Let $k$ be a field, let $I, J \subset k[x_1, \dots, x_n]$ be homogeneous ideals such that $I \subset J$, and let $\text{reg}(I), \text{reg}(J)$ be the Castelnuovo-Mumford regularity of $I, J$. We define the dimension of an ideal $I$ as $\dim (I) = \dim (k[x_1, \dots, x_n] / I)$.
Question 1: Are there known conditions on $I \subset J$ such that $\text{reg}(I) \geq \text{reg}(J)$?
In particular, I am interested in monomial ideals. In this case there is a simple counterexample, let $(x^2), (x^2, y^2) \subset k[x, y]$. Then $\text{reg} (x^2) = 2$ but $\text{reg} (x^2, y^2) = 3$. However, for all examples I could come up with where $\dim (I) = \dim (J)$ the inequality is true. So for monomial ideals I would rephrase my question as:
Question 2: If $I \subset J$ are monomial ideals with $\dim (I) = \dim (J)$, then $\text{reg}(I) \geq \text{reg}(J)$?
Is it somewhat obvious that for monomial ideals you can extend a resolution of $I$ to an resolution of $J$ if the dimensions agree?