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<title>Noncommutative algebraic geometry</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Noncommutative algebraic geometry</span></span>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist" style="width: 20.5em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><span style="font-size: 8pt; font-weight: none"><a href="Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → Ring theory</span><br><a href="Ring_theory" title="Ring theory">Ring theory</a></th></tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Basic concepts</div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><b><a href="Ring_(mathematics)" title="Ring (mathematics)">Rings</a></b>

<dl><dd>• <a href="Subring" title="Subring">Subrings</a></dd>

<dd>• <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">Ideal</a></dd>

<dd>• <a href="Quotient_ring" title="Quotient ring">Quotient ring</a>

<dl><dd>• <a href="Fractional_ideal" title="Fractional ideal">Fractional ideal</a></dd>

<dd>• <a href="Total_ring_of_fractions" title="Total ring of fractions">Total ring of fractions</a></dd></dl></dd>

<dd>• <a href="Product_of_rings" title="Product of rings">Product of rings</a></dd>

<dd>• <a href="Free_product_of_associative_algebras" title="Free product of associative algebras">Free product of associative algebras</a></dd>

<dd>• <a href="Tensor_product_of_algebras" title="Tensor product of algebras">Tensor product of algebras</a></dd></dl>

<p><b><a href="Ring_homomorphism" title="Ring homomorphism">Ring homomorphisms</a></b>

</p>

<dl><dd>• <a href="Kernel_(algebra)#Ring_homomorphisms" title="Kernel (algebra)">Kernel</a></dd>

<dd>• <a href="Inner_automorphism#Ring_case" title="Inner automorphism">Inner automorphism</a></dd>

<dd>• <a href="Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius endomorphism</a></dd></dl>

<p><b><a href="Algebraic_structure" title="Algebraic structure">Algebraic structures</a></b>

</p>

<dl><dd>• <a href="Module_(mathematics)" title="Module (mathematics)">Module</a></dd>

<dd>• <a href="Associative_algebra" title="Associative algebra">Associative algebra</a></dd>

<dd>• <a href="Graded_ring" title="Graded ring">Graded ring</a></dd>

<dd>• <a href="Involutive_ring" class="mw-redirect" title="Involutive ring">Involutive ring</a></dd>

<dd>• <a href="Category_of_rings" title="Category of rings">Category of rings</a>

<dl><dd>• <a href="Integer" title="Integer">Initial ring</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>

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</math></span><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"></span></dd>

<dd>• <a href="Zero_ring" title="Zero ring">Terminal ring</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }">

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</mrow>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }</annotation>

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</math></span><img src="./e45ab495cb8cfbac68a9322af662c3d6c7dbe494.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.686ex; height:2.843ex;" alt="{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }" loading="lazy"></span></dd></dl></dd></dl>

<p><b>Related structures</b>

</p>

<dl><dd>• <a href="Field_(mathematics)" title="Field (mathematics)">Field</a>

<dl><dd>• <a href="Finite_field" title="Finite field">Finite field</a></dd></dl></dd>

<dd>• <a href="Non-associative_ring" class="mw-redirect" title="Non-associative ring">Non-associative ring</a>

<dd>• <a href="Jordan_ring" class="mw-redirect" title="Jordan ring">Jordan ring</a></dd></dl></dd>

<dd>• <a href="Semiring" title="Semiring">Semiring</a>

<dl><dd>• <a href="Semifield" title="Semifield">Semifield</a></dd></dl></dd></dl></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="Commutative_algebra" title="Commutative algebra">Commutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><b><a href="Commutative_ring" title="Commutative ring">Commutative rings</a></b>

<dl><dd>• <a href="Integral_domain" title="Integral domain">Integral domain</a>

<dl><dd>• <a href="Integrally_closed_domain" title="Integrally closed domain">Integrally closed domain</a></dd>

<dd>• <a href="GCD_domain" title="GCD domain">GCD domain</a></dd>

<dd>• <a href="Unique_factorization_domain" title="Unique factorization domain">Unique factorization domain</a></dd>

<dd>• <a href="Principal_ideal_domain" title="Principal ideal domain">Principal ideal domain</a></dd>

<dd>• <a href="Euclidean_domain" title="Euclidean domain">Euclidean domain</a></dd>

<dd>• <a href="Field_(mathematics)" title="Field (mathematics)">Field</a>

<dl><dd>• <a href="Finite_field" title="Finite field">Finite field</a></dd></dl></dd>

<dd>• <a href="Polynomial_ring" title="Polynomial ring">Polynomial ring</a></dd>

<dd>• <a href="Formal_power_series_ring" class="mw-redirect" title="Formal power series ring">Formal power series ring</a></dd></dl></dd></dl>

<p><b><a href="Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></b>

</p>

<dl><dd>• <a href="Algebraic_number_field" title="Algebraic number field">Algebraic number field</a></dd>

<dd>• <a href="Integers_modulo_n" class="mw-redirect" title="Integers modulo n">Integers modulo <span class="texhtml mvar" style="font-style:italic;">n</span></a></dd>

<dd>• <a href="Ring_of_integers" title="Ring of integers">Ring of integers</a></dd>

<dd>• <a href="P-adic_integer" class="mw-redirect" title="P-adic integer"><i>p</i>-adic integers</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation>

</semantics>

</math></span><img src="./dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}" loading="lazy"></span></dd>

<dd>• <a href="P-adic_number" title="P-adic number"><i>p</i>-adic numbers</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation>

</semantics>

</math></span><img src="./35f44bc6894c682710705f3ea74f33042e0acc3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}" loading="lazy"></span></dd>

<dd>• <a href="Pr%C3%BCfer_group#The_Prüfer_group_as_a_ring" title="Prüfer group">Prüfer <i>p</i>-ring</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} (p^{\infty })}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} (p^{\infty })}</annotation>

</semantics>

</math></span><img src="./14af623e08c241266c125ad927dd35086ec8ce90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.404ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} (p^{\infty })}" loading="lazy"></span></dd></dl></div></div></td>

</tr><tr><td class="sidebar-content">

<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="Noncommutative_algebra" class="mw-redirect" title="Noncommutative algebra">Noncommutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><b><a href="Noncommutative_ring" title="Noncommutative ring">Noncommutative rings</a></b>

<dl><dd>• <a href="Division_ring" title="Division ring">Division ring</a></dd>

<dd>• <a href="Semiprimitive_ring" title="Semiprimitive ring">Semiprimitive ring</a></dd>

<dd>• <a href="Simple_ring" title="Simple ring">Simple ring</a></dd>

<dd>• <a href="Commutator_(ring_theory)" class="mw-redirect" title="Commutator (ring theory)">Commutator</a></dd></dl>

</p><p><b><a href="Free_algebra" title="Free algebra">Free algebra</a></b>

</p><p><b><a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a></b>

</p>

<dl><dd>• <a href="Geometric_algebra" title="Geometric algebra">Geometric algebra</a></dd></dl>

<b><a href="Operator_algebra" title="Operator algebra">Operator algebra</a></b></div></div></td>

</tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">

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<p><b>Noncommutative algebraic geometry</b> is a branch of <a href="Mathematics" title="Mathematics">mathematics</a>, and more specifically a direction in <a href="Noncommutative_geometry" title="Noncommutative geometry">noncommutative geometry</a>, that studies the geometric properties of formal duals of <a href="Non-commutative" class="mw-redirect" title="Non-commutative">non-commutative</a> <a href="Algebra" title="Algebra">algebraic objects</a> such as <a href="Ring_(mathematics)" title="Ring (mathematics)">rings</a> as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative <a href="Stack_quotient" class="mw-redirect" title="Stack quotient">stack quotients</a>).

</p><p>For example, noncommutative algebraic geometry is supposed to extend a notion of an <a href="Scheme_(mathematics)" title="Scheme (mathematics)">algebraic scheme</a> by suitable gluing of spectra of noncommutative rings; depending on how literally and how generally this aim (and a notion of spectrum) is understood in noncommutative setting, this has been achieved in various level of success. The noncommutative ring generalizes here a commutative <a href="Ring_of_regular_functions" class="mw-redirect" title="Ring of regular functions">ring of regular functions</a> on a <a href="Scheme_(mathematics)" title="Scheme (mathematics)">commutative scheme</a>. Functions on usual spaces in the traditional (commutative) <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> have a product defined by <a href="Pointwise_multiplication" class="mw-redirect" title="Pointwise multiplication">pointwise multiplication</a>; as the values of these functions <a href="Commutative_property" title="Commutative property">commute</a>, the functions also commute: <i>a</i> times <i>b</i> equals <i>b</i> times <i>a</i>. It is remarkable that viewing noncommutative associative algebras as algebras of functions on "noncommutative" would-be space is a far-reaching geometric intuition, though it formally looks like a fallacy.

</p><p>Much of the motivation for noncommutative geometry, and in particular for the noncommutative algebraic geometry, is from physics; especially from quantum physics, where the <a href="Observables" class="mw-redirect" title="Observables">algebras of observables</a> are indeed viewed as noncommutative analogues of functions, hence having the ability to observe their geometric aspects is desirable.

</p><p>One of the values of the field is that it also provides new techniques to study objects in commutative algebraic geometry such as <a href="Brauer_group" title="Brauer group">Brauer groups</a>.

</p><p>The methods of noncommutative algebraic geometry are analogs of the methods of commutative algebraic geometry, but frequently the foundations are different. Local behavior in commutative algebraic geometry is captured by <a href="Commutative_algebra" title="Commutative algebra">commutative algebra</a> and especially the study of <a href="Local_ring" title="Local ring">local rings</a>. These do not have a ring-theoretic analogue in the noncommutative setting; though in a categorical setup one can talk about <a href="Stack_(mathematics)" title="Stack (mathematics)">stacks</a> of local categories of <a href="Quasicoherent_sheaf" class="mw-redirect" title="Quasicoherent sheaf">quasicoherent sheaves</a> over noncommutative spectra. Global properties such as those arising from <a href="Homological_algebra" title="Homological algebra">homological algebra</a> and <a href="K-theory" title="K-theory">K-theory</a> more frequently carry over to the noncommutative setting.

</p>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Classical_approach:_the_issue_of_non-commutative_localization">Classical approach: the issue of non-commutative localization</h3></div>

<p>Commutative algebraic geometry begins by constructing the <a href="Spectrum_of_a_ring" title="Spectrum of a ring">spectrum of a ring</a>. The points of the algebraic variety (or more generally, <a href="Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a>) are the <a href="Prime_ideal" title="Prime ideal">prime ideals</a> 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="Weyl_algebra" title="Weyl algebra">Weyl algebra</a> of polynomial <a href="Differential_operator" title="Differential operator">differential operators</a> on affine space: The Weyl algebra is a <a href="Simple_ring" title="Simple ring">simple ring</a>. Therefore, one can for instance attempt to replace a prime spectrum by a <a href="Primitive_spectrum" class="mw-redirect" title="Primitive spectrum">primitive spectrum</a>: there are also the theory of <a href="Non-commutative_localization" class="mw-redirect" title="Non-commutative localization">non-commutative localization</a> as well as <a href="Descent_theory" class="mw-redirect" title="Descent theory">descent theory</a>. This works to some extent: for instance, <a href="Dixmier" class="mw-redirect" title="Dixmier">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="Lie_algebra" title="Lie algebra">Lie algebra</a>. Another work in a similar spirit is <a href="Michael_Artin" title="Michael Artin">Michael Artin</a>’s notes titled “noncommutative rings”,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> which in part is an attempt to study <a href="Representation_theory" title="Representation theory">representation theory</a> from a non-commutative-geometry point of view. The key insight to both approaches is that <a href="Irreducible_representations" class="mw-redirect" title="Irreducible representations">irreducible representations</a>, or at least <a href="Primitive_ideal" title="Primitive ideal">primitive ideals</a>, can be thought of as “non-commutative points”.

</p>

<div class="mw-heading mw-heading3"><h3 id="Modern_viewpoint_using_categories_of_sheaves">Modern viewpoint using categories of sheaves</h3></div>

<p>As it turned out, starting from, say, primitive spectra, it was not easy to develop a workable <a href="Sheaf_theory" class="mw-redirect" title="Sheaf theory">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="Pierre_Gabriel" title="Pierre Gabriel">Pierre Gabriel</a>'s thesis and partly justified by the <a href="Gabriel%E2%80%93Rosenberg_reconstruction_theorem" title="Gabriel–Rosenberg reconstruction theorem">Gabriel–Rosenberg reconstruction theorem</a> (after <a href="Pierre_Gabriel" title="Pierre Gabriel">Pierre Gabriel</a> and <a href="Alexander_L._Rosenberg" title="Alexander L. Rosenberg">Alexander L. Rosenberg</a>) that a commutative scheme can be reconstructed, up to isomorphism of schemes, solely from the <a href="Abelian_category" title="Abelian category">abelian category</a> of <a href="Quasicoherent_sheaf" class="mw-redirect" title="Quasicoherent sheaf">quasicoherent sheaves</a> on the scheme. <a href="Alexander_Grothendieck" title="Alexander Grothendieck">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="Yuri_Manin" title="Yuri Manin">Yuri Manin</a>. There are, a bit weaker, reconstruction theorems from the derived categories of (quasi)coherent sheaves motivating the <b>derived noncommutative algebraic geometry</b> (see just below).

</p>

<div class="mw-heading mw-heading3"><h3 id="Derived_algebraic_geometry">Derived algebraic geometry</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Derived_noncommutative_algebraic_geometry" title="Derived noncommutative algebraic geometry">Derived noncommutative algebraic geometry</a></div>

<p>Perhaps the most recent approach is through the <a href="Deformation_theory" class="mw-redirect" title="Deformation theory">deformation theory</a>, placing non-commutative algebraic geometry in the realm of <a href="Derived_algebraic_geometry" title="Derived algebraic geometry">derived algebraic geometry</a>.

</p><p>As a motivating example, consider the one-dimensional <a href="Weyl_algebra" title="Weyl algebra">Weyl algebra</a> over the <a href="Complex_number" title="Complex number">complex numbers</a> <b>C</b>. This is the quotient of the free ring <b>C</b><<i>x</i>, <i>y</i>> by the relation

</p>

<p>This ring represents the polynomial differential operators in a single variable <i>x</i>; <i>y</i> stands in for the differential operator ∂<sub><i>x</i></sub>. This ring fits into a one-parameter family given by the relations <span class="nowrap"><i>xy</i> - <i>yx</i> = α</span>. 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, <b>C</b>[<i>x</i>, <i>y</i>]. Geometrically, the polynomial ring in two variables represents the two-dimensional <a href="Affine_space" title="Affine space">affine space</a> <b>A</b><sup>2</sup>, 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="Symbol_of_a_differential_operator" class="mw-redirect" title="Symbol of a differential operator">symbol of a differential operator</a> and that <b>A</b><sup>2</sup> is the <a href="Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a> of the affine line. (Studying the Weyl algebra can lead to information about affine space: The <a href="Dixmier_conjecture" title="Dixmier conjecture">Dixmier conjecture</a> about the Weyl algebra is equivalent to the <a href="Jacobian_conjecture" title="Jacobian conjecture">Jacobian conjecture</a> about affine space.)

</p><p>In this line of the approach, the notion of <i><a href="Operad" title="Operad">operad</a></i>, a set or space of operations, becomes prominent: in the introduction to (<a href="#CITEREFFrancis2008">Francis 2008</a>), Francis writes:

</p>

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</style><blockquote class="templatequote rquote"><p>We begin the study of certain <i>less</i> commutative algebraic geometries. … algebraic geometry over <a href="En-ring" title="En-ring"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}_{n}}">

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</math></span><img src="./b51c26b17c2dd33bd88d7476e7c6e099bcccd416.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.446ex; height:2.509ex;" alt="{\displaystyle {\mathcal {E}}_{n}}" loading="lazy"></span>-rings</a> can be thought of as interpolating between some derived theories of noncommutative and commutative algebraic geometries. As <i>n</i> increases, these <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}_{n}}">

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<div class="mw-heading mw-heading2"><h2 id="Proj_of_a_noncommutative_ring">Proj of a noncommutative ring</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Noncommutative_projective_geometry" title="Noncommutative projective geometry">Noncommutative projective geometry</a></div>

<p>One of the basic constructions in commutative algebraic geometry is the <a href="Proj_construction" title="Proj construction">Proj construction</a> of a <a href="Graded_commutative_ring" class="mw-redirect" title="Graded commutative ring">graded commutative ring</a>. This construction builds a <a href="Projective_algebraic_variety" class="mw-redirect" title="Projective algebraic variety">projective algebraic variety</a> together with a <a href="Very_ample_line_bundle" class="mw-redirect" title="Very ample line bundle">very ample line bundle</a> whose <a href="Homogeneous_coordinate_ring" title="Homogeneous coordinate ring">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="Jean-Pierre_Serre" title="Jean-Pierre Serre">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="Topos_theory" class="mw-redirect" title="Topos theory">topos theory</a> promoted by <a href="Alexander_Grothendieck" title="Alexander Grothendieck">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 <b>C</b>-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 <a href="Localization_of_a_category" title="Localization of a category">localization</a> 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="Non-commutative_projective_geometry" class="mw-redirect" title="Non-commutative projective geometry">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.

</p>

<ul><li><a href="Q-category" title="Q-category">Q-category</a></li>

<li><a href="Quasi-free_algebra" title="Quasi-free algebra">quasi-free algebra</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">M. Artin, <a rel="nofollow" class="external text" href="http://www-math.mit.edu/~etingof/artinnotes.pdf">noncommutative rings</a></span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><a href="Michael_Artin" title="Michael Artin">M. Artin</a>, J. J. Zhang, Noncommutative projective schemes, <a href="Advances_in_Mathematics" title="Advances in Mathematics">Advances in Mathematics</a> 109 (1994), no. 2, 228–287, <a rel="nofollow" class="external text" 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="Compositio_Mathematica" title="Compositio Mathematica">Compositio Mathematica</a> 125 (2001), 327–344 <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1023/A:1002470302976">doi</a></li>

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</style><cite id="CITEREFFrancis2008" class="citation cs2">Francis, John (2008), <a rel="nofollow" class="external text" href="https://sites.math.northwestern.edu/~jnkf/writ/thezrev.pdf"><i>Derived algebraic geometry over <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {E}}_{n}}">

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</math></span><img src="./b51c26b17c2dd33bd88d7476e7c6e099bcccd416.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.446ex; height:2.509ex;" alt="{\displaystyle {\mathcal {E}}_{n}}" loading="lazy"></span>-rings</i></a> <span class="cs1-format">(PDF)</span> (Ph.D. thesis), Massachusetts Institute of Technology, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2717524">2717524</a>, <a href="ProQuest" title="ProQuest">ProQuest</a> <a rel="nofollow" class="external text" href="https://www.proquest.com/docview/304382161">304382161</a></cite></li>

<li>O. A. Laudal, Noncommutative algebraic geometry, Rev. Mat. Iberoamericana 19, n. 2 (2003), 509--580; <a rel="nofollow" class="external text" href="http://projecteuclid.org/euclid.rmi/1063050166">euclid</a>.</li>

<li><a href="Fred_Van_Oystaeyen" title="Fred Van Oystaeyen">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="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7923-3575-9</bdi></li>

<li>M. Kontsevich, A. Rosenberg, Noncommutative smooth spaces, The Gelfand Mathematical Seminars, 1996--1999, 85--108, Gelfand Math. Sem., Birkhäuser, Boston 2000; <a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9812158">arXiv:math/9812158</a></li>

<li>A. L. Rosenberg, Noncommutative schemes, <a href="Compositio_Mathematica" title="Compositio Mathematica">Compositio Mathematica</a> 112 (1998) 93--125, <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1023/A:1000479824211">doi</a>; Underlying spaces of noncommutative schemes, preprint MPIM2003-111, <a rel="nofollow" class="external text" href="http://www.mpim-bonn.mpg.de/preprints/send?bid=1947">dvi</a>, <a rel="nofollow" class="external text" href="http://www.mpim-bonn.mpg.de/preprints/send?bid=1948">ps</a>; <a href="Mathematical_Sciences_Research_Institute" class="mw-redirect" title="Mathematical Sciences Research Institute">MSRI</a> lecture <i>Noncommutative schemes and spaces</i> (Feb 2000): <a rel="nofollow" class="external text" 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 rel="nofollow" class="external text" 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 rel="nofollow" class="external text" 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 rel="nofollow" class="external text" href="http://www.mpim-bonn.mpg.de/preprints/send?bid=57">dvi</a>, <a rel="nofollow" class="external text" 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="Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik" class="mw-redirect" title="Journal für die reine und angewandte Mathematik">Journal für die reine und angewandte Mathematik</a> 505 (1998), 73-118, <a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/9802041">math.AG/9802041</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li>A. Bondal, D. Orlov, Semi-orthogonal decomposition for algebraic varieties_, PreprintMPI/95–15, <a rel="nofollow" class="external text" href="https://arxiv.org/abs/alg-geom/9506012">alg-geom/9506006</a></li>

<li>Tomasz Maszczyk, Noncommutative geometry through monoidal categories, <a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0611806">math.QA/0611806</a></li>

<li>S. Mahanta, On some approaches towards non-commutative algebraic geometry, <a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0501166">math.QA/0501166</a></li>

<li>Ludmil Katzarkov, <a href="Maxim_Kontsevich" title="Maxim Kontsevich">Maxim Kontsevich</a>, Tony Pantev, Hodge theoretic aspects of mirror symmetry, <a rel="nofollow" class="external text" href="https://arxiv.org/abs/0806.0107">arxiv/0806.0107</a></li>

<li>Dmitri Kaledin, Tokyo lectures "Homological methods in non-commutative geometry", <a rel="nofollow" class="external text" href="http://imperium.lenin.ru/~kaledin/tokyo/final.pdf">pdf</a>, <a rel="nofollow" class="external text" href="http://imperium.lenin.ru/~kaledin/tokyo/final.tex">TeX</a>; and (similar but different) <a rel="nofollow" class="external text" href="http://imperium.lenin.ru/~kaledin/seoul">Seoul lectures</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li>MathOverflow, <a rel="nofollow" class="external text" href="https://mathoverflow.net/q/10512">Theories of Noncommutative Geometry</a></li>

<li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/noncommutative+algebraic+geometry">noncommutative algebraic geometry</a> at the <a href="NLab" title="NLab"><i>n</i>Lab</a></li>

<li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/equivariant+noncommutative+algebraic+geometry">equivariant noncommutative algebraic geometry</a> at the <a href="NLab" title="NLab"><i>n</i>Lab</a></li>

<li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/noncommutative+scheme">noncommutative scheme</a> at the <a href="NLab" title="NLab"><i>n</i>Lab</a></li>

<li><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/Kapranov%27s%20noncommutative%20geometry">Kapranov's noncommutative geometry</a> at the <a href="NLab" title="NLab"><i>n</i>Lab</a></li></ul></div><div><div class="zim-footer">

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