[Paper Review] Non-commutative resolutions and Grothendieck groups

Hailong Dao, Osamu Iyama+2 more|arXiv (Cornell University)|May 21, 2012

Algebraic structures and combinatorial models22 references3 citations

TL;DR

This paper investigates non-commutative resolutions (NCRs) of singularities in commutative noetherian normal domains by analyzing the Grothendieck group and class group of the ring. It establishes that the existence of an NCR imposes strong finiteness conditions on these groups, and in particular, implies rational singularities for standard graded Cohen-Macaulay algebras under mild assumptions.

ABSTRACT

Let $R$ be a noetherian normal domain. We investigate when $R$ admits a faithful module whose endomorphism ring has finite global dimension. This can be viewed as a non-commutative desingularization of $\Spec(R)$. We show that the existence of such modules forces stringent conditions on the Grothendieck group of finitely generated modules over $R$. In some cases those conditions are enough to imply that $\Spec(R)$ has only rational singularities.

Motivation & Objective

• To determine necessary conditions for a noetherian normal domain R to admit a non-commutative resolution (NCR), i.e., a faithful module M such that End_R(M) has finite global dimension.
• To investigate how the existence of such NCRs constrains the algebraic K-theory and class group structure of R.
• To establish that for standard graded Cohen-Macaulay algebras over C, the existence of an NCR forces the ring to have only rational singularities if the punctured spectrum is already rational.
• To explore the relationship between finiteness of the Grothendieck group K0(R) and the rationality of singularities in local rings.
• To provide a negative example in characteristic zero where CH₀(X)Q is finite but H^d(X, O_X) ≠ 0, suggesting potential counterexamples to a conjectural converse of the main theorem.

Proposed method

• Use algebraic K-theory and Grothendieck group techniques to relate the finiteness of K₀(R) and Cl(R) to the existence of NCRs.
• Apply results from algebraic geometry, including the Leray-Serre spectral sequence and cohomological vanishing theorems, to analyze the cohomology of projective varieties.
• Leverage the fact that the non-Gorenstein locus of a module M giving an NCR has codimension at least 2 in a normal domain, via Proposition 3.8.
• Use the isomorphism between CH₀(X)Q and CH₀(X-D)Q for a divisor D to reduce cohomological vanishing to the behavior on a divisor.
• Apply Lemma 3.9, which states that if CH₀(X)Q is supported on a divisor, then H^d(X, O_X) = 0 for a smooth projective variety X of dimension d.
• Use the fact that rational singularities are characterized by H^d(X, O_X) = 0 for a resolution f: X̃ → X, via [34, Theorem 2.2].

Experimental results

Research questions

• RQ1 Under what conditions on a normal domain R does there exist a faithful module M such that End_R(M) has finite global dimension?
• RQ2 How does the finiteness of the Grothendieck group K₀(R) or the class group Cl(R) constrain the singularities of R?
• RQ3 For a standard graded Cohen-Macaulay algebra R over C, does the existence of an NCR imply that Spec(R) has only rational singularities if Spec(R) – {m} is already rational?
• RQ4 Can the finiteness of CH₀(X)Q for a smooth projective variety X imply that its homogeneous coordinate ring has rational singularities?
• RQ5 Is the converse of the main result true: if K₀(R) is finitely generated, must R have rational singularities?

Key findings

• If R is a semilocal normal domain with an NCR, then its class group Cl(R) is finitely generated.
• If R is semilocal and M is a module that is locally a generator outside a closed subscheme of dimension ≤1 and gives an NCR, then K₀(R) is finitely generated.
• For a standard graded Cohen-Macaulay algebra R over C with only rational singularities outside the irrelevant ideal, the existence of an NCR implies that Spec(R) has only rational singularities.
• The cohomological condition H^d(X, O_X) = 0 for a smooth projective variety X of dimension d is equivalent to CH₀(X)Q being supported on a divisor, under the assumptions of Lemma 3.9.
• The existence of a K3 surface X over Q̄ with finite-dimensional CH₀(X)Q but non-vanishing H²(X, O_X) suggests a potential counterexample to the converse of the main theorem in positive characteristic.
• The result establishes a new characterization of rational singularities for surfaces: a surface singularity is rational if and only if it admits an NCR.