Let \(A\) be a ring of finite dimension, and let \(P\) be a projective \(A\)-module of finite rank. \(P\) is called cancellative if \(P\oplus A^m \cong Q\oplus A^m\) for some \(A\)-module \(Q\) implies that \(P\cong Q\). In case \(k\) is an algebraically closed field for which \(n!\) is invertible, and \(A\) a \(k\)-algebra of dimension \(n\), it is an open question (due to Mohan Kumar) whether any projective \(A\)-module of rank \(n-1\) is cancellative. The author gives an overview of known cases for which cancellativity holds, and gives an affirmative answer of Kumar’s question for some specific rings, in particular:
Let \(R\) be an affine \(k\)-algebra of dimension \(n-1\) and \(k\) algebraically closed such that \(n!\) is invertible in \(k\).
If \(A=R[T,T^{-1}] \), or \(A=R[T,f_1/f,\ldots,f_r/f]\) where \(f(T)\) is monic and \(f,f_1,\ldots,f_r\in R[T]\) is an \(R[T]\)-regular sequence, then \(A^{n-1}\) is cancellative. Furthermore, in the case that \(k=\bar{\mathbb{F}}_p\), then any projective \(A\)-algebra \(P\) of dimension \(n-1\) is cancellative.
(Note of the reviewer: in the abstract of the paper, the ring \(R\) is meant to be as described above.)