Some derived equivalences between noncommutative schemes and algebras (Q2869242)

This is a nice, to the point, survey of the most common noncommutative algebraic geometry. Let \(X\) be a smooth algebraic variety. If there is a commutative algebra \(R\) such that \(\text{Coh} X\cong\text{mod}R\), then \(X\) is by definition an affine variety. If \(X\) is not affine, it is sometimes possible to find a noncommutative algebra \(R\) such that \(\mathcal D^b(\text{coh}X)\cong\mathcal D^b(\text{mod}R)\). If \(\mathcal E\) is a tilting generator and \(R=\text{End}_X(\mathcal E)\), then \(\mathcal D^b(\text{coh}X)\cong\mathcal D^b(\text{mod}R)\). This survey show similar derived equivalences between noncommutative schemes and noncommutative algebras.NEWLINENEWLINEIf \(R\) is a not necessarily commutative algebra, \textit{the noncommutative scheme} associated to \(R\) is defined as \(\text{Spec}_{\text{nc}}R=\text{Mod} R\) and \(\text{spec}_{\text{nc}}R=\text{mod}R\) (the capital letters means the general category, the small letters denotes the subcategory of finitely presented modules). The category of graded modules are considered similarly, denoted \(\text{GrMod}\), and \(\text{Tors}A\subset\text{GrMod}A\) is the full subcategory consisting of torsion modules. \(\text{Tors}A\) is dense in \(\text{GrMod}A\) and so the quotient category \(\text{GrMod}A/\text{Tors}A\) can be formed. If \(A\) is graded right coherent, also the quotient \(\text{grmod}A/\text{tors}A\) can be formed. Then, if \(A\) is not necessarily commutative, the noncommutative projective scheme associated to \(A\) is defined by \(\text{Proj}_{\text{nc}}A=\text{GrMod}A/\text{Tors}A\) and \(\text{proj}_{\text{nc}}A=\text{grmod}A/\text{tors}A\). \(\mathcal M(n)\) denotes the image of \(M(n)\in\text{GrMod}A\) in \(\text{Proj}_{\text{nc}}A\). Also, \(\text{Hom}_{\text{Proj}_{\text{nc}}A}(\mathcal M,\mathcal N)\) is denoted \(\text{Hom}_{\mathcal A}(\mathcal M,\mathcal N)\).NEWLINENEWLINEThe author gives the definition and fundamental properties of Artin-Schelter (AS) regular algebras as noncommutative algebraic geometry was motivated by the classification of these: Since commutative AS-regular algebras are the weighted commutative polynomial algebras, and projective spaces are projective schemes associated to commutative polynomial algebras generated in degree 1, it is reasonable to define a quantum projective space as \(\text{Proj}_{\text{nc}}S\) for some AS-regular algebra \(S\). It is pointed out that the classification of 3-dimensional AS-regular algebras of finite GK dimension by Artin-Tate-Van den Bergh and Stephenson, using commutative methods, was the starting point of noncommutative algebraic geometry.NEWLINENEWLINEEvery commutative projective space is derived equivalent to a finite dimensional algebra of finite global dimension, called a Beilinson algebra. The noncommutative generalization says that, if \(S\) is a graded right coherent AS-regular algebra of dimension \(d\geq 1\) and of Gorenstein parameter \(\ell\), then \(\{\mathcal S,\mathcal S(1),\dots,\mathcal S(l-1)\}\) is a full strong exceptional sequence for \(\mathcal D^b(\text{proj}_{\text{nc}}S)\) where \(\mathcal S(i)\) is the image of \(S(i)\in\text{grmod}S\) in \(\text{proj}_{\text{nc}}S\). Moreover, \(R=\text{End}_{\mathcal S}(\bigoplus_{l=0}^{\ell-1}\mathcal S(i))\) is a finite dimensional algebra of global dimension \(d-1\) such that \(\mathcal D^b(\text{proj}_{\text{nc}}S)\cong\mathcal D^b(\text{spec}_{\text{nc}}R)\). One of this articles main objectives, is to extend this result.NEWLINENEWLINEFano algebras are introduced, and with this the concept of ample and quasi-ample objects in an abelian category \(\mathcal C\). If \(\mathcal C\) is a \(\text{Hom}\)-finite \(k\)-linear abelian category and \((\mathcal O,s)\) is an ample pair for \(\mathcal C\), then \(A=\bigoplus_{i\in\mathbb N}\text{Hom}(s^{-i}\mathcal O,\mathcal O)\cong\bigoplus_{i\in\mathbb N}\text{Hom}_{\mathcal C}(\mathcal O,s^i\mathcal O)\) is a graded right coherent algebra such that \(\mathcal C\cong\text{proj}_{\text{nc}}A\). This in turn is used to define the Veronese algebra, the Beilinson algebra, and the extended result is achieved: If \(S\) is an AS-regular algebra over \(R\) of dimension \(d\geq 1\) and of Gorenstein parameter \(\ell\), thenNEWLINENEWLINE(1) \(\nabla S\) (defined from the above) is a quasi-Fano algebra of dimension \(d-1\), (2) \(S^{[\ell]}\cong\prod(\nabla S)^\tau\) for some graded algebra automorphism \(\tau\in\text{Aut}_k\prod(\nabla S)\), (3) \(\mathcal D(\text{Proj}_{\text{nc}}S)\cong\mathcal D(\text{Spec}_{\text{nc}}\nabla S)\). Also, if \(S\) is graded right coherent, then \(\nabla S\) is Fano and \(\mathcal D^b(\text{proj}_{\text{nc}}S)\cong\mathcal D^b(\text{spec}_{\text{nc}}\nabla S)\).NEWLINENEWLINEFinally, the McKay correspondence is studied as a noncommutative resolution of singularities, illustrating the above results and giving explicit examples of noncommutative algebraic computations.NEWLINENEWLINEThe survey article is very well written and gives a good understanding of the fundamental principles of this particular definition of noncommutative algebraic geometry. Also, it illustrates the eventual weaknesses of classification by derived categories.NEWLINENEWLINEFor the entire collection see [Zbl 1256.14001].