# Castelnuovo-Mumford regularity

Let $R=K[x_1,\dots,x_{10}]$, where $K$ is a field. Consider $$I=(x_1x_7,x_1x_{10},x_2x_8,x_3x_9,x_4x_{10},x_1x_5x_9,x_2x_6x_{10},x_1x_4x_5x_8,x_2x_5x_6x_9,x_3x_6x_7x_{10})$$ which is a squarefree monomial ideal. I know by CoCoA that $\mathrm{reg}(R/I)=3$. Also, I can prove that $\mathrm{reg}(R/I)\geq 3$. But, I want to prove the equality. What should I do?

• What is reg() ? – user18119 Jul 16 '13 at 11:38
• It is Castelnuovo-Mumford regulaity, i.e. $\mathrm{max}\{j-i:\beta_{i,j}(R/I)\neq 0\}$. – user86498 Jul 16 '13 at 11:42