Diagonal F-splitting and Symbolic Powers of Ideals

Let \(J\) be any ideal in a strongly \(F\)-regular, diagonally \(F\)-split ring \(R\) essentially of finite type over an \(F\)-finite field. We show that \(J^{s+t} \subseteq \tau(J^{s - \epsilon}) \tau(J^{t-\epsilon})\) for all \(s, t, \epsilon > 0\) for which the formula makes sense. We use this to show a number of novel containments between symbolic and ordinary powers of prime ideals in this setting, which includes all determinantal rings and a large class of toric rings in positive characteristic. In particular, we show that \(P^{(2hn)} \subseteq P^n\) for all prime ideals \(P\) of height \(h\) in such rings.