Connected Graded Gorenstein Algebras with Enough Normal Elements

AbstractWe generalize [12, 1.1 and 1.2] to the following situation.Theorem 1.Let A be a connected graded noetherian algebra of injective dimension d such that every nonsimple graded prime factor ring of A contains a homogeneous normal element of positive degree. Then:(1)A is Auslander–Gorenstein and Cohen–Macaulay.(2)A has a quasi-Frobenius quotient ring.(3)Every minimal prime ideal P is graded andGKdimA/P=d.(4)If, moreover, A has finite global dimension, then A is a domain and a maximal order in its quotient division ring.To prove the above we need the following result, which is a generalization of [3, 2.46(ii)].Theorem 2.Let A be a connected graded noetherian AS-Gorenstein algebra of injective dimension d. Then:(1)The last term of the minimal injective resolution of AAis isomorphic to a shift of A*.(2)For every noetherian graded A-module M,Extd(M,A)is finite dimensional over k