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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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