2004/01/10 by Z. Reichstein, Zinovy Reichstein, Reichstein, Z. +5 · 1 citation
Mathematics · #14A22 #14J50 #14K05 #16W50 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Rings and Algebras (math.RA) #math.AG #math.RA #msc:14A22 #msc:14J50 #msc:14K05 #msc:16W50
paper · pdf · doi:10.48550/arxiv.math/0401098
Some new material has been added in Section 1; to appear in Advances in Mathematics
openalex publication_date 2004/01/10 · arxiv created 2005/05/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce the notion of a projectively simple ring, which is an infinite-dimensional graded k-algebra A such that every 2-sided ideal has finite codimension in A (over the base field k). Under some (relatively mild) additional assumptions on A, we reduce the problem of classifying such rings (in the sense explained in the paper) to the following geometric question, which we believe to be of independent interest. Let X is a smooth irreducible projective variety. An automorphism f: X -> X is called wild if it X has no proper f-invariant subvarieties. We conjecture that if X admits a wild automorphism then X is an abelian variety. We prove several results in support of this conjecture; in particular, we show that the conjecture is true if X is a curve or a surface. In the case where X is an abelian variety, we describe all wild automorphisms of X. In the last two sections we show that if A is projectively simple and admits a balanced dualizing complex, then Proj(A) is Cohen-Macaulay and Gorenstein.