Socle Degrees, Resolutions, and Frobenius Powers

We first describe a situation in which every graded Betti number in the tail of the resolution of R J may be read from the socle degrees of R J . Then we apply the above result to the ideals J and J [q]; and thereby describe a situation in which the graded Betti numbers in the tail of the resolution of R/J [q] are equal to the graded Betti numbers in the tail of a shift of the resolution of R/J. Let (R,m) be a Noetherian graded algebra over a field of positive characteristic p, with irrelevant ideal m. Let J be an m-primary homogeneous ideal in R. Recall that if q = p, then the e Frobenius power of J is the ideal J [q] generated by all j with j ∈ J . Recall, also, that the socle of R J is the ideal (J : m) J of R J . The socle degrees of RJ are the degrees of any homogeneous basis for the graded vector space soc RJ . The basic question is: Question 0.1. How do the socle degrees of R J [q] vary with q? The question of finding a linear bound for the top socle degree of R/J [q] has been considered by Brenner in [1] from the point of view of finding inclusion-exclusion criteria for tight closure. An answer to Question 0.1 would provide insight into the tight closure of J , and possibly a handle on Hilbert-Kunz functions. We are particularly interested in how the socle degrees of Frobenius powers encode homological information about the ideal J . For example, the answer to Question 0.1 is well-known in the case when J has finite projective dimension: if the socle degrees of R J are {σi | 1 ≤ i ≤ s}, then the socle degrees of R J [q] are {qσi− (q− 1)a | 1 ≤ i ≤ s}, where a is the a-invariant of R. When R is a complete intersection, the converse is established in [6].