Products of commutators of transvections over local rings
Let \(R\) be a commutative local ring, and \(M\) the maximal ideal of \(R\). The natural map \(\pi\colon R\to R/M\) will induce the natural map \(R^n\to(R/M)^n\). For any matrix \(A\) in \(\text{GL}_nR\), write \(\text{res }A=n-\dim(\pi P)\), where \(P=\{v\in R^n\mid Av=v\}\) is the fixed submodule of \(A\). An invertible matrix \(A\) over \(R\) is a transvection if \(\text{res }A=1\) and \(\det A=1\). The authors show that if \(|R/M|>3\), then every matrix in \(\text{SL}_nR\) (\(n\geq 2\)) is a product of at most \([n/2]+2\) commutators of transvections and the length \([\text{res }A/2]+2\) cannot be reduced for \(n=2\).
scientific article; zbMATH DE number 1864768
scientific article