (a) Pic(R) = Pic(R[M ]),