On the rates of growth of the homologies of Veronese subrings

Subadditivity of syzygies of Koszul algebras

“…For Koszul algebras Backelin [3] and Kempf [21] proved inequalities (1) t S i pRq ď 2i for 1 ď i ď pd S R . Cases when equality hold are described in [2].…”

“…In Section 5 we prove (2) t S i`1 pRq ď t S i pRq`2 for 1 ď i ď e´dim R`1 when pi`1q is invertible in k; since then t S 1 pRq " 2, this inequality refines (1) under additional hypotheses. It is itself the special case b " 1 of the following inequality: (3) t S a`b pRq ď t S a pRq`t S b pRq for a, b ě 1 with a`b ď pd S R . We conjecture that (3) holds without restrictions on the characteristic of k and review evidence gleaned from a number of sources.…”

On the log-concavity of Hilbert series of Veronese subrings and Ehrhart series

“…The n'th Veronese subring of R is the graded ring R n = ⊕ i≥0 R in . The behaviour of Veronese subrings for large n has been studied by Backelin [2] and Eisenbud et al [14]. The Hilbert function of R is defined by H (R, m) = dim k R m , for each nonnegative integer m, and by a theorem of Hilbert [11,Theorem 4.1.3], H (R, m) is a polynomial in m of degree d for m sufficiently large.…”

Linear equations over noncommutative graded rings

“…The rate is equal to 1 if and only if R is Koszul. If an algebra has finite rate, then its Veronese subring of sufficiently high order is Koszul [7]. The following proposition (part (b)) was originally proved for commutative algebras in [16,Proposition 1.2].…”