Let $R=k[x_1,\ldots,x_n]$ be a usual graded ring. Let $I$ , $J$ be monomial ideals.
Definition: $reg(I)=\max\{j-i| \beta_{i,j}(I) \neq 0 \}$.
We know that $reg(I+J) \le reg(I)+reg(J)-1$.
Can we write $reg(IJ)$ in terms of $reg(I)$ and $reg(J)$ ?
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1It's well known that reg behaves badly on the product of ideals. – user26857 Sep 16 '15 at 21:31