On the growth of deviations

Abstract

The deviations of a graded algebra are a sequence of integers that determine the Poincaré series of its residue field and arise as the number of generators of certain DG algebras. In a sense, deviations measure how far a ring is from being a complete intersection. In this paper we study extremal deviations among those of algebras with a fixed Hilbert series. In this setting, we prove that, like the Betti numbers, deviations do not decrease when passing to an initial ideal and are maximized by the Lex-segment ideal. We also prove that deviations grow exponentially for Golod rings and for certain quadratic monomial algebras.

Syzygies in Hilbert schemes of complete intersections

“…We have HP(F ) HP Exp q(ζ), R by Proposition 2.5 (5). Since HP(E h ) HP(E k ) and HP Exp q(ζ), R = HP(E k ) + 1 we conclude that HP(F ) and HP(E h ) differ by an integer.…”

“…A consequence of Theorem 4.3 and the results of [5] is the fact that an expansive subscheme of P n has maximal deviations in its Hilbert scheme.…”

“…We may assume, as in Lemma 4.1, that I :x n+1 = I. Let I = I+(x n+1 ) (x n+1 )⊆ S and L = Lex( I) ⊆ S. By[5, Theorem 3.4] we have ε i ( S/ I) ≤ ε i ( S/ L) for all i ≥ 2. It follows from[3, Proposition 7.1.6] that ε i (S/I) ≤ ε i (S/L) for all i ≥ 2, where L = LS ⊆ S. The ideals L and Exp(p) are strongly stable, and this implies that S/L and S/Exp(p) are Golod rings by[22, Theorem 4].…”

Graded algebras with cyclotomic Hilbert series

“…Then R is a 2-dimensional standard graded domain which is Cohen-Macaulay in every characteristic (one checks via some characteristic-free Gröbner basis computation that the system of parameters {s 8 , t 8 } is a regular sequence for R). Since the h-polynomial of R is (1+x) 3 , one has that R is cyclotomic; however, it is not a complete intersection.…”

“…Many properties of the graded algebra are reflected into the 1 numerator of such expression. In the last few years, several authors have been investigating the behaviour of the roots of this polynomial [7,16,3], often focusing on the combinatorially interesting case when such roots are all real. For the combinatorial consequences of real-rootedness, we direct the interested reader to the survey [5].…”