18 - Rigid Noncommutative Dualizing Complexes

Summary

Let A be a NC noetherian ring, with enveloping ring Aen.A NC DC over A is a complex R ∈ D(Aen) satisfying the conditions stated earlier. The NC square of R is a complex Sq(R) ∈ D(Aen). A NC rigidDC over A is a pair (R,ρ), where R is a NC DC and ρ : R → Sq(R) is an isomorphism in D(Aen). We prove that a rigid NC DC (R,ρ) is unique up to a unique rigid isomorphism. If the ring A admits a filtration such that the graded ring Gr(A) is noetherian connected and has a balanced DC, then A has a rigid DC. This material is due to Van den Bergh. If the graded ring Gr(A) is AS regular, then the rigid NC DC of A is R = A(μ)[n], where μ is a ring automorphism of A and n is an integer. The automorphism ν := μ−1 is called the Nakayama automorphism of A. Such a ring A is called an n-dimensional twisted Calabi--Yau ring. We state and prove the Van den Bergh Duality Theorem for Hochschild (co)homology and give an example of a Calabi--Yau category of fractional dimension.