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Non-commutative resolutions and Grothendieck groups. (English) Zbl 1327.13053
M. Auslander initiated the use of noncommutative algebras of finite dimension by studying the representation theory of Cohen-Macaulay rings, that is, more generally orders. Two classes of noncommutative algebras were singled out, the Auslander algebras and the non-singular orders. The representation theory of finite-dimensional representations is encoded in the structure of their Auslander algebras. All Cohen-Macaulay modules are projective, so that the study of the non-singular orders are more basic.
The study of such algebras gives applications in algebraic geometry. A first example is Van den Bergh’s definition of Noncommutative Crepant Resolution, NCCR: Let $$R$$ be a commutative noetherian normal domain. Then a reflexive $$R$$-module $$M$$ is said to give a NCCR of $$\text{Spec}R$$ if $$\Lambda=\text{End}_R(M)$$ is a nonsingular $$R$$-order, which means that $$\Lambda_{\mathfrak p}=\text{End}_{R_{\mathfrak p}}(M_{\mathfrak p},M_{\mathfrak p})$$ is a maximal Cohen-Macaulay $$R_{\mathfrak p}$$-module for each $$\mathfrak p\in\text{Spec}R$$.
This article considers the slightly simpler concept of NCR, that is Noncommutative resolution: A finitely generated module $$M$$ over a commutative noetherian ring $$R$$ is called a NCR of $$\text{Spec}(R)$$ if $$M$$ is faithful and $$\text{End}_R(M)$$ has finite global dimension. NCRs exist when $$R$$ is artinian, or reduced and one-dimensional. This article treats the question on which rings $$R$$ that have an NCR. The conditions are given in the terms of the Grothendieck group of the category of finitely generated $$R$$-modules, and its subcategories. The existence of an NCR forces strong constraints on the singularities of $$R$$. The formulations of the conditions on $$R$$ in terms of the Grothendieck group leads to influence and use of results from algebraic K-theory. This leads to one of the main results stating that for surface singularities over an algebraically closed field, the existence of a NCR characterise rational singularities. Thus the rationality of a surface singularity can be tested on the existence of a NCR.
The main results of the article, more or less verbatim, is as follows:
Theorem 2.5. Let $$R$$ be a semilocal ring and assume that $$M$$ gives a NCR of $$R$$. Let $$\mathcal C_M$$ be the full subcategory of mod$$R$$ consisting of $$X$$ satisfying $$\text{supp}X\subset\text{NG}(M).$$ Then $$K_0(R)/\langle\mathcal C_M\rangle$$ is a finitely generated abelian group. ($$K_0(R)$$ denotes the Grothendieck group of $$R$$, $$\text{NG}(M)$$ is the nongenerating locus of $$M$$.)
Theorem 3.11. Let $$R$$ be a normal, Cohen-Macaulay standard graded algebra over a subfield $$k$$ of $$\mathbb C$$. Let $$\mathfrak m$$ be the irrelevant ideal of $$R$$. Suppose that $$\text{Spec}R\setminus\{\mathfrak m\}$$ has only rational singularities. Suppose moreover that there exists an $$R$$-module $$M$$ giving a NCR. Then $$\text{Spec}R$$ has only rational singularities.
A few relevant, explicit examples are given, and all in all this article gives nice results from the noncommutative algebraic geometry to the commutative. Also, a really nice historical survey is given in the introduction, and a good list of references ends the article.

MSC:
 13D15 Grothendieck groups, $$K$$-theory and commutative rings 14B05 Singularities in algebraic geometry 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 16G30 Representations of orders, lattices, algebras over commutative rings 18G20 Homological dimension (category-theoretic aspects)
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References:
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