Let $R = k[x_1\ldots x_n]$, and $a,b$ are vectors with integer entries, whose all entries of $a$ are non-negative, and say sum of coordinates of $b$ is $0$. Let $I$ be a monomial ideal generated by $x^a, x^{a+b}, \ldots, x^{a + kb}$ whenever $a + kb$ has non-negative integer entries. Is this true that regularity of $I$ is at most two times the degree of $a$? More generally, if we have $c$ pairs of vectors $a_1, b_1, \ldots, a_c, b_c$ and consider monomial ideal generated by the powers of translations of the $a_i$ by $b_i$. Is this true that the regularity of $I$ is at most two times the sum of degrees of $a_i$?

## 1 Answer

Doesn't a "yes" answer for your first question follow straightforwardly from facts about the lcm lattice?

See e.g. Nevo's "Regularity of edge ideals of $C_4$ free graphs via the topology of the lcm lattice" equation (1) for a clear statement of the relationship of regularity to the lcm lattice. In particular, (excluding the case where I has a single generator)

$\mathrm{reg}\ I \leq \max \deg m,$ where $m$ varies over proper least common multiples of generators of $I$.

In this case, the lcm of $x^{a+ib}, x^{a+jb}$ and $x^{a+\ell b}$ where $i<j<\ell$ is the same as that of $x^{a+ib}$ and $x^{a+\ell b}$, hence the greatest degree of an lcm is that of the lcm of $x^a$ and $x^{a+kb}$. By the zero sum condition, this is at most twice the degree of $x^a$.

There's almost certainly a more direct way to see the fact from Nevo's paper that I used.

A "yes" answer to the latter question should then follow via the Kalai-Meshulam Theorem (see "Intersections of Leray complexes and regularity of monomial ideals" Theorem 1.4; I also discuss this theorem in my own paper "Matchings, coverings and CM regularity").