Projectively simple rings |
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Authors: | Z. Reichstein D. Rogalski J.J. Zhang |
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Affiliation: | a Department of Mathematics, University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z2 b Department of Mathematics, MIT, Cambridge, MA 02139-4307, USA c Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195, USA |
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Abstract: | An infinite-dimensional N-graded k-algebra A is called projectively simple if dimkA/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 . 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 is Cohen-Macaulay and Gorenstein. |
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Keywords: | 16W50 14A22 14J50 14K05 |
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