On the growth of deviations

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 increase 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.