Maximal shifts below the Taylor bound

Abstract

Let $I\subseteq S$ be a monomial ideal with minimal generator degrees between $e$ and $d$, where $2\le e\le d$. We prove that if $t_a(S/I)<ea$, then $$t_{a+b}(S/I)\le t_a(S/I)+\lceil \frac{(2d-1)b}{2}\rceil$$ for $a\ge 2$, $b\ge 0$, and $a+b\le {\mathrm{pd}}_S(S/I)$. Consequently, $$t_{a+b}(S/I)\le ae+db-1-\lfloor \frac{b}{2}\rfloor .$$ In particular, if $t_2(S/I)<2e$, then $$t_i(S/I)\le \lceil \frac{(2d-1)i}{2}\rceil -2(d-e)$$ for $2\le i\le {\mathrm{pd}}_S(S/I)$. We also obtain a regularity bound and give a quadratic family in which our estimate for the last shift is smaller than every bound obtained from ordinary subadditivity. The constant $2d-1$ is sharp for $d=2$.

8 pages