In this survey, noncommutative algebraic geometry is interpreted as the category of quasi-coherent sheaves on an affine noncommutative scheme, having in view the category \(\operatorname{Mod}_A\) of (right) \(A\)-modules. That is, one considers directly the categories of sheaves of modules as this is an effective way to study algebraic varieties in the ordinary commutative situation. The author emphasises that quasi-coherent sheaves are independent of the choice of the topology on schemes, they are sheaves in any natural topology, and so they are claimed to be the objects that best reflect the algebraic structures of schemes. The next task is to glue together the noncommutative affine schemes, and there are different possibilities. The author finds that the most fruitful possibility is connected with the category of sheaves together with natural generalizations dictated by homological algebra. This leads to the derived category of quasi-coherent sheaves and the category of perfect complexes on a noncommutative scheme.
Let \(X\) be a scheme over a field \(\Bbbk\) that is quasi-compact, quasi-separated, has a finite covering by affine schemes whose intersections are affine with the same properties. To \(X\) is associated the unbounded derived category of complexes of \(\mathcal O_X\)-modules with quasi-coherent cohomology \(\mathcal D_\text{Qcoh}.\) It is proved that this category has enough compact objects, and that the triangulated subcategory of compact objects coincides with the category \(\mathit{Perf}-X\) of perfect complexes: A complex is perfect if it is locally isomorphic to a bounded complex of locally free sheaves of finite type. This category can be generated by a single object, called a classical generator. Then the minimal full triangulated subcategory of \(\mathit{Perf}-X\) containing this object and closed under taking direct summands coincides with the whole category \(\mathit{Perf}-X\).
Given the existence of a classical generator \(E\in\mathit{Perf}-X,\) the author gives a new view on the derived category \(\mathcal D_\text{Qcoh}(X),\) as well as the triangulated category \(\mathit{Perf}-X\). With this, \(\mathcal D_\text{Qcoh}(X)\) is equivalent to the unbounded derived category of differential graded (DG) modules \(D(\mathcal R)\) over some differential graded algebra \(\mathcal R\), and the triangulated category of perfect complexes \(\mathit{Perf}-X\) is equivalent to the category of perfect DG modules \(\mathit{Perf}-\mathcal R.\) The algebra \(\mathcal R\) is given as the DG algebra if endomorphisms \(\operatorname{End}(E)\) of the given generator as its lift to a differential graded category \(\mathscr{P}\mathit{erf}-X.\) This is in some sense the game-changer in this paper: The differential graded category \(\mathscr{P}\mathit{erf}-X\) is a natural enhancement of the category \(\mathit{Perf}-X,\) in particular \(\mathit{Perf}-X\) is equivalent to the homotopy category \(\mathcal H^0(\mathscr{P}\mathit{erf}-X).\) A differential graded (DG) category \(\mathcal A\) is a category whose morphisms have the structure of complexes of \(\Bbbk\)-vector spaces. Passing to their zero cohomology spaces, one obtain a \(\Bbbk\)-linear category \(\mathcal H^0(\mathcal A)\) with the same set of objects, called the homotopy category for the DG category \(\mathcal A.\) With an equivalence \(\epsilon:\mathcal H^0(\mathcal A)\overset\sim\rightarrow\mathcal T,\;(\mathcal A,\epsilon)\) is called a DG enhancement for the category \(\mathcal T.\)
The triangulated category \(\mathcal D_\text{Qcoh}(X)\) has several natural enhancements, all naturally quasi-equivalent to each other. A convenient model can be chosen from the xlass of quasi-equivalent DG categories. A DG enhancement of the category \(\mathcal D_\text{Qcoh}(X)\) induces a DG enhancement of the triangulated subcategory \(\mathit{Perf}-X\), and this is denoted \(\mathscr P\mathit{erf}-X.\)
It is proved that \(\mathscr P\mathit{erf}-X\) is quasi-equivalent to a category of the form \(\mathscr P\mathit{erf}-\mathcal R\) with \(\mathcal R\) the DG algebra of endomorphisms of some generator \(E\in\mathscr P\mathit{erf}-X.\) Note that when \(X\) is quasi-compact and quasi-separated, \(\mathcal R\) is cohomologically bounded; it has only a finite number of non-trivial cohomology spaces. This leads to the following definition, stated verbatim:
By a derived noncommutative scheme \(\mathscr X\) ia meant a \(\Bbbk\)-linear DG category of the form \(\mathit{Perf}-\mathcal R,\) where \(\mathcal R\) is a cohomologically bounded DG algebra over \(\Bbbk.\) \(D(\mathcal R)\) is called the derived category of quasi-coherent sheaves on this noncommutative scheme, and the triangulated category \(\mathcal P\mathit{erf}-\mathcal R\) is called the category of perfect complexes on \(\mathscr X.\) There is an equivalence of categories \(\mathcal H^0(\mathscr P\mathit{erf}-\mathcal R)\cong \mathit{Perf}-\mathcal R,\) and by considering the DG categories of the form \(\mathscr P\mathit{erf}-\mathcal R\) the author is able to glue together noncommutative schemes from affine pieces and to obtain noncommutative derived schemes.
The paper studies noncommutative derived schemes, and properties as smoothness, regularity, and properness are extended to such. Morphisms between NC schemes are defined as quasi-functors between DG categories, so there are more morphisms between schemes in the noncommutative sense, forming a DG category where the morphisms can be added and one can define maps between morphisms. Also, for noncommutative schemes the concepts of compactification, resolution of singularities and the Serre functor, are defined.
As a preparation, this article studies some properties of derived noncommutative schemes and compare them to the corresponding properties for commutative schemes. Much space is devoted to a glueing together of noncommutative schemes. This concept does not exist in the commutative world, and gives a lot of applications. The most important is a geometric realization of derived noncommutative schemes, arising naturally. First of all as a natural geometric realization of an abstract algebraic structure, and also because many noncommutative schemes are constructed from a usual geometry with a given geometric realization. These are sometimes linked to admissible subcategories \(\mathcal N\subset\mathscr P\mathit{erf}-\mathcal X\) of categories of perfect complexes on smooth projective schemes \(X.\) Then \(\mathcal N\) is a DG category \(\mathcal N\subset\mathscr P\mathit{erf}-\mathcal R\) and \(\mathcal N\) is smooth and proper. The embedding \(\mathcal N\subset\mathscr P\mathit{erf}-\mathcal X\) is a particular case of a geometric realization of smooth, proper, noncommutative schemes, and is called pure.
Given two DG categories \(\mathscr A,\mathscr B\) and a \(\mathscr B^\circ-\mathscr A\)-bimodule \(\mathsf T\) a DG category \(\mathscr A\vdash_{\mathsf T}\mathscr B\) is defined, and called the gluing via \(\mathsf T.\) This defines gluing of noncommutative schemes \(\mathscr X, \mathscr Y\) under conditions on \(\mathsf T\) and base extension conditions are considered.
In previous work [D. Orlov, Adv. Math. 302, 59–105 (2016; Zbl 1368.14031)], the author proved that if \(\mathscr X, \mathscr Y\) comes from admissible subcategories in categories of perfect complexes on smooth projective schemes, then their gluing \(\mathscr X\vdash_{\mathsf T}\mathscr Y\) via a perfect bimodule \(\mathsf T\) can be realized in the same way, and the article gives conditions for when geometric realizations are stable under base change. Also, noncommutative schemes give more information than commutative schemes. Phenomena called phantoms and quasi-phantoms are discussed. They are smooth and proper noncommutative schemes \(\mathscr X\) for which the \(K\)-theory \(K_\ast(\mathscr X)\) is completely trivial. Krull-Schmidt partners are discussed; these are smooth and proper noncommutative schemes \(\mathscr X\) and \(\mathscr X'\) for which there exists a smooth proper noncommutative scheme \(\mathscr Y\) with the condition that some gluings \(\mathscr X\vdash_{\mathsf T}\mathscr Y\) and \(\mathscr X'\vdash_{\mathsf T}\mathscr Y\) are isomorphic. The paper gives a procedure for constructing smooth and proper noncommutative schemes that are Krull-Schmidt partners for the usual schemes and have the same additive invariants.
The final section of the article studies geometric realization of finite-dimensional algebras. An explicit construction is given, and the particular case of algebras that are endomorphism of a vector-bundle on a (projective) scheme gives really nice results, linking the definitions in this article to other versions (tilting theory) of noncommutative geometry.
This article gives an important survey of Orlov’s noncommutative geometry, it gives a very thorough illustration of the ideas and the complete theory, and it includes new techniques and thereby new important results.