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1. Universal resolutions. A recent result of Iyengar  addresses this problem on the level of resolutions 12 . 1. Let fA: A ---+ R be a DC algebra resolution over Q with structure map ryA: Q ---+ A, and let fU: U ---+ M be a DC module resolution of Mover A. 8, if Q is noetherian and finite projective Q-rnodules are free, then a resolution of Mover R is finitistically determined by matrix data over Q, namely, the multiplication tables of the algebra A and the module U, and the differentials in these finite complexes.
I Q9Q k) . Proof. The fact that R is faithfully flat over R, and mR is its maximal ideal yields the relations on both ends, so we argue for those in the middle. 2. For (3), use the long exact sequence of Tor Q' (-, k) applied to the exact sequence 0 -+ I' -+ Q' -+ R -+ O. 4. 5. There is an a E lR, such that f3;;(M) ::; an for n ~ 1. 0 A local ring homomorphism cp: R -+ R', such that mR' is the maximal ideal of R', may be lifted - in more than one way - to a morphism of DG algebras K'P: KR -+ K R' (we use a 'functorial' notation, because in the cases treated below the choice of a specific lifting will be of no consequence, while it helps to distinguish K'P from K R Q9 R cp: K R -+ K R Q9 R R').
A weaker property, called 'strongly22 exponential growth' in , is established when m3 = 0 by Lescot : for each M there is a real number "( > 1, such that f3;;(M) 2: "(n for n 2: O. This special case is significant, as Anick and Gulliksen  prove that each P~(t) is rationally related to P~ (t) for some artinian k-algebra (8, n, k) with n3 = O. Rationality. The study of infinite resolutions over local rings was triggered by a question, variously and appropriately linked to the names of Kaplansky, Kostrikin, Serre, and Shafarevich: Does P~(t) represent a rational function?
Non-self-adjoint operator algebras in Hilbert space