# $m$-regularity of sheaves

This is Lemma 1.4 on Green and Lazarsfeld's Some results on the syzygies of finite sets and algebraic curves. Let $$X$$ be a closed subscheme of $$\mathbb{P}^r$$. Suppose the ideal sheaf $$\mathcal{I}$$ of $$X$$ is 3-regular, meaning that $$H^i(\mathbb{P}^r,\mathcal{I}(3-i))=0$$ for $$i>0$$. Then for $$p\leq codim(X,\mathbb{P}^r)$$, $$(N_p)$$ holds for $$X\subset\mathbb{P}^r$$ iff $$Tor^S_p(R,k)_{p+2}=0$$, where $$S=k[x_0,\cdots,x_r]$$, $$R=\bigoplus\limits_{l\geq 0}H^0(\mathbb{P}^r,\mathcal{O}/\mathcal{I}(l))$$. The proof is short. First observe that $$m$$-regularity implies that $$\Gamma_*(\mathcal{I})$$ is generated by elements of degree $$\leq m$$. Apply this to the sheafification $$0\to\mathcal{E}_{r+1}\to\cdots\to\mathcal{E}_1\to\mathcal{I}\to 0$$ of a minimal resolution of $$I=\Gamma_*(\mathcal{I})$$. One finds that $$\mathcal{E}_i$$ is $$(i+2)$$-regular, which means that $$Tor_i^S(R,k)_j=0$$ for $$j\geq i+3$$ and therefore $$(N_p)$$ only requires $$Tor^S_p(R,k)_{p+2}=0$$. I don't know how to deduce the $$(i+2)$$-regularity of $$\mathcal{E}_i$$. For example, let $$r=2$$. Consider the homogeneous ideal $$I=(xy^2,z^3)$$. Let $$E_1=S(-3)\oplus S(-3)$$, with the map $$E_1\to I$$ given by $$(xy^2,z^3)$$. Then the kernel is $$S(-6)$$ with the inclusion to $$S(-3)\oplus S(-3)$$ given by $$(-z^3,xy^2)^T$$. This is a minimal resolution but $$\mathcal{O}(-6)$$ is not 4-regular. Can anyone tell me what's wrong with my observation?

• This just mean that your ideal $\mathscr{I}$ is not 3-regular -- in fact $H^1(\mathscr{I}(2))\cong H^2(\mathscr{O}(-4))\neq 0$.
– abx
Apr 4 '20 at 18:48