Maximal shifts below the Taylor bound

Abstract

Let $I\subseteq S$ be a monomial ideal with minimal generator degrees between $e$ and $d$, where $2\leq e\leq d$. We prove that if $t_a(S/I)<ea$, then $$ t_{a+b}(S/I)\leq t_a(S/I)+ \left\lceil\frac{(2d-1)b}{2}\right\rceil $$ for $a\geq2$, $b\geq0$, and $a+b\leq\operatorname{pd}_S(S/I)$. Consequently, $$ t_{a+b}(S/I)\leq ae+db-1-\left\lfloor\frac b2\right\rfloor. $$ In particular, if $t_2(S/I)<2e$, then $$ t_i(S/I)\leq \left\lceil\frac{(2d-1)i}{2}\right\rceil-2(d-e) $$ for $2\leq i\leq\operatorname{pd}_S(S/I)$. 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 $d=2$.