Good and bad Koszul algebras and their Hochschild homology

A systematic study of the homological behavior of finitely and linearly presented modules over Koszul algebras is started. In particular, it is shown that the generating series of the linear Betti numbers of these modules over some precise special Koszul algebras are rationally related to the generating series for the Betti numbers of all local commutative noetherian rings. It is also shown that there are very small commutative Koszul algebras (4 generators and 4, 5 or 6 relations) which are bad in the sense that rational generating series for Betti numbers of linearly presented modules over them cannot be put on a common denominator. Finally, we do some explicit calculations that indicate that if a Koszul algebra is not a local complete intersection, then the generating series for its Hochschild (and also cyclic homology) is an irrational function. This supports the conjecture that the answer to our Question 1 on pp. 185–186 of Roos (Varna (1986) 173–189; Lecture Notes in Mathematics, vol. 1352, Springer, Berlin, 1988) is positive.