Castelnuovo-Mumford Regularity over Scrolls and Splitting Criteria

We introduce and study a notion of Castelnuovo-Mumford regularity suitable for scrolls obtained as projectivisations of sums of line bundles on $\mathbb P^m$. We show that this is a natural generalisation of the well known regularity on projective and multiprojective spaces and we prove Horrocks-type splitting criteria for vector bundles.