We study Zariski cancellation problem for noncommutative algebras that are not necessarily domains.2000 Mathematics Subject Classification. Primary 16P99, 16W99.

The notion of a noncommutative quasi-resolution is introduced for a noncommutative noetherian algebra with singularities, even for a non-Cohen-Macaulay algebra. If A is a commutative normal Gorenstein domain, then a noncommutative quasi-resolution of A naturally produces a noncommutative crepant resolution (NCCR) of A in the sense of Van den Bergh, and vice versa. Under some mild hypotheses, we prove that (i) in dimension two, all noncommutative quasi-resolutions of a given noncommutative algebra are Morita equivalent, and (ii) in dimension three, all noncommutative quasi-resolutions of a given noncommutative algebra are derived equivalent. These assertions generalize important results of Van den Bergh, Iyama-Reiten and Iyama-Wemyss in the commutative and central-finite cases.In dimension three (respectively, two), this conjecture was proved by Bridgeland [Br] in 2002 (respectively, by Kapranov-Vasserot [KV] in 2000). The conjecture is still open in higher dimensions. Noticed by Van den Bergh [VdB1] in the study of one-dimensional fibres and by Bridgeland-King-Reid [BKR] in the study of the McKay correspondence for dimension d ≤ 3 that both D b (coh(Y 1 )) and D b (coh(Y 2 )) are equivalent to the derived category of certain noncommutative rings. Motivated by Conjecture 0.1 and work of [BKR, Br, VdB1], Van den Bergh [VdB2] introduced the notation of a noncommutative crepant resolution (NCCR) of a commutative normal Gorenstein domain A (in the original reference, the author used the notation R). Let us recall the definition of a NCCR given in [IR, Section 8] which is quite close to the original definition of Van den Bergh [VdB2, Definition 4.1]. As usual, CM stands for Cohen-Macaulay. Definition 0.2. Let R be a noetherian commutative CM ring and let A be a module-finite R-algebra.(1) [Au] A is called an R-order if A is a maximal CM R-module. An R-order A is called non-singular if gldim A p = Kdim R p for all p ∈ Spec(R).

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