Characteristic-free bounds for the CastelnuovoMumford regularity

We study bounds for the Castelnuovo–Mumford regularity of homogeneous ideals in a polynomial ring in terms of the number of variables and the degree of the generators. In particular, our aim is to give a positive answer to a question posed by Bayer and Mumford in What can be computed in algebraic geometry? (Computational algebraic geometry and commutative algebra, Symposia Mathematica, vol. XXXIV (1993), 1–48) by showing that the known upper bound in characteristic zero holds true also in positive characteristic. We first analyse Giusti’s proof, which provides the result in characteristic zero, giving some insight into the combinatorial properties needed in that context. For the general case, we provide a new argument which employs the Bayer–Stillman criterion for detecting regularity. Introduction The Castelnuovo–Mumford regularity is an important invariant in commutative algebra and algebraic geometry, which gives an estimate of the complexity of computing a minimal free resolution. It is common in the literature to attempt to find bounds for this invariant and, in general, the expected results range quite widely, from the well-behaved examples coming from algebraic geometry, as suggested by the Eisenbud–Goto conjecture [EG84], to the worst case provided by the example of Mayr and Meyer [MM82]. Clearly, when the assumptions are quite unrestrictive, the regularity can be very large. If one works with a homogeneous ideal I in a polynomial ring R = K[X1, . . . ,Xn] over a field K, a very natural question is to ask whether the regularity can be limited just by knowing that the ideal is generated in degree less than or equal to some positive integer d. What was known to this point were bounds depending on the characteristic of the base field K. If charK = 0, as observed in [BM93, Proposition 3.8], from the work of Giusti [Giu84] and Galligo [Gal79, Gal73], one can derive the bound reg(I) (2d)2 , (A) which seems to be sharp (see again [MM82]). On the other hand, in any characteristic, it has been proved in [BM93] using ‘straightforward cohomological methods’ that reg(I) (2d)(n−1)!, (B) but in the same paper it is asked whether (A) holds in general independently of the characteristic. The main purpose of this paper is to give a positive answer to this question. The main effort in extending the result to positive characteristic is that this proof utilises the combinatorial structure of the generic initial ideal in characteristic zero. Received 8 October 2003, accepted in final form 3 January 2005. 2000 Mathematics Subject Classification 03P10 (primary), 13D02 (secondary).