Noncommutative algebraic geometry

<h2 id="history">History</h2>
<h3>Classical approach: the issue of non-commutative localization</h3>
<p>Commutative algebraic geometry begins by constructing the <a href="/facts/Spectrum_of_a_ring/2k2nDE5w">spectrum of a ring</a>. The points of the algebraic variety (or more generally, <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a>) are the prime ideals of the ring, and the functions on the algebraic variety are the elements of the ring. A noncommutative ring, however, may not have any proper non-zero two-sided prime ideals. For instance, this is true of the <a href="/facts/Weyl_algebra/PADSD0k9">Weyl algebra</a> of polynomial differential operators on affine space: The Weyl algebra is a <a href="/facts/Simple_ring/5ndCExsb">simple ring</a>. Therefore, one can for instance attempt to replace a prime spectrum by a <a href="/facts/Primitive_spectrum/V7hTJve2">primitive spectrum</a>: there are also the theory of <a href="/facts/Non-commutative_localization/3jmll7q7">non-commutative localization</a> as well as <a href="/facts/Descent_theory/YTMNz2vH">descent theory</a>. This works to some extent: for instance, <a href="/facts/Dixmier/eIgzkHvX">Dixmier</a>'s <i>enveloping algebras</i> may be thought of as working out non-commutative algebraic geometry for the primitive spectrum of an enveloping algebra of a <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebra</a>. Another work in a similar spirit is <a href="/facts/Michael_Artin/Tfk0f6rK">Michael Artin</a>’s notes titled “noncommutative rings”,<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> which in part is an attempt to study <a href="/facts/Representation_theory/3ydLtaGH">representation theory</a> from a non-commutative-geometry point of view. The key insight to both approaches is that <a href="/facts/Irreducible_representations/Hu5geBgk">irreducible representations</a>, or at least <a href="/facts/Primitive_ideal/V7hTJve2">primitive ideals</a>, can be thought of as “non-commutative points”.
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<h3>Modern viewpoint using categories of sheaves</h3>
<p>As it turned out, starting from, say, primitive spectra, it was not easy to develop a workable <a href="/facts/Sheaf_theory/MM3hcRw4">sheaf theory</a>. One might imagine this difficulty is because of a sort of quantum phenomenon: points in a space can influence points far away (and in fact, it is not appropriate to treat points individually and view a space as a mere collection of the points).
</p><p>Due to the above, one accepts a paradigm implicit in <a href="/facts/Pierre_Gabriel/g95YjYqI">Pierre Gabriel</a>'s thesis and partly justified by the <a href="/facts/Gabriel%E2%80%93Rosenberg_reconstruction_theorem/R0R2cxC8">Gabriel–Rosenberg reconstruction theorem</a> (after <a href="/facts/Pierre_Gabriel/g95YjYqI">Pierre Gabriel</a> and <a href="/facts/Alexander_L._Rosenberg/enf1uQGS">Alexander L. Rosenberg</a>) that a commutative scheme can be reconstructed, up to isomorphism of schemes, solely from the <a href="/facts/Abelian_category/4WN473Yc">abelian category</a> of <a href="/facts/Quasicoherent_sheaf/h2NoUN7e">quasicoherent sheaves</a> on the scheme. <a href="/facts/Alexander_Grothendieck/jLQGPY7M">Alexander Grothendieck</a> taught that to do geometry one does not need a space, it is enough to have a category of sheaves on that would be space; this idea has been transmitted to noncommutative algebra by <a href="/facts/Yuri_Manin/gDlvYGCq">Yuri Manin</a>. There are, a bit weaker, reconstruction theorems from the derived categories of (quasi)coherent sheaves motivating the derived noncommutative algebraic geometry (see just below).
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<h3>Derived algebraic geometry</h3>
<p class="note">Main article: <a href="/facts/Derived_algebraic_geometry/FxXzQExB">Derived algebraic geometry</a></p>
<p>Perhaps the most recent approach is through the <a href="/facts/Deformation_theory/tVLS3kkG">deformation theory</a>, placing non-commutative algebraic geometry in the realm of <a href="/facts/Derived_algebraic_geometry/FxXzQExB">derived algebraic geometry</a>.
</p><p>As a motivating example, consider the one-dimensional <a href="/facts/Weyl_algebra/PADSD0k9">Weyl algebra</a> over the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> C. This is the quotient of the free ring C<<i>x</i>, <i>y</i>> by the relation
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<i>xy</i> - <i>yx</i> = 1.
<p>This ring represents the polynomial differential operators in a single variable <i>x</i>; <i>y</i> stands in for the differential operator ∂<i>x</i>. This ring fits into a one-parameter family given by the relations <i>xy</i> - <i>yx</i> = α. When α is not zero, then this relation determines a ring isomorphic to the Weyl algebra. When α is zero, however, the relation is the commutativity relation for <i>x</i> and <i>y</i>, and the resulting quotient ring is the polynomial ring in two variables, C[<i>x</i>, <i>y</i>]. Geometrically, the polynomial ring in two variables represents the two-dimensional <a href="/facts/Affine_space/tlvI3oX1">affine space</a> A2, so the existence of this one-parameter family says that <i>affine space admits non-commutative deformations to the space determined by the Weyl algebra.</i> This deformation is related to the <a href="/facts/Symbol_of_a_differential_operator/E80ugWJb">symbol of a differential operator</a> and that A2 is the <a href="/facts/Cotangent_bundle/dewBXXxr">cotangent bundle</a> of the affine line. (Studying the Weyl algebra can lead to information about affine space: The <a href="/facts/Dixmier_conjecture/PL1q1AxU">Dixmier conjecture</a> about the Weyl algebra is equivalent to the <a href="/facts/Jacobian_conjecture/J2NbRNk4">Jacobian conjecture</a> about affine space.)
</p><p>In this line of the approach, the notion of <i><a href="/facts/Operad/RfUyH38R">operad</a></i>, a set or space of operations, becomes prominent: in the introduction to (Francis 2008), Francis writes:
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<p>We begin the study of certain <i>less</i> commutative algebraic geometries. … algebraic geometry over <a href="/facts/En-ring/C1BeC4BR">
  
    
      
        
          
            
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-rings</a> can be thought of as interpolating between some derived theories of noncommutative and commutative algebraic geometries. As <i>n</i> increases, these 
  
    
      
        
          
            
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-algebras converge to the <a href="/facts/Derived_algebraic_geometry/FxXzQExB">derived algebraic geometry</a> of Toën-Vezzosi and <a href="/facts/Jacob_Lurie/QUNLubLt">Lurie</a>.</p>
<h2 id="proj-of-a-noncommutative-ring">Proj of a noncommutative ring</h2>
<p class="note">Main article: <a href="/facts/Noncommutative_projective_geometry/Bh7cbKIv">Noncommutative projective geometry</a></p>
<p>One of the basic constructions in commutative algebraic geometry is the <a href="/facts/Proj_construction/gcvPmjpU">Proj construction</a> of a <a href="/facts/Graded_commutative_ring/AIUjkSaP">graded commutative ring</a>. This construction builds a <a href="/facts/Projective_algebraic_variety/aB2XdB7B">projective algebraic variety</a> together with a <a href="/facts/Very_ample_line_bundle/XRQKvXGD">very ample line bundle</a> whose <a href="/facts/Homogeneous_coordinate_ring/6Av7AlsQ">homogeneous coordinate ring</a> is the original ring. Building the underlying topological space of the variety requires localizing the ring, but building sheaves on that space does not. By a theorem of <a href="/facts/Jean-Pierre_Serre/1xSr7BCh">Jean-Pierre Serre</a>, quasi-coherent sheaves on Proj of a graded ring are the same as graded modules over the ring up to finite dimensional factors. The philosophy of <a href="/facts/Topos_theory/zzttCIwC">topos theory</a> promoted by <a href="/facts/Alexander_Grothendieck/jLQGPY7M">Alexander Grothendieck</a> says that the category of sheaves on a space can serve as the space itself.  Consequently, in non-commutative algebraic geometry one often defines Proj in the following fashion: Let <i>R</i> be a graded C-algebra, and let Mod-<i>R</i> denote the category of graded right <i>R</i>-modules. Let <i>F</i> denote the subcategory of Mod-<i>R</i> consisting of all modules of finite length. Proj <i>R</i> is defined to be the quotient of the abelian category Mod-<i>R</i> by <i>F</i>. Equivalently, it is a localization of Mod-<i>R</i> in which two modules become isomorphic if, after taking their direct sums with appropriately chosen objects of <i>F</i>, they are isomorphic in Mod-<i>R</i>.
</p><p>This approach leads to a theory of <a href="/facts/Non-commutative_projective_geometry/Bh7cbKIv">non-commutative projective geometry</a>. A non-commutative smooth projective curve turns out to be a smooth commutative curve, but for singular curves or smooth higher-dimensional spaces, the non-commutative setting allows new objects.
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<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Derived_noncommutative_algebraic_geometry/LJ6Cp9MY">Derived noncommutative algebraic geometry</a></li>
<li><a href="/facts/Q-category/pQuXG91o">Q-category</a></li>
<li>quasi-free algebra</li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Michael_Artin/Tfk0f6rK">M. Artin</a>, J. J. Zhang, Noncommutative projective schemes, <a href="/facts/Advances_in_Mathematics/kXtcl4pM">Advances in Mathematics</a> 109 (1994), no. 2, 228–287, <a href="https://dx.doi.org/10.1006/aima.1994.1087">doi</a>.</li>
<li>Yuri I. Manin, Quantum groups and non-commutative geometry, CRM, Montreal 1988.</li>
<li>Yuri I Manin, Topics in noncommutative geometry, 176 pp. Princeton 1991.</li>
<li>A. Bondal, M. van den Bergh, Generators and representability of functors in commutative and noncommutative geometry, Moscow Mathematical Journal 3 (2003), no. 1, 1–36.</li>
<li>A. Bondal, D. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, <a href="/facts/Compositio_Mathematica/Igc7zupB">Compositio Mathematica</a> 125 (2001), 327–344 <a href="https://dx.doi.org/10.1023/A:1002470302976">doi</a></li>
<li>Francis, John (2008), <a href="https://sites.math.northwestern.edu/~jnkf/writ/thezrev.pdf"><i>Derived algebraic geometry over 
  
    
      
        
          
            
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-rings</i></a> (PDF) (Ph.D. thesis), Massachusetts Institute of Technology, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2717524">2717524</a>, <a href="/facts/ProQuest/yoBIhGBy">ProQuest</a> <a href="https://www.proquest.com/docview/304382161">304382161</a></li>
<li>O. A. Laudal, Noncommutative algebraic geometry, Rev. Mat. Iberoamericana 19, n. 2 (2003), 509--580; <a href="http://projecteuclid.org/euclid.rmi/1063050166">euclid</a>.</li>
<li><a href="/facts/Fred_Van_Oystaeyen/rplN7Iqk">Fred Van Oystaeyen</a>, Alain Verschoren, Non-commutative algebraic geometry, Springer Lect. Notes in Math. 887, 1981.</li>
<li>Fred van Oystaeyen, Algebraic geometry for associative algebras, Marcel Dekker 2000. vi+287 pp.</li>
<li>A. L.  Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, MIA 330, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-7923-3575-9</li>
<li>M. Kontsevich, A. Rosenberg, Noncommutative smooth spaces,  The Gelfand Mathematical Seminars, 1996--1999,  85--108, Gelfand Math. Sem., Birkhäuser, Boston 2000; <a href="https://arxiv.org/abs/math/9812158">arXiv:math/9812158</a></li>
<li>A. L. Rosenberg, Noncommutative schemes, <a href="/facts/Compositio_Mathematica/Igc7zupB">Compositio Mathematica</a> 112 (1998) 93--125, <a href="https://dx.doi.org/10.1023/A:1000479824211">doi</a>; Underlying spaces of noncommutative schemes, preprint MPIM2003-111, <a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=1947">dvi</a>, <a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=1948">ps</a>; <a href="/facts/Mathematical_Sciences_Research_Institute/Kvc4tGVl">MSRI</a> lecture <i>Noncommutative schemes and spaces</i> (Feb 2000): <a href="http://www.msri.org/publications/ln/msri/2000/interact/rosenberg/1/index.html">video</a></li>
<li>Pierre Gabriel, Des catégories abéliennes, Bulletin de la Société Mathématique de France 90 (1962), p. 323-448, <a href="http://www.numdam.org/item?id=BSMF_1962__90__323_0">numdam</a></li>
<li>Zoran Škoda, Some equivariant constructions in noncommutative algebraic geometry, Georgian Mathematical Journal 16 (2009), No. 1, 183--202, <a href="https://arxiv.org/abs/0811.4770">arXiv:0811.4770</a>.</li>
<li>Dmitri Orlov, Quasi-coherent sheaves in commutative and non-commutative geometry, Izv. RAN. Ser. Mat., 2003,  vol. 67, issue 3, 119–138 (MPI preprint version <a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=57">dvi</a>, <a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=56">ps</a>)</li>
<li>M. Kapranov, Noncommutative geometry based on commutator expansions, <a href="/facts/Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik/YCcRSaAQ">Journal für die reine und angewandte Mathematik</a> 505 (1998), 73-118, <a href="https://arxiv.org/abs/math/9802041">math.AG/9802041</a>.</li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li>A. Bondal, D. Orlov, Semi-orthogonal decomposition for algebraic varieties_, PreprintMPI/95–15, <a href="https://arxiv.org/abs/alg-geom/9506012">alg-geom/9506006</a></li>
<li>Tomasz Maszczyk, Noncommutative geometry through monoidal categories, <a href="https://arxiv.org/abs/math/0611806">math.QA/0611806</a></li>
<li>S. Mahanta, On some approaches towards non-commutative algebraic geometry, <a href="https://arxiv.org/abs/math/0501166">math.QA/0501166</a></li>
<li>Ludmil Katzarkov, <a href="/facts/Maxim_Kontsevich/7No1RaTQ">Maxim Kontsevich</a>, Tony Pantev, Hodge theoretic aspects of mirror symmetry, <a href="https://arxiv.org/abs/0806.0107">arxiv/0806.0107</a></li>
<li>Dmitri Kaledin, Tokyo lectures "Homological methods in non-commutative geometry", <a href="http://imperium.lenin.ru/~kaledin/tokyo/final.pdf">pdf</a>, <a href="http://imperium.lenin.ru/~kaledin/tokyo/final.tex">TeX</a>; and (similar but different) <a href="http://imperium.lenin.ru/~kaledin/seoul">Seoul lectures</a></li></ul>
<h2 id="external-links">External links</h2>
<ul><li>MathOverflow, <a href="https://mathoverflow.net/q/10512">Theories of Noncommutative Geometry</a></li>
<li><a href="https://ncatlab.org/nlab/show/noncommutative+algebraic+geometry">noncommutative algebraic geometry</a> at the <a href="/facts/NLab/lpPvRmgo"><i>n</i>Lab</a></li>
<li><a href="https://ncatlab.org/nlab/show/equivariant+noncommutative+algebraic+geometry">equivariant noncommutative algebraic geometry</a> at the <a href="/facts/NLab/lpPvRmgo"><i>n</i>Lab</a></li>
<li><a href="https://ncatlab.org/nlab/show/noncommutative+scheme">noncommutative scheme</a> at the <a href="/facts/NLab/lpPvRmgo"><i>n</i>Lab</a></li>
<li><a href="https://ncatlab.org/nlab/show/Kapranov%27s%20noncommutative%20geometry">Kapranov's noncommutative geometry</a> at the <a href="/facts/NLab/lpPvRmgo"><i>n</i>Lab</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>M. Artin, noncommutative rings <a href="http://www-math.mit.edu/~etingof/artinnotes.pdf" target="_blank">http://www-math.mit.edu/~etingof/artinnotes.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Noncommutative algebraic geometry open-in-new

Noncommutative algebraic geometry