Simple Bundles of Rank Three

In [9, 10], Hartshorne gives an account of the known vector bundles of "small" rank on P" and also on the punctured spectrum of a regular local ring. Specifically, he remarks on various constructions of rank two vector bundles on P", for n ^ 4, and Horrocks' examples [12] of rank three bundles on P as well. The result of Evans and Griffith [7] on syzygies shows that the existence of rank two indecomposable bundles always gives rise to rank three indecomposable bundles on P", for n ^ 3, in the following manner. Let k be a field and suppose that & corresponds to an indecomposable vector bundle on PjJ of rank two. We may suppose that $ is generated by its global sections since we may otherwise accomplish this after twisting S a suitable number of times. Let E be the module of sections of $. Then according to Evans and Griffith [7] one has that ExtJ^is, R) ± 0, where R = k[X0,1,,..., Xn]. A nontrivial graded extension of the form 0-*R~*F-*E-*0 gives rise to a vector bundle !F (associated to the module F) of rank three (cf. Corollary 3.2 to see that F is necessarily indecomposable). Depending on the behavior of E and ExtJ^jE, R) we also show in Section 3 how to construct vector bundles of rank four which are indecomposable. However, a construction of this sort patently gives rise to proper subbundles. The main point of this note is to demonstrate another way in which vector bundles of rank two can be used as the building blocks for bundles of higher rank. Specifically, in Section 2 we show how to obtain simple vector bundles (no proper subbundle of positive rank) of rank three from indecomposable rank two bundles, and in Section 3 we describe how these simple rank three bundles further yield indecomposable bundles of rank four. Although our results hold for graded modules over polynomial rings, we shall confine our discussion to the punctured spectrum of a regular local ring since our methods are purely those of local algebra.