Projectively simple rings

By Zinovy Reichstein, D. Rogalski, James J. Zhang. 2004

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Abstract An infinite-dimensional N -graded k-algebra A is called projectively simple if dim k A / I ∞ for every nonzero two-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 dim X ⩽ 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 proj A is Cohen–Macaulay and Gorenstein.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/f72646887f5c213cef9cdfb8e915e07ddcc167ad