Regularity of Tor for weakly stable ideals

It is proved that if $I$ and $J$ are weakly stable ideals in a polynomial ring $R=k[x_1,\dots ,x_n]$, with $k$ a field, then the regularity of ${\text{Tor}}_i^R(R\mathrm{/}I,R\mathrm{/}J)$ has the expected upper bound. We also give a bound for the regularity of ${\text{Ext}}_R^i(R\mathrm{/}I,R)$ for $I$ a weakly stable ideal.