Non-commutative resolutions of toric varieties
Let \(R\) be a local or graded, commutative, Noetherian ring. Auslander-Buchsbaum and Serre have proved that \(R\) is regular if and only if \(R\) has finite global dimension. \textit{J. Dixmier} [Anais Acad. Brasil. Ci. 35, 491--519 (1963; Zbl 0143.05302)] made finite global dimension a part of the definition of regularity for non-commutative rings, and the understanding of the connection between regularity and finite global dimension has been refined by Auslander, Artin, Shelter, Van den Bergh and others. In particular, if \(R\) is a regular ring of finite type over a perfect field \(k\), then the ring of differential operators \(D_k(R)\) has finite global dimension. The proof of this result was given by \textit{J.-E. Roos} [C. R. Acad. Sci., Paris, Sér. A 274, 23--26 (1972; Zbl 0227.16021)] in characteristic zero, and \textit{S. P. Smith} [J. Algebra 107, 98--105 (1987; Zbl 0617.13007)] in characteristic \(p\). This paper establishes the finite global dimension of several non-commutative algebras associated to a normal toric algebra \(R\) over a field \(k\). Such \(R\) is commutative, but only rarely regular. The algebras under consideration in this article are (1) \(\operatorname{End}_R(\mathbb{A})\), where \(\mathbb{A}\) is a complete sum of conic modules. Note that a conic module is a combinatorial defined \(R\)-module given below, and a direct sum of conic modules is complete if every conic module is isomorphic to a summand. (2) \(\operatorname{End}_R(R^{1/q})\), for sufficiently large \(q\). \(R^{1/q}\) is the \(k\)-algebra spanned by \(q\)-th roots of monomials in \(R\). When \(k\) is perfect and \(q\) is a power of the characteristic of \(k\), \(R^{1/q}\) is the ring of \(q\)-th roots of elements in \(R\). (3) The ring of \(k\)-linear differential operators \(D_k(R)\) of \(R\), assuming \(k\) is perfect with positive characteristic. The main result of the paper is an explicit projective resolution for each simple module for the algebra \(\operatorname{End}_R(\mathbb{A})\). This proves that the global dimension of \(\operatorname{End}_R(\mathbb{A})\) equals the dimension of \(R\). Then similar results follow for \(\operatorname{End}_R(R^{1/q})\), and it is proved that this ring is Morita equivalent to \(\operatorname{End}_R(\mathbb{A})\). Taking direct limits, the result for differential operators follows. The authors claim that the proofs are elementary, direct, self-contained, and constructive. This statement is most probably given because \textit{Š. Špenko} and \textit{M. Van den Bergh} [Invent. Math. 210, No. 1, 3--67 (2017; Zbl 1375.13007)] have given the finiteness of global dimension in (1) and (2) by less constructive proofs, and the authors of the present article admits that they beat them to publication. Now, as the algebras in question have finite global dimension, they should be regarded as regular in the non-commutative sense. The two first are said to be non-commutative resolutions of the toric algebra \(R\), as defined by \textit{A. Bondal} and \textit{D. Orlov} [in: Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20--28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press; Singapore: World Scientific/distributor. 47--56 (2002; Zbl 0996.18007)]. The final result considers when the non-commutative resolutions are crepant. This ensures that the module category of the resolution is sufficiently close to the module category of the original algebra. The conclusion is that \(\operatorname{End}_R(\mathbb{A})\) is a non-commutative crepant resolution if and only if the toric algebra \(R\) is simplicial. For the proofs, a conic module of a normal toric algebra \(R\) is a fractional monomial ideal of \(R\) whose monomial support is given by shifting the defining cone of \(R\). A conic module has rank one and is Cohen Macaulay, and each toric \(R\) has finitely many conic modules up to isomorphism. The conic modules can be parametrized by polyhedral chambers of constancy inside a hyperplane arrangement. For each chamber of constancy, the combinatorics of the faces are used to construct a chain complex \(K^\bullet_\Delta\) of sums of conic modules. These complexes satisfy the acyclicity Lemma which state that the complex \(\Hom_R(A_{\Delta^\prime},K^\bullet_\Delta)\) is either acyclic or a resolution of the ground field \(k\), depending on whether or not \(A_\Delta\simeq A_{\Delta^\prime}\). The definition of a complete conic module is motivated by the ring \(R^{1/n}\) spanned by formal \(n\)th roots of monomials in \(R\), which is a complete sum of conic modules for large \(n\). For a complete sum of conic modules \(\mathbb{A}\) and a chamber of constancy \(\Delta\), the acyclicity Lemma implies that the complex \(\Hom_R(\mathbb{A},K^\bullet_\Delta)\) is a finite projective resolution of a simple \(\operatorname{End}_R(\mathbb A)\)-module. The authors then prove that every finitely generated \(\operatorname{End}_R(\mathbb{A})\)-module has a finite projective resolution, i.e. finite global dimension. The authors explore the relation with the Frobenius map. Kunz's Theorem states that regularity of rings can be characterized by flatness of the Frobenious, and this leads to the development of tight closure and F-regularity. It gives a relationship to singularities in the minimal model program and then with rings of differential operators. The results of the present article indicate that under the given hypothesis, the Frobenius provides a way to construct functorial resolutions of singularities. The property that a \(\mathbb{C}\)-algebra admits a non-commutative crepant resolution is related to rational singularities. Also, the property of strong-F-regularity is a prime characteristic analogue of rational singularities. This leads to a question that for a strongly F-regular ring \(R\) with finite F-representative type, \(\operatorname{End}_R(R^{1/p^e})\) always has finite global dimension. The authors claim that the strength of the article is that the exposition is fairly direct and self-contained, widely accessible, including to commutative algebraists and algebraic geometers. This reviewer find this true, and recommends the article both as a theoretical minimum and as a different view to non-commutative resolutions.
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