Asymptotic behavior of the socle of Frobenius powers

Abstract

Let (R, m) be a local ring of prime characteristic p and q a varying power of p. We study the asymptotic behavior of the socle of R/I [q] where I is an m -primary ideal of R. In the graded case, we define the notion of diagonal F -threshold of R as the limit of the top socle degree of R/m [q] over q when q → ∞. Diagonal F -threshold exists as a positive number (rational number in the latter case) when:(1) R is either a complete intersection or R is F -pure on the punctured spectrum;(2) R is a two dimensional normal domain. In the latter case, we also discuss its geometric interpretation and apply it to determine the strong semistability of the syzygy bundle of (x d , y d , z d ) over the smooth projective curve in P 2 defined by x n + y n + z n = 0. The rest of this paper concerns a different question about how the length of the socle of R/I [q] vary as q varies. We give explicit calculations of the length of the socle of R/m [q] for a class of hypersurface rings which attain the minimal Hilbert-Kunz function. We finally show, under mild conditions, the growth of such length function and the growth of the second Betti numbers of R/m [q] differ by at most a constant, as q → ∞.

On the degrees of relations onx1d1,…,xnVraciu1 2015Journal of Algebra5060View full textAdd to dashboardCite

“…If n i=1 i ≤ 2 in Eq. (14), then the sum i≤n, i =0 v i consists of at least one term, and we get E p (r 1 , . .…”

“…If n i=1 i ≤ 1, it follows that i≤n, i =0 v i from Eq. (14) consists of at least two terms, and since u 1 = . .…”

“…. = u 1 = 1, or when n = 3, u 3 = 2, and u 1 , (14) consists of at least two terms, and at least one of them is ≥ min{v 1 , v 2 } (when n = 3 then one of the two terms must be v 1 or v 2 ; when n ≥ 4 we must have v 1 ≤ v 2 ≤ . .…”

“…We remind the reader that given two ideals a, J ⊂ m in a local ring (R, m) of characteristic p > 0, such that a ⊆ √ J, the F-threshold of a with respect to J is defined as When a = J = m, c m (m) is called the diagonal F-threshold of R. It is observed in [14] that when a = m, and R is a standard graded ring, ν J m (q) is the top socle degree of R/J [q] . Therefore we can apply the result of Theorem 4.1 to calculate the diagonal F-threshold of diagonal hypersurface rings.…”

“…If a ⊆ √ J, then c J (a) = lim q=p e →∞ max{N |a N ⊂J [q] } q is the F-threshold of a with respect to J. The diagonal F-threshold of a ring is obtained when a = J = m. Diagonal F-thresholds of certain rings were studied in [15,14,4]. Question 1.1 has also been answered in the case when n = 2 and char(k) = p > 0 in [7].…”