3 problems

Let $A_{\mathbb{K}}$ be the polynomial ring over a ground field $\mathbb{K}$, and let $I$ be a $p$-stable ideal of $A_{\mathbb{K}}$. Write $\beta_{ij}(I)$ for its graded Betti numb…

Cellular replacement conjecture. The labelled regular cell complex $Z$ can be constructed from $Y$ by replacing each vertex $\mathbf{b}_j$ with a copy $\mathbf{b}_jX$; moreover, th…

Pardue's regularity conjecture. The regularity $reg(I)$ is equal to the maximum of those numbers.