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On the noncommutative Bondal-Orlov conjecture. (English) Zbl 1325.14007
The noncommutative algebraic geometry, based on derived algebraic geometry, developed by Van den Bergh leads to the concept of Noncommutative Crepant Resolutions (NCCRs). This beautiful article brings this to its explicit form, and proves a very essential conjecture due to Bondal and Orlov in low dimensions:
If $$Y_1$$ and $$Y_2$$ are two crepant resolutions of $$X$$, then $$Y_1$$ is derived equivalent to $$Y_2$$. In the study of one-dimensional fibres and in the McKay Correspondence for dimension $$d\leq 3$$, $$Y_1$$ and $$Y_2$$ are derived equivalent to certain noncommutative rings. This means that to show $$Y_1$$ derived equivalent to $$Y_2$$ is equivalent to proving that the noncommutative rings are derived equivalent.
The authors really give all necessary definitions: If $$R$$ is Cohen-Macaulay (CM), $$\Lambda$$ a module-finite $$R$$-algebra, then
(1)
$$\Lambda$$ is called an $$R$$-order if $$\Lambda$$ is maximal CM as $$R$$-module. It is called nonsingular if gl.dim $$\Lambda_{\mathfrak p}=\dim R_{\mathfrak p}$$ for all primes $$\mathfrak p\subset R$$.
(2)
A noncommutative Crepant Resolution (NCCR) of $$R$$ is $$\Gamma=\text{End}_R(M)$$ with $$M$$ a non-zero reflexive $$R$$-module such that $$\Gamma$$ is a non-singular $$R$$-order.
When $$R$$ is CM and not necessarily Gorenstein, it can be many NCCR’s of $$R$$, and these are related to cluster tilting objects in the category CM $$R$$.
This article considers a specialization of a more general conjecture: (Noncommutative Bondal-Orlov): If $$R$$ is a normal Gorenstein domain, then all NCCRs of $$R$$ are derived equivalent. This conjecture is generalized to some cases with CM rings, and the main result is (literally):
Theorem. Let $$R$$ be a $$d$$-dimensional CM equi-codimensional normal domain with a canonical module $$\omega_R$$.
(1)
If $$d=2$$, then all NCCRs of $$R$$ are Morita equivalent.
(2)
If $$d=3$$, then all NCCRs of $$R$$ are derived equivalent.
An important point with this theorem, is that unlike other conditions in the literature, this condition is decided on the base singularity $$R$$, and not on a fibre product.
Again, all necessary preliminaries are given, except possibly the theory of tilting modules. However, relevant references (equally good as the present article) to this are given. Even the definition of CM modules. Then the main result is explicitly proved by giving projective resolutions, and so the theorem follows directly from general results in tilting theory.