[Paper Review] Noncommutative resolutions and rational singularities

J. T. Stafford, Michel Van den Bergh|ArXiv.org|Dec 1, 2006

Algebraic structures and combinatorial models10 references4 citations

TL;DR

This paper establishes that the centre of a homologically homogeneous, finitely generated $k$-algebra over an algebraically closed field of characteristic zero has rational singularities. Using the framework of noncommutative crepant resolutions—defined as homologically homogeneous algebras of the form $\operatorname{End}_R(M)$ for reflexive $R$-modules $M$—the authors prove that such resolutions imply rational singularities in the commutative case, affirming a conjecture in noncommutative algebraic geometry.

ABSTRACT

Let k be an algebraically closed field of characteristic zero. We show that the centre of a homologically homogeneous, finitely generated k-algebra has rational singularities. In particular if a finitely generated normal commutative k-algebra has a noncommutative crepant resolution, as introduced by the second author, then it has rational singularities.

Motivation & Objective

• To determine whether the existence of a noncommutative crepant resolution implies that the underlying commutative singularity has rational singularities.
• To extend the classical result that crepant resolutions imply rational singularities to the noncommutative setting.
• To clarify the relationship between homologically homogeneous algebras and the singularities of their centres.
• To investigate the necessity of characteristic zero by constructing counterexamples in positive characteristic.
• To provide a theoretical foundation for the Bondal-Orlov conjecture via noncommutative resolutions.

Proposed method

• Define a noncommutative crepant resolution as a homologically homogeneous $k$-algebra $\Delta = \operatorname{End}_R(M)$, where $R$ is a normal affine $k$-domain and $M$ is a reflexive $R$-module.
• Use the theory of tame orders and dualizing complexes to analyze the homological properties of $\Delta$.
• Apply results from noncommutative algebraic geometry, particularly the theory of dualizing complexes and Gelfand-Kirillov dimension.
• Establish that if $\Delta$ is homologically homogeneous, then its centre $Z(\Delta)$ has rational singularities via cohomological and module-theoretic arguments.
• Construct counterexamples in positive characteristic to show that the result fails when $\operatorname{char}(k) > 0$, using fixed rings under group actions.
• Use graded module theory and the structure of twisted polynomial rings to demonstrate non-homogeneity and failure of rational singularities in positive characteristic.

Experimental results

Research questions

• RQ1 Does the existence of a noncommutative crepant resolution imply that the commutative base ring has rational singularities?
• RQ2 What conditions ensure that the centre of a homologically homogeneous algebra has rational singularities?
• RQ3 Can rational singularities be characterized in terms of noncommutative resolutions?
• RQ4 Why does the result fail in positive characteristic, and what structural properties break down?
• RQ5 To what extent do noncommutative crepant resolutions reflect the geometry of their commutative counterparts?

Key findings

• The centre of any homologically homogeneous, finitely generated $k$-algebra over an algebraically closed field of characteristic zero has rational singularities.
• If a normal affine $k$-domain $R$ admits a noncommutative crepant resolution, then $R$ has rational singularities.
• The result fails in positive characteristic: a counterexample is constructed using a $\mathbb{Z}/2\mathbb{Z}$-action on a regular ring, yielding a homologically homogeneous algebra whose centre is not rational.
• The dualizing complex of a noncommutative crepant resolution is not projective, which implies that the algebra is not homologically homogeneous in certain graded settings.
• A counterexample shows that finite global dimension of $\operatorname{End}_R(M)$ does not imply homological homogeneity, even when $R$ is normal and $M$ is reflexive.
• The failure in positive characteristic is linked to the fact that fixed rings need not be direct summands of the original ring, breaking key homological assumptions.