17 - Balanced Dualizing Complexes over NC Graded Rings

Summary

The focus of this chapter is on balanced NC dualizing complexes (DC). Let A be a noetherian connected NC graded ring over the base field K, with enveloping ring Aen = A ⊗K Aop. A complex R ∈ D(Aen,gr) is called a graded NC DC if its cohomology is bounded and finite both sides; it has finite graded injective dimension on both sides; and it has NC derived Morita property (see abstract of Chapter 13) on both sides. A balanced NC DC over A is a pair (R,β), where R is a graded NC DC over A with symmetric derived m-torsion, and β : RΓm(R) → A*is an isomorphism in D(Aen,gr). A balanced DC (R,β) is unique up to a unique isomorphism, and it satisfies the NC Graded Local Duality Theorem. We prove that A has a balanced DC iff A satisfies the χ condition and has finite local cohomological dimension. If A is an Artin--Schelter (AS) regular graded ring, then it has a balanced DC R = A(φ,-l)[n], a twist of the bimodule A by an automorphism φ and integers -l and n.