2 problems

Assume that ${\Bbbk}$ has characteristic $2$, let $R={\Bbbk}[x,y,a,b]$, set $Q=(x^3,y^3)$ and $f=xya-(x^2+y^2)b$, and let $n=2^s$ for some $s\in\mathbb N$. Characteristic-two upper…

Let $S$ be a standard graded polynomial ring and let $I\subset S$ be a graded ideal with linear powers, meaning that every power $I^k$ has a linear free resolution. Let $\alpha(I)$…