Geometric langlands seminar page university of chicago. Jacob lurie submitted on 11 feb 2007, last revised 19 sep 2007 this version, v5 abstract. Derived categories of torsors for abelian varieties for curves c 1,c. Furthermore, it obscures the physical origins of dng. We study moduli spaces of boundary conditions in 2d topological field theories. Very basic noncommutative geometry vanderbilt university. Moduli problems in derived noncommutative geometry by. Triangulated categories appear in many subjects, such as algebraic geometry. The artinlurie representability criterion makes manifest the. The need for a noncommutative algebraic geometry is apparent in classical invariant.
The derived category of finitedimensional representations of a quiver is an important and somewhat unusual example of a smooth proper dg category. Homological methods in noncommutative geometry 2008 pdf. The theory, called noncommutative geometry, rests on two essential points. I work primarily at the intersection of kontsevich and artinzhang style noncommutative algebraic geometry, where i study the derived category of quasicoherent sheaves on noncommutative projective schemes via the framework of differential graded categories. Algebraic geometry, noncommutative algebra and derived categories. It has now been four decades since david mumford wrote that algebraic geometry seems to have acquired the reputation of being esoteric, exclusive, and. This text is a survey of derived algebraic geometry.
In derived higher noncommutative algebraic geometry one instead considers the derived category of quasicoherent sheaves, or more precisely its dgenhancement or ainfinityenhancement. It is known that a tilting generator on an algebraic variety x gives a derived equivalence between x and a certain noncommutative algebra. A guided tourthrough thegarden of noncommutative motives. In this book, we discuss several types of geometric objects in the usual sense of. By noncommutative algebraic geometry i mean the study of triangulated categories and their enhancements, dg and a 1categories. I explain a method to construct a tilting generator from an ample line bundle. Derived noncommutative schemes, geometric realizations, and finite dimensional algebras preprint pdf available august 2018 with 290 reads how we measure reads. Moduli problems in derived noncommutative geometry pranav. Though it is not clear to me at all how much information is loss by applying these functors some information is. A study of formal moduli problems in the setting of commutative and noncommutative derived algebraic geometry. Thus to understand its relation with classical geometry one should. Dmitri kaledin, tokyo lectures homological methods in noncommutative geometry, pdf, tex. Many natural concepts from usual commutative algebraic geometry also generilize to noncommutative schemes. We show that the infinity category of spectra admits an essentially unique monoidal structure such that the tensor product preserves colimits in each variable, and thereby recover the classical smashproduct operation on spectra.
I am interested in derived categories, the grothendieck ring of varieties, algebraic ktheory and applications of these in birational algebraic geometry, in particular to rationality questions and in studying singularities. As such, they are the fundamental building blocks of derived algebraic geometry. Connections between commutative and noncommutative algebraic. Derived algebraic geometry is the specialization of higher geometry and homotopical algebraic geometry to the infinity,1category of simplicial commutative rings or sometimes, coconnective commutative dgalgebras. Noncommutative geometry is the study of noncommutative algebras as if they were algebras of functions on spaces, like the commutative algebras associated to a ne algebraic varieties, di erentiable manifolds, topological spaces, and measure spaces.
Just as schemes, algebraic spaces and stacks are simplicial sheaves admitting some kind of atlases, the rst stepwill give usuptohomotopysimplicialsheaves, among which thesecond stepwill single out thederived spacesstudied by derived algebraic geometry. It is known that a tilting generator on an algebraic variety x gives a derived equivalence between x and a certain non. Lecture notes on noncommutative algebraic geometry and noncommutative tori s mahanta lectures on derived and triangulated categories b noohi examples of noncommutative manifolds. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Noncommutative, derived and homotopical methods in geometry. An ordinary associative ring can be viewed as an algebra object of the category of abelian groups a, where a is endowed with the structure of a monoidal category via the tensor product. Homological methods in noncommutative geometry tokyo, 20072008 3 i algebraic ktheory. Noncommutative algebraic geometry is a branch of mathematics, and more specifically a. Some descent theorems for quasicoherent sheaves and quasicoherent stacks in the setting of spectral algebraic geometry. In noncommutative algebraic geometry one represents a scheme by an abelian category of quasicoherent sheaves on the scheme, and looks at more general abelian categories as being categories of quasicoherent sheaves on a noncommutative space in derived higher noncommutative algebraic geometry one instead considers the derived category of.
Derived noncommutative algebraic geometry wikipedia. My dissertation is titled geometry of derived categories on noncommutative projective schemes. In this paper, we present an infinitycategorical version of the theory of monoidal categories. Recall that every commutative algebra is of the form. Noncommutative geometry has as its limiting case the classical geometry, but geometry expressed in algebraic terms. Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras over, simplicial commutative rings or. Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of noncommutative algebraic objects such as rings as well as geometric objects derived from them e. An ordinary commutative ring rcan be viewed as a commutative algebra object in the category of abelian. Derived categories were invented by grothendieck and verdier around 1960, not very long after the old homological algebra of derived functors between abelian categories was established. Our goal in this paper is to introduce the theory of e 1ring spectra from an 1categorical point of view.
Noncommutative geometry and algebra my main interest is the noncommutative world in all its aspects. Noncommutative geometry ncg is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions possibly in some generalized sense. In this paper we discuss some properties of derived noncommutative schemes and draw various. This approach represents spaces by categories enriched in cochain complexes dgcategory approach or enriched in spectra. Pdf derived noncommutative schemes, geometric realizations. In mathematics, derived noncommutative algebraic geometry, the derived version of noncommutative algebraic geometry, is the geometric study of derived categories and related constructions of triangulated categories using categorical tools. Both lie theory and algebraic geometry have been at the center of the 20thcentury mathematical studies.
Complex tori and spherical manifolds j plazas dbranes in noncommutative field theory r j szabo readership. Because the approach developed by rosenberg himself aims at representation theory, so i would discuss the relationship with belinson bernstein and deligne. In noncommutative algebraic geometry, we follow ideas of algebraic geometry. Some basic examples include the bounded derived category of coherent sheaves on a smooth variety, called its derived category, or the derived category. Jacob luries home page home ias school of mathematics. I shall apply it to construct a noncommutative algebraic geometry. Recalling duality duality is one of the fundamental concepts in mathematics. Lequivalence has been introduced in my joint work with kuznetsov s6.
Some of these people are also interested in the noncommutative world. The purpose of this book is to extend this correspondence to the noncommutative case in the framework of real analysis. Khalkhali noncommutative bundles and instantons in tehran g. In derived higher noncommutative algebraic geometry one instead. Aug 07, 2018 derived noncommutative schemes, geometric realizations, and finite dimensional algebras.
The institute is located at 17 gauss way, on the university of california, berkeley campus, close to. The existence of many natural spaces for which the classical settheoretic tools. Noncommutative algebraic geometry uw math department. I will discuss some recent progress on this question, in the form of an explicit description of the possible calabiyau deformations of the polynomial ring. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. As a consequence, in derived algebraic geometry the full cotangent complex and not. Derived noncommutative geometry is a subject related to the commutative derived algebraic geometry of c. Geometricity for derived categories of algebraic stacks. Pdf noncommutative geometry download ebook for free.
Though it is not clear to me at all how much information is loss by applying these functors some information is, for sure. Marcolli renormalization of noncommutative quantum field theory h. In noncommutative algebraic geometry one represents a scheme by an abelian category of quasicoherent sheaves on the scheme, and looks at more general abelian categories as being categories of quasicoherent sheaves on a noncommutative space in derived higher noncommutative algebraic geometry one instead considers the derived category of quasicoherent sheaves, or more precisely its. My primary research interests lie in the interactions of complexalgebraic geometry with lie theory and representation theory in the spirit of noncommutative geometry, derived algebraic geometry and mathematical physics. Moduli problems in derived noncommutative geometry. Moduli problems in derived noncommutative geometry abstract we study moduli spaces of boundary conditions in 2d topological field theories. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization. Hence it is a generalization of ordinary algebraic geometry where instead of commutative rings, derived schemes are locally modelled on simplicial commutative rings. Let xbe a projective algebraic variety over c, for example x pn c. Noncommutative algebraic geometry amnon yekutieli bgu duality 2 27. A noncommutative algebra is an associative algebra in which the multiplication is not. Bertrand toen, derived algebraic geometry, arxiv1401. Topics in noncommutative geometry, buenos aires, july 26 to august 6, 2010. Algebraic geometry, noncommutative algebra and derived.
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