Finite regularity and Koszul algebras

We determine the positively graded commutative algebras over which the resi due field module the homogeneous maximal ideal has finite Castelnuovo-Mumfo rd regularity: they are the polynomial rings in finitely many indeterminate s over Koszul algebras; this proves a conjecture of Avramov and Eisenbud. W e also show that if the residue field of a finitely generated graded algebr as has finite regularity, then so do all finitely generated graded modules.