GEOMETRIC IDEALIZER RINGS

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Abstract

Let B = B(X, L, sigma) be the twisted homogeneous coordinate ring of a projective variety X over an algebraically closed field k. We construct and investigate a large class of interesting and highly noncommutative noetherian subrings of B. Specifically, let Z be a closed subscheme of X and let I subset of B be the corresponding right ideal of B. We study the subalgebra

R = k + I

of B. Under mild conditions on Z and sigma is an element of Aut(k)(X), R is the idealizer of I in B: the maximal subring of B in which I is a two-sided ideal.

Our main result gives geometric conditions on Z and sigma that determine the algebraic properties of R. We say that

{sigma(n()Z)}

is critically transverse if for any closed subscheme Y of Z, for vertical bar n vertical bar >> 0 the subschemes Y and sigma(n)(Z) are homologically transverse. We show that if {sigma(n)(Z)} is critically transverse, then R is left and right noetherian, has finite left and right cohomological dimension, is strongly right noetherian but not strongly left noetherian, and satisfies right chi(d) (where d = codim Z) but fails left chi(1). This generalizes results of Rogalski in the case that Z is a point in P-d. We also give an example of a right noetherian ring with infinite right cohomological dimension, partially answering a question of Stafford and Van den Bergh.

Further, we study the geometry of critical transversality and show that it is often generic behavior, in a sense that we make precise.

Original languageEnglish
Pages (from-to)457-500
Number of pages44
JournalTransactions of the American Mathematical Society
Volume363
Issue number1
DOIs
Publication statusPublished - Jan 2011

Keywords

  • NONCOMMUTATIVE PROJECTIVE GEOMETRY
  • KLEIMAN-BERTINI THEOREM
  • DUALIZING COMPLEXES
  • ALGEBRAS
  • SURFACES
  • AMPLENESS
  • SCHEMES

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