## November 12, 2016

### Castelnuovo-Mumford regularity

Let $X$ be a projective variety. Then a coherent sheaf $F$ is called $m$-regular by Castelnuovo-Mumford if
$$H^i(F(m-i))=0.$$
It looks strange that one does not simply require
$$H^i(F(m))=0.$$
But then $m$-regularity would not imply $(m+1)$-regularity. For example, $\Omega^1_{\mathcal P^n}$ with $n>1$ is $(-1)$-regular, but not $0$-regular.

For properties of Castelnuovo-Mumford regularity, see
Mumford Lectures on Curves on an Algebraic Surface, lecture 14.

From MSE:2008517.