Finite regularity and Koszul algebras

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