Castelnuovo–Mumford regularity

Remarks

Let $f \in S$ be homogeneous of degree $d \ge 1$ and $M = S/(f)$. The minimal resolution is $0 \to S(-d) \to S \to M \to 0$, so $\beta_{0,0} = 1$ and $\beta_{1,d} = 1$. Hence $\operatorname{reg} M = \max(0, d - 1) = d - 1$.

Minimality of the resolution is needed. Take $S = k[x]$ and $M = S/(x)$, so that $\operatorname{reg} M = 0$. Add the exact summand $S(-5) \to S(-5)$ in homological degrees $1$ and $0$. The result is a non-minimal resolution with the pair $(i, j) = (0, 5)$, and it would give the wrong value $5$.

An equivalent form uses local cohomology with respect to $\mathfrak{m} = (x_1, \ldots, x_n)$: $\operatorname{reg} M = \max_i \set{ a_i(M) + i }$, where $a_i(M)$ is the largest degree $d$ with $H^i_{\mathfrak{m}}(M)_d \ne 0$, and the maximum is over the $i$ with $H^i_{\mathfrak{m}}(M) \ne 0$. For a nonzero proper homogeneous ideal $I$, one has $\operatorname{reg} I = \operatorname{reg}(S/I) + 1$.