Regularity of trace rings of generic matrices (Q1115940)

Let \(T_{m,n}\) be the trace ring of m generic \(n\times n\) matrices over a field of characteristic zero and let \(n\leq 4\). It is proved that \(T_{m,n}\) has finite global dimension if and only if \(n=1\), \(m=1\) or \(T_{m,n}=T_{2,2}\), \(T_{3,2}\), or \(T_{2,3}\). The authors conjecture that gldim \(T_{m,n}=\infty\) if \(n\geq 5\), \(m\geq 2\).