On bounds for Castelnuovo’s index of regularity

Abstract

For coherent sheaves on a projective space Castelnuovo's index of regularity was first defined by D. Mumford [11], who attributes the idea to G. Castelnuovo. In fact, he used it to show that certain twists of a coherent sheaf are generated by its g lo b a l sec tio n s . In a m o re algebraic setting Castelnuovo's regularity was defined by D. Eisenbud and S. Goto [2] and A. Ooishi [12], see (2.2). I t c o m e s out that Castelnuovo's index of regularity gives an upper bound for the maximal degree of the syzygies in a m inim al free resolution, see (2.8) fo r th e precise statem ent. This fact is very im portant for the complexity o f a program for a numerical computation of syzygies. As shown by examples of D . Bayer and M. Stillm an, see [1], Castelnuovo's regularity can be o f exponential growth with respect to the K rull dim ension. While these examples have rather wild singularities, one might hope to get better bounds in the case of tame singularities. This point of view is pursued further in the papers [2], [10], [12], [16], [17], [18]. O n e o f o u r m a in results, see (3.5), supplies an upper bound for reg R, Castelnuovo's regularity, o f a graded k-algebra R th a t is Cohen-Macaulay or locally Cohen-Macaulay and unm ixed . I n a geometric context, w e get a satisfactory bound in the case of a Cohen-Macaulay projective v a rie ty . In particular, with (3.5) a) we solve a problem posed in [10] in the affirmative. The same result was shown independently a n d b y a different argum ent b y J . Stuckrad and W. Vogel in [18]. In the case of a Cohen-Macaulay ring R it follows, see [2], that