. R A ] 1 0 Ja n 20 04 PROJECTIVELY SIMPLE RINGS

We introduce the notion of a projectively simple ring, which is an infinite-dimensional N-graded k-algebra A such that dimk A/I < ∞ for every nonzero 2-sided ideal I ⊂ A. We show that if a projectively simple ring A is strongly noetherian, is generated in degree 1, and has a point module, then A is equal in large degree to a twisted homogeneous coordinate ring B = B(X,L, σ). Here X is a smooth projective variety, σ is an automorphism of X with no proper σ-invariant subvariety (we call such automorphisms wild), and L is a σ-ample line bundle. We conjecture that if X admits a wild automorphism then every irreducible component of X is an abelian variety. We prove several results in support of this conjecture; in particular, we show that the conjecture is true if dimX ≤ 2. In the case where X is an abelian variety, we describe all wild automorphisms of X. Finally, we show that if A is projectively simple and admits a balanced dualizing complex, then projA is Cohen-Macaulay and Gorenstein.