[Paper Review] Remarks on non-commutative crepant resolutions of complete intersections

Hailong Dao|ArXiv.org|Nov 24, 2009

Commutative Algebra and Its Applications11 references7 citations

TL;DR

This paper investigates the existence of non-commutative crepant resolutions (NCCRs) over complete intersection singularities using homological algebra. It proves that NCCRs do not exist for 3-dimensional $$-factorial hypersurfaces or even-dimensional complete intersections of dimension $\geq 4$, despite commutative crepant resolutions existing in these cases. The key insight links NCCR existence to Tor-rigidity and Serre conditions, offering a homological obstruction framework applicable over arbitrary fields.

ABSTRACT

We study obstructions to existence of non-commutative crepant resolutions, in the sense of Van den Bergh, over local complete intersections.

Motivation & Objective

• To determine the existence conditions for non-commutative crepant resolutions (NCCRs) over local complete intersection rings.
• To identify homological obstructions—particularly Tor-rigidity and Serre conditions—to NCCR existence.
• To explain why certain non-commutative resolutions fail to be crepant in positive characteristic, especially in the context of Frobenius pullbacks.
• To provide a homological framework for constructing or ruling out NCCRs using module-theoretic conditions like S_n and Tor-rigidity.
• To extend known results on NCCRs in dimension 2 and 3 to higher-dimensional complete intersections.

Proposed method

• Uses homological algebra over commutative Gorenstein local rings, focusing on reflexive modules and their endomorphism rings.
• Applies Tor-rigidity as a key condition: a module $M$ is Tor-rigid if $\operatorname{Tor}_i^R(M,N)=0$ implies $\operatorname{Tor}_j^R(M,N)=0$ for all $j\geq i$.
• Employs Serre's condition $(S_n)$ and freeness in codimension $n$ to analyze depth and module structure.
• Leverages the structure of hypersurfaces $R = S/(f)$ with $S$ regular or unramified, and lifts modules from $R$ to $S$ to study deformation of NCCRs.
• Uses exact sequences and local cohomology to relate depth conditions on $M$ over $R$ to those on $N$ over $S$, especially via the long exact sequence in local cohomology.
• Applies Nakayama’s Lemma and projective dimension arguments to show finite global dimension of endomorphism rings.

Experimental results

Research questions

• RQ1 Under what conditions does a non-commutative crepant resolution (NCCR) exist for a complete intersection singularity?
• RQ2 Why do NCCRs fail to exist for 3-dimensional $\mathbb{Q}$-factorial hypersurfaces, despite commutative crepant resolutions existing?
• RQ3 Can Tor-rigidity and Serre conditions be used to construct or obstruct NCCRs in higher dimensions?
• RQ4 How do NCCRs behave under deformation from a hypersurface to a regular ambient ring?
• RQ5 Why do Frobenius pullback resolutions fail to be crepant in positive characteristic, and how does this relate to Tor-rigidity?

Key findings

• For a 3-dimensional local hypersurface $R$ that is $\mathbb{Q}$-factorial, no NCCR exists, even if commutative crepant resolutions exist.
• For even-dimensional complete intersections of dimension $\geq 4$ with isolated singularities, no NCCR exists.
• If $M$ is a reflexive $R$-module with $\mathcal{N}$ the set of non-Tor-rigid indecomposable MCM modules, and $\mathcal{N} \subset \operatorname{pe}^n(M)$, then $\operatorname{Hom}_R(M,M)$ has finite global dimension at most $\dim R + n$, and if it is maximal Cohen-Macaulay, it is an NCCR.
• If $R = S/(f)$ with $S$ regular and $M$ satisfies $(S_3)$ and is free in codimension 2, then $\operatorname{Hom}_R(M,M)$ has finite global dimension only if $\operatorname{Hom}_S(N,N)$ does for any lift $N$ of $M$, implying obstruction to NCCR existence.
• In positive characteristic, the non-commutative resolution $A = \operatorname{Hom}_R({}^eR,{}^eR)$ is not a NCCR because ${}^eR$ is Tor-rigid, and thus $A$ fails to be crepant.