Flat and stably flat modules (Q1818874)

The context of this paper is complete cohomology and modules are of type \((FP)_\infty\). Let \(\text{LH}{\mathbf F}\) denote the class of hierarchically decomposable groups and \(\text{H}_1{\mathbf F}\) the subclass of groups acting on a finite dimensional contractible CW-complex, in such a way that the setwise stabiliser of each cell is equal to the pointwise stabiliser and is finite. Let \(G\) be an \(\text{H}_1{\mathbf F}\)-group and \(R\) a strongly \(G\)-graded \(k\)-algebra such that the base ring \(R_1\) is coherent of finite global dimension. If \(N\) is an \(R\)-module, then it is proved that \(N\) is stably flat if and only if \(N\) has finite projective dimension. This result is a generalization of a result of \textit{D. J. Benson} [J. Algebra 193, No. 1, 260-287 (1997; Zbl 0886.20002)] but avoids complicated diagram chasing. The second result of importance states that for a finite group \(G\) and a strongly \(G\)-graded \(k\)-algebra \(R\) such that \(R_1\) has finite global dimension, if \(N\) is a flat \(R\)-module which is projective as an \(R_1\)-module, then \(N\) is projective.