$$H^i(F(m-i))=0.$$

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.

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