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Suppose that I and J are homogeneous ideals of $S=K[x_1,...,x_n]$ and set $d =\dim Tor_1(S/I,S/J)$. prove that for $d\geq d\leq 2$ $$reg(IJ)^sat reg(IJ)^{sat} \leq reg I+reg J$$

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# Castelnuvo-Mumford regularity of Saturation

Suppose that I and J are homogeneous ideals of $S=K[x_1,...,x_n]$ and set $d =\dim Tor_1(S/I,S/J)$. prove that for $d\geq 2$ $$reg(IJ)^sat \leq reg I+reg J$$