Arnold-Kuiper-Massey theorem

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###Context###
#### Topology
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[[!include topology - contents]]
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#### Complex geometry
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[[!include complex geometry - contents]]
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=--

#Contents#
* table of contents
{:toc}

## Statements

+-- {: .num_prop}
###### Proposition
**(Arnold-Kuiper-Massey theorem)**

The [[4-sphere]] is the [[quotient space]] of the [[complex projective plane]] by the [[finite group of order 2|O(1)]]-[[action]] by [[complex conjugation]] (on homogeneous coordinates):

$$
  \mathbb{C}P^2 /  \mathrm{O}(1)
  \simeq
  S^4
$$

=--

([Arnold 71](#Arnold71), [Massey 73](#Massey73), [Kuiper 74](#Kuiper74), [Arnold 88](#Arnold88))

In fact, this is is the beginning of a small pattern indexed by the [[real normed division algebras]]:

+-- {: .num_prop}
###### Proposition

The [[7-sphere]] is the [[quotient space]] of the (right-)[[quaternionic projective plane]] by the left multiplication [[action]] by [[U(1)]] $\subset$ [[Sp(1)]]:

$$
  \mathbb{H}P^2 / \mathrm{U}(1)
  \simeq
  S^7
$$

=--

([Arnold 99](#Arnold99), [Atiyah-Witten 01, Sec. 5.5](#AtiyahWitten01))

+-- {: .num_prop}
###### Proposition

The [[13-sphere]] is the [[quotient space]] of the (right-)[[octonionic projective plane]] by the left multiplication [[action]] by [[Sp(1)]]:

$$
  \mathbb{O}P^2 / \mathrm{Sp}(1)
  \simeq
  S^{13}
$$

=--

([Atiyah-Berndt 02](#AtiyahBerndt02))

## References

### AKM-theorem for the complex projective plane

The original proof that the [[4-sphere]] is a quotient of the [[complex projective plane]] by an action of [[cyclic group of order 2|Z/2]]:

* {#Arnold71} [[Vladimir Arnold]], _On disposition of ovals of real plane algebraic curves, involutions of four-dimensional manifolds and arithmetics of integer quadratic forms_, Funct. Anal. and Its Appl., 1971, V. 5, N 3, P. 1-9.

* {#Massey73} [[William Massey]], _The quotient space of the complex projective space under conjugation is a 4-sphere_, Geometriae Didactica 1973 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/massey5.pdf))

* {#Kuiper74} [[Nicolaas Kuiper]], _The quotient space of $\mathbb{C}P(2)$ by complex conjugation is the 4-sphere_, Mathematische Annalen, 1974 ([doi:10.1007/BF01432386](https://doi.org/10.1007/BF01432386))

* {#Arnold88} [[Vladimir Arnold]], _Ramified covering $\mathbb{C}P^2  \to S^4$, hyperbolicity and projective topology_, Siberian Math. Journal 1988, V. 29, N 5, P.36-47

See also

* José Seade, Section V.5 in: _On the Topology of Isolated Singularities in Analytic Spaces_, Progress in Mathematics, Birkhauser 2006 ([ISBN:978-3-7643-7395-5](https://www.springer.com/gp/book/9783764373221))

* J. A. Hillman, _An explicit formula for a branched covering from $\mathbb{C}P^2$ to $S^4$_ ([arXiv:1705.05038](https://arxiv.org/abs/1705.05038))

The [[SO(3)]]-[[equivariant function|equivariant]] enhancement:

* Le, [[José Seade]], [[Alberto Verjovsky]], _Quadrics, orthogonal actions and involutions in complex projective space_, L'Enseignement Mathématique, t.  49 (2003) ([e-periodica:001:2003:49::488](https://www.e-periodica.ch/digbib/view?pid=ens-001:2003:49::488#488))

### Generalization to the quaternionic projective plane

The generalization to the [[7-sphere]] being a [[U(1)]]-quotient of the [[quaternionic projective plane]] is due to

* {#Arnold99} [[Vladimir Arnold]], _Relatives of the Quotient of the Complex Projective Plane by the Complex Conjugation_, Tr. Mat. Inst. Steklova, 1999, Volume 224, Pages 56–67; English translation: Proceedings of the Steklov Institute of Mathematics, 1999, 224, 46–56 ([mathnet:tm691](http://mi.mathnet.ru/eng/tm691))

and independently due to

* {#AtiyahWitten01} [[Michael Atiyah]], [[Edward Witten]], Section 5.5 of: _$M$-Theory dynamics on a manifold of $G_2$-holonomy_, Adv. Theor. Math. Phys. 6 (2001) ([arXiv:hep-th/0107177](http://arxiv.org/abs/hep-th/0107177), [doi:10.4310/ATMP.2002.v6.n1.a1]( https://dx.doi.org/10.4310/ATMP.2002.v6.n1.a1))

  > (in the context of [[M-theory on G2-manifolds]])

### Generalization to the octonionic projective plane

Another proof of these cases and further generalization to the [[13-sphere]] being an [[Sp(1)]]-quotient of the [[octonionic projective plane]]:

* {#AtiyahBerndt02} [[Michael Atiyah]], [[Jürgen Berndt]], *Projective planes, Severi varieties and spheres*, in:  Surv. Differ. Geom. VIII, Papers in Honor of Calabi, Lawson, Siu and Uhlenbeck (International Press, Somerville, MA, 2003) 1-27 ([arXiv:math/0206135](https://arxiv.org/abs/math/0206135), [doi:10.4310/SDG.2003.v8.n1.a1](https://dx.doi.org/10.4310/SDG.2003.v8.n1.a1))

[[!redirects AKM theorem]]
[[!redirects AKM-theorem]]


Aron C. Wall

* [personal page](http://www.wall.org/~aron/)

## Selected writings

Argument that the image of *[[wormhole]] traversal* under [[AdS-CFT duality]] is *[[quantum teleportation]]*:

* [[Ping Gao]], [[Daniel Louis Jafferis]], [[Aron C. Wall]], *Traversable Wormholes via a Double Trace Deformation*, Journal of High Energy Physics **2017** 151 (2017) [[arXiv:1608.05687](https://arxiv.org/abs/1608.05687), doi:10.1007/JHEP12(2017)151]

On [[holographic tensor networks]] and [[quantum error correcting codes]] in full [[AdS-CFT]]:

* [[Ning Bao]], [[Geoffrey Penington]], [[Jonathan Sorce]], [[Aron C. Wall]], _Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT_,  JHEP 2019:69 ([arXiv:1812.01171](https://arxiv.org/abs/1812.01171))

* [[Ning Bao]], [[Geoffrey Penington]], [[Jonathan Sorce]], [[Aron C. Wall]], _Holographic Tensor Networks in Full AdS/CFT_ ([arXiv:1902.10157](https://arxiv.org/abs/1902.10157))

category: people

[[!redirects Aron Wall]]


Aron J. Beekman

* [personal page](https://abeekman.nl/)

## Selected writings

On [[Abrikosov vortices]] in [[superconductors]] as [[strings]]:

* {#BeekmanZaanen11} [[Aron J. Beekman]], [[Jan Zaanen]], *Electrodynamics of Abrikosov vortices: the field theoretical formulation*, Front. Phys. **6** (2011) 357–369 [[doi:10.1007/s11467-011-0205-0](https://doi.org/10.1007/s11467-011-0205-0)]

category: people

[[!redirects Aron Beekman]]


Arpon Raksit

* [personal page](https://www.mit.edu/~arpon/)

## Selected writings

On the [[J-homomorphism]]:

* [[Arpon Raksit]], _Vector fields and the J-homomorphism_, 2014 ([pdf](http://stanford.edu/~arpon/math/files/vfields.pdf), [[Raksit_JHomomorphism.pdf:file]])

On [[transchromatic characters]] in [[global equivariant homotopy theory]]:

* {#Raksit15} [[Arpon Raksit]], _Characters in global equivariant homotopy theory_, 2015 ([pdf](https://www.math.harvard.edu/media/raksit.pdf), [[Raksit_Characters.pdf:file]])

category: people


arrangement of hyperplanes

#Contents#
* table of contents
{:toc}

## Idea

An arrangement of [[hyperplanes]] is a finite set of hyperplanes in a (finite-dimensional) linear, affine or projective space. Usually (such as for [[configuration spaces of points]]) one studies in fact the __[[complement]] of the union of the hyperplanes__, and its topological and other properties. This space is the basis of many interesting [[fiber bundles]] appearing in [[conformal field theory]], study of [[hypergeometric functions]] (Aomoto, Gelfand, Varchenko), [[quantum groups]] etc., for more on which see at
*[[Knizhnik-Zamolodchikov equation]]*.

## References

### General

* Wikipedia [arrangement of hyperplanes](http://en.wikipedia.org/wiki/Arrangement_of_hyperplanes)

* [[eom]]: [arrangement of hyperplanes](http://eom.springer.de/A/a110700.htm)
* Peter Orlik, Hiroaki Terao, _Arrangements of Hyperplanes_, Grundlehren der Mathematischen Wissenschaften __300__, Springer 1992, MR1217488

* [[I. M. Gelfand]], M. M. [[Kapranov]], [[A. Zelevinsky]], _Discriminants, resultants and multidimensional determinants_, Birkhäuser 1994, 523 pp.
* Corrado De Concini, Claudio Procesi, _Topics in hyperplane arrangements, polytopes and box-splines_, Universitext 223, Springer 2010.

Stanley's survey focuses instead more on combinatorics of the intersection poset of an arrangement (as well as arrangements in the case of vector space over a finite field):

* Richard P. Stanley, _An introduction to hyperplane arrangements_, in: Geometric combinatorics, 389–496, IAS/Park City Math. Ser., 13, Amer. Math. Soc. 2007, [pdf](http://www.math.umn.edu/~ezra/PCMI2004/stanley.jcp.pdf), [errata](http://www-math.mit.edu/~rstan/arrangements/errata.pdf)

* Richard Randell, _Morse theory, Milnor fibers and minimality of hyperplane arrangements_, [math.AG/0011101](http://arxiv.org/abs/math/0011101)

* [[Daniel C. Cohen]], Michael Falk, Richard Randell, _Discriminantal bundles, arrangement groups, and subdirect products of free groups_, [arxiv/1008.0417](http://arxiv.org/abs/1008.0417)

### Relating to configuration spaces of points

See the references on *[Braid representatioons via twisted de Rham cohomology of configuration spaces](Knizhnik-Zamolodchikov+equation#BraidRepresentationsViaTwisteddRCohomologyOfConfigurationSpaces)*

[[!redirects arrangements of hyperplanes]]
[[!redirects hyperplane arrangement]]


arrow

The term _arrow_ is sometimes used as a synonym for _[[morphism]]_, _[[map]]_, and also for directed edge (in a [[directed graph]] or [[quiver]]).

In [[computer science]] it may also refer to a concept generalizing [[monad (in computer science)|monads]], see at _[[arrow (in computer science)]]_.

Abbreviations for the class of all arrows of a category $\mathsf{C}$ used in the literature include $\mathrm{Arr}(\mathsf{C})$, $\mathrm{Ar}(\mathsf{C})$, and $\mathrm{Mor}(\mathsf{C})$.  Note that this is the class of [[objects]] of the [[arrow category]] of $\mathsf{C}$, and the same notations are sometimes used for that whole category.

category: disambiguation

[[!redirects arrows]]


arrow (in computer science)

#Contents#
* table of contents
{:toc}

## Idea

In [[computer science]], the notion of *arrows* [[Hughes 2000](#Hughes00)] generalizes that of _[[monads in computer science]]_.

More concretely, the notion of *arrow* abstracts that of the [[hom-functor|hom-]][[profunctor]] of the [[Kleisli category]] $\mathrm{Kl}(T)$ for a [[strong monad]] $T \colon \mathbf C \to \mathbf C$ as an endoprofunctor on $\mathbf C$.

Therefore, an arrow $A \colon \mathbf C^{op} \times \mathbf C \to \mathbf{Set}$ can be thought as a putative replacement of $\mathrm{Hom}_{\mathbf C} \colon \mathbf C^{op} \times \mathbf C \to \mathbf{Set}$. In fact an arrow comes equipped with a transformation $\mathrm{arr} : \mathrm{Hom}_{\mathbf C} \Rightarrow A$ that lift morphisms of $\mathbf C$ to the 'augmented' morphisms given by $A$, and with a transformation $\ggg : A \circ A \Rightarrow A$ that behaves like a composition operation. Then one requires that $\ggg$ is associative and that $\mathrm{arr}(1_a)$ is the unit of this operation.

These data makes $A$ a [[promonad]] on $\mathbf C$, i.e. a monad on $\mathbf C$ in the bicategory of [[profunctors]]. Since every monad on $\mathbf{Set}$ (or on any reasonable enriching base where programmers work) is strong, arrows traditionally generalize the hom-profunctor of [[strong monads]]. The additional requirement that $A$ is a [[strong profunctor]] (and the laws it is required to satisfy) imply that an arrow is a [[strong monad|strong monad]] in the bicategory of [[profunctors]] ([Asada10](#Asada10)).

## Definition
\begin{definition}
  An arrow $A$ on a [[monoidal category]] $\mathbf C$ is a monoid in the category of [[strong profunctor|strong endoprofunctors]] on $\mathbf C$, i.e. it's a functor $\mathbf C^{op} \times \mathbf C \to \mathbf{Set}$ equipped with the following structure:

1. A natural family of morphisms $\mathrm{arr} : \mathbf C(a,b) \to A(a,b)$ (the unit of the monoid),
2. A natural family of morphisms called *composition* $\ggg : A(a,b) \times A(b,c) \to A(a,c)$ (the multiplication of the monoid), which satisfy an associative law and for which $\mathrm{arr}_{a,a}(1_a)$ is the unit,
3. A family of morphisms called *strength* $s_{a,b,m} : A(a,b) \to A(m \otimes a, m \otimes b)$, which is [[dinatural transformation|dinatural]] in $m$ and natural in $a, b$, and satisfies coherence laws that make $(A, s)$ a [[strong profunctor]].

\end{definition}

## References

Arrows originate in Section 2 of

* {#Hughes00} [[John Hughes]], *Generalising monads to arrows*, Sci. Comput. Program. **37** 1-3 (2000) 67-111 [[doi:10.1016/S0167-6423(99)00023-4](https://doi.org/10.1016/S0167-6423%2899%2900023-4), [pdf](http://www.cse.chalmers.se/~rjmh/Papers/arrows.pdf)]

The 'justification' of arrows as [[strong monads]] in $\mathbf{Prof}$ is given in

* {#Asada10} Kazuyuki Asada, _Arrows are strong monads_, 2010 ([pdf](http://www.riec.tohoku.ac.jp/~asada/papers/arrStrMnd.pdf))

Comparisons of monads with [[applicative functors]] (also known as idioms) and with [[arrows (in computer science)]] are in

* [[Sam Lindley]], [[Philip Wadler]], [[Jeremy Yallop]], _Idioms are Oblivious, Arrows are Meticulous, Monads are Promiscuous_, Electron. Notes Theor. Comput. Sci. 229(5), pp. 97--117, 2011 ([doi:10.1016/j.entcs.2011.02.018](https://doi.org/10.1016/j.entcs.2011.02.018))
* [[Exequiel Rivas]], _Relating Idioms, Arrows and Monads from Monoidal Adjunctions_, MSFP@FSCD 2018, pp. 18--33 ([arXiv:1807.04084](https://arxiv.org/abs/1807.04084), [doi:10.4204/EPTCS.275.3](https://doi.org/10.4204/EPTCS.275.3))

[[!redirects arrows (in computer science)]]

[[!redirects arrows in computer science]]


arrow (∞,1)-category

#Contents#
* table of contents
{:toc}

## Defintion

For $\mathcal{C}$ an [[(∞,1)-category]], its **arrow $(\infty,1)$-category** is the [[(∞,1)-category of (∞,1)-functors]]

$$
  \mathcal{C}^{\Delta^1}
  \coloneqq
  Func_\infty(\Delta^1, \mathcal{C})
  \,.
$$

## Related concepts

* [[slice (∞,1)-category]]

* [[arrow category]]

* [[arrow (∞,1)-topos]]


arrow (∞,1)-topos

#Contents#
* table of contents
{:toc}

## Definition

For $\mathbf{H}$ an [[(∞,1)-topos]], its _arrow $(\infty,1)$-topos_ is its [[arrow (∞,1)-category]].

## Properties

For $(X_{def} \stackrel{\iota_X}{\to} X_{bulk})$ and $(A_{def} \stackrel{\mathbf{F}}{\to} A_{bulk})$ in $\mathbf{H}^{\Delta^1}$, there is an [[(∞,1)-pullback]] diagram

$$
  \array{
    \mathbf{H}^{\Delta^1}(\iota_X, \mathbf{F})
    &\stackrel{}{\to}&
    \mathbf{H}(X_{def}, A_{def})
    \\
    \downarrow &\swArrow& \downarrow
    \\
    \mathbf{H}(X_{bulk}, A_{bulk})
    &\to&
    \mathbf{H}(X_{def}, A_{bulk})
  }
  \,.
$$

Note: 'def' and 'bulk' indicate terms of fields in a [[QFT with defects]].

## Related concepts

* [[slice (∞,1)-topos]]

* [[relative cohomology]]

* [[Sierpinski topos]]


arrow category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

Every [[category]] $C$ gives rise to an _arrow category_ $Arr(C)$ such that the [[objects]] of $Arr(C)$ are the [[morphisms]] (or _arrows_, hence the name) of $C$.

## Definition

For $C$ any [[category]], its **arrow category** $Arr(C)$ is the category such that:

* an [[object]] $a$ of $Arr(C)$ is a [[morphism]] $a\colon a_0 \to a_1$ of $C$;
* a [[morphism]] $f\colon a \to b$ of $Arr(C)$ is a [[commutative square]]
  $$ \array {
     a_0 & \overset{f_0}\to & b_0 \\
     \llap{a}\downarrow & & \rlap{b}\downarrow \\
     a_1 & \underset{f_1}\to & b_1
     } $$
  in $C$;
* [[composition]] in $Arr(C)$ is given simply by placing commutative squares side by side to get a commutative oblong.

This is isomorphic to the [[functor category]]
$$
  Arr(C) := Funct(I,C) = [I,C] = C^I
$$
for $I$ the [[interval category]] $\{0 \to 1\}$.  $Arr(C)$ is also written $[\mathbf{2},C]$, $C^{\mathbf{2}}$, $[\Delta[1],C]$, or $C^{\Delta[1]}$, since $\mathbf{2}$ and $\Delta[1]$ (for the $1$-[[simplex]]) are common notations for the interval category.

## Properties

\begin{proposition}
The arrow category $Arr(C)$ is equivalently the [[comma category]] $(id/id)$ for the case that $id\colon C \to C$ is the [[identity functor]].
\end{proposition}
\begin{remark}
$Arr(C)$ plays the role of a [[directed homotopy theory|directed]] [[homotopy|path object]] for categories in that functors
$$
  X \to Arr(Y)
$$
are the same as [[natural transformations]] between functors between $X$ and $Y$.
\end{remark}

\begin{example}\label{SliceCategoriesAndArrowCategory}
**(arrow category is [[Grothendieck construction]] on [[slice categories]])**
\linebreak
For $\mathcal{S}$ any category, let
$$
  \mathcal{S}_{(-)}
  \,\colon\,
  \mathcal{S}
  \longrightarrow
  Cat
$$
be the [[pseudofunctor]] which sends

* an [[object]] $B \,\in\, \mathcal{S}$ to the [[slice category]] $\mathcal{S}_{/B}$,

* a [[morphism]] $f \colon B \to B'$ to the left [[base change]] [[functor]] $f_! \,\colon\, \mathcal{C}_{B} \to \mathcal{C}_{/B'}$ given by post-[[composition]] in $\mathcal{C}$.

The [[Grothendieck construction]] on this functor is the [[arrow category]] $Arr(\mathcal{S})$ of $\mathcal{S}$:

$$
  Arr(\mathcal{S})
  \;\;\;
    \simeq
  \;\;\;
  \int_{B \in \mathcal{S}}
  \mathcal{S}_{/B}
  \mathrlap{\,.}
$$

This follows readily by unwinding the definitions.
In the refinement to the [[Grothendieck construction for model categories]] (here: [[slice model categories]] and [[model structures on functors]]) this equivalence is also considered for instance in [Harpaz & Prasma (2015), above Cor. 6.1.2](Grothendieck+construction+for+model+categories#HarpazPrasma15).

The correponding [[Grothendieck fibration]] is also known as the *[[codomain fibration]]*.
\end{example}

## Related concepts

* [[path space object]]

* [[slice category]], [[undercategory]]

* [[arrow (∞,1)-category]]

* [[arrow (∞,1)-topos]]

* [[twisted arrow category]]

[[!redirects arrow category]]
[[!redirects arrow categories]]
[[!redirects category of morphisms]]
[[!redirects categories of morphisms]]


arrow structure

# Arrow frames, arrow languages, arrow models and arrow logics
* automatic table of contents goes here
{:toc}

## Idea

These structures are intended to allow one to talk about 'all such objects
as may be represented in a picture by arrows', (Venema).

## Arrow frames

+-- {: .un_defn}
###### Definition

An **arrow frame** is a [[frame (modal logic)|frame]], $\mathfrak{F} = (W, C, R, I )$, such that $C$ is a ternary relation, (so $C\subseteq
W \times W \times W$), $R$ is a binary relation, and $I$ a unary one, so $I\subseteq W$.
=--

## Gloss

The interpretation intended for these relations is similar to the structure of a [[groupoid]], but from a relational point of view.

*  For $C(a,b,c)$ think $a = b\star c$, 'the' _composite_ of $b$ and $c$, but, of course, $a$ need not be uniquely defined by this;

* For $R(a,b)$, think $b$ is a 'reverse' arrow for $a$;

* For $I(a)$ i.e. $a\in I$, think $a$ is an 'identity arrow'.

This defines a frame by a relational signature.  It does not specify any formulae to be satisfied, nor is there a [[geometric models for modal logics|valuation]] around  to give some 'meaning' to the structure.

## Arrow languages

The arrow language follows the general form of the [[modal logic|modal languages]], but note that there are several 'arities' of relation in the structure, so we need several different types of modal operators.

+-- {: .un_defn}
###### Definition

The **arrow language** is defined by the rule

$$\phi ::= p \mid \bot \mid \neg \phi \mid \phi_1 \vee \phi_2 \mid \phi\circ \psi \mid \otimes\phi\mid 1',$$

where the $p$ are the propositional variables, as usual, and in addition

* $1'$ is 'identity' and is a nullary modality or modal constant;

* $\otimes$ (an unfortunate notation, but which seems fairly standard in the sources used for this) is the 'converse' operator and is a 'diamond', i.e. a unary operator, and

*  $\circ$ is the 'composition operator' and is a dyadic operator.
=--

The possible interpretations or readings of these include

$1'$ is SKIP;

$\otimes \phi$ is $phi$  conversely; and

$\phi\circ \psi$ is 'first $\phi$ then $\psi$'.

## Arrow models

The usual rules for modal semantics apply.  An **arrow model**, $\mathfrak{M}$, consists of an arrow frame, $\mathfrak{F} = (W, C, R, I )$, (and a valuation $V$, which plays a somewhat behind the scenes role in this general situation).  In what follows, $a\in W$,

*  $\mathfrak{M},a \models 1'$ if and only if $Ia$;

*  $\mathfrak{M},a \models \otimes\phi$ if and only if $\mathfrak{M},b \models\phi$ for some $b$ with $R a b$;

*  $\mathfrak{M},a \models\phi\circ \psi$ if and only if $\mathfrak{M},b \models\phi$ and $\mathfrak{M},c \models\psi$ for some $b,c$ with $C a b c$.

## References

A general treatment of these ideas can be found in

* P. [[Blackburn]], M. de Rijke and Y. [[Venema]], _Modal Logic_, Cambridge Tracts in Theoretical Computer Science, vol. 53, 2001,

whilst a short introduction is

*  Y. [[Venema]], [_A crash course in Arrow Logic_](http://staff.science.uva.nl/~yde/papers/arrow.pdf),  in: M Marx, L Pólos and M Masuch (editors), Arrow Logic and Multi-Modal Logic,
Studies in Logic, Language and Information, CSLI Publications, Stanford (1996) 3--34.

[[!redirects arrow structure]]
[[!redirects arrow structures]]
[[!redirects arrow logic]]
[[!redirects arrow logics]]


arsmath

I'm Walt.  I blog at [Ars Mathematica](http://arsmathematica.net).

category:people


Arthur Bartels

* [website](http://wwwmath.uni-muenster.de/u/bartelsa/)

category: people


Arthur Besse

Arthur L. Besse is a collective of mathematicians who worked together closely in round-table meetings at various institutions in France, focused mainly on topics in [[differential geometry]]. His middle name is Lancelot, but this is never written. The collective has consisted of: Marcel Berger, Hermann Karcher, Jean-Pierre Bourguignon, Geneviève Averous, Nigel Hitchin, Jerry Kazdan, Pierre Pansu, Paul Gauduchon, Dennis DeTurck, Lionel Bérard-Bergery, Andrei Derdzinski, Josette Houillot, Norihito Koiso, Albert Polombo, John A. Thorpe, Jacques Lafontaine, Liane Valère. From the preface of (besse) we find the acknowledgement:

"Enfin, qu'il me soit permis de saluer ici mon prédécesseur et homonyme Jean Besse, de Zürich, qui s'est illustré dans la théorie des fonctions d'une variable complexe..." (Finally, here let me salute my predecessor and namesake Jean Besse of Zürich, who has illustrated himself in the theory of the functions of a complex variable...)

at which point the author(s) refer to the article:

Besse, Jean. _Sur le domaine d'existence d'une fonction analytique_, Commentarii Mathematici Helvetici 10 (1938).

## selected publications

* Arthur L. Besse {#besse}, _Einstein Manifolds_, Springer-Verlag 1987.

## related $n$Lab entries

* [[holonomy]], [[special holonomy]]
* [[curvature]]
* [[spectral theory]] of the [[Laplacian]]
* [[Alexandrov space]]
* [[Riemannian geometry]]
* [[subriemannian geometry]]
* [[Einstein manifolds]]
* [[quaternionic manifold]], [[hyperKähler manifold]], [[quaternionic-Kähler manifold]]

category: people


Arthur Cayley

_Arthur Cayley_ (16 August 1821 – 26 January 1895) is one of the principal and most prolific mathematicians of 19-th century, especially in [[algebra]]. He also worked on questions in [[algebraic geometry]], [[graph theory]], ...

* [Wikipedia entry](http://en.wikipedia.org/wiki/Arthur_Cayley)

## Selected writings

Introducing the [[octonions]]:

* {#Cayley1845} [[Arthur Cayley]], _On certain results relating to quaternions_,  The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science Series 3 Volume 26, 1845 - Issue 171 ([doi:10.1080/14786444508562684](https://doi.org/10.1080/14786444508562684))

Cayley's 1848 paper on elimination theory contains probably the first serious application of [[homological algebra]], including [[Koszul complexes]]. It is reproduced and commented in the book

* [[I. M. Gelfand]], [[Mikhail Kapranov|M. M. Kapranov]], A. Zelevinsky, _Discriminants, [[resultants]], and multidimensional determinants_, Birkhäuser 1994, 523 pp.

## Related $n$Lab entries

* [[resultant]]

* [[hyperdeterminant]]

* [[Cayley form]]

* [[weak Cayley table]]

* [[Cayley-Hamilton theorem]]

[[!redirects Cayley]]

category: people


Arthur Eddington

[[!redirects Eddington]]

* [Wikipedia entry](http://en.wikipedia.org/wiki/Arthur_Eddington)

category: people


Arthur Greenspoon

* [[Eric Sharpe]], [[Arthur Greenspoon]], _[Advances in String Theory: The First Sowers Workshop in Theoretical Physics](http://www.ams.org/bookstore-getitem/item=AMSIP-44)_, AMS 2008


Arthur Gretton

* [webpage](https://www.is.mpg.de/~bs)

## Selected writings

On [[Bochner's theorem]] generalized to [[non-abelian groups]] in the context of [[kernel methods]]:

* [[Kenji Fukumizu]], [[Bharath Sriperumbudur]], [[Arthur Gretton]], [[Bernhard Schölkopf]], *Characteristic Kernels on Groups and Semigroups*, Advances in Neural Information Processing Systems 21 : 22nd Annual Conference on Neural Information Processing Systems 2008 ([NIPS 2008](https://neurips.cc/Conferences/2008/)), 473-480 ([mpg:5466](http://www.is.mpg.de/publications/5466), [pdf](http://www.gatsby.ucl.ac.uk/~gretton/papers/FukSriGreSch09.pdf))

category: people


Arthur H. Stone

> Not to be confused with [[Arthur Lewis Stone]].

Arthur Harold Stone was a [[general topology|general topologist]].
He got his PhD degree in 1941 from Princeton University,
advised by [[Solomon Lefschetz]].

## Selected writings

On [[numerable covers]]:

* {#Stone} [[Arthur H. Stone]], _Paracompactness and product spaces_, Bulletin of the American Mathematical Society 54:10 (1948), 977–983 ([doi:10.1090/S0002-9904-1948-09118-2](https://doi.org/10.1090/S0002-9904-1948-09118-2))

## Related entries

* [[numerable cover]]

[[!redirects Arthur Harold Stone]]


Arthur Jaffe

* [Wikipedia entry](http://en.wikipedia.org/wiki/Arthur_Jaffe)

## Selected writings

On [[non-perturbative quantum field theory]]:

* [[Gerard ’t Hooft]], [[Arthur Jaffe]], [[Gerhard Mack]], [[P. K. Mitter]], [[Raymond Stora]] (eds.) *Nonperturbative Quantum Field Theory*,  NATO Science Series B **185** (1988) [[doi:10.1007/978-1-4613-0729-7](https://doi.org/10.1007/978-1-4613-0729-7)]

On the relation between the practice of [[mathematics]] and [[physics]] (for historical context see [Hitchin 20, Sec. 8](Michael+Atiyah#Hitchin20)):

* {#JaffeQuinn93} [[Arthur Jaffe]], [[Frank Quinn]], _"Theoretical Mathematics": Towards a cultural synthesis of mathematics and theoretical physics_, Bulletin of the AMS, Volume 29,Number 1, July 1993 ([arXiv:math/9307227](http://arxiv.org/abs/math/9307227))

  > we claim that the role of rigorous [[proof]] in mathematics is functionally [[analogy|analogous]] to the role of [[experiment]] in the natural sciences ([Jaffe-Quinn 93, p. 2](JaffeQuinn93))

On the [[mass gap]]-problem in [[Yang-Mills theory]]:

* {#JaffeWitten00} [[Arthur Jaffe]], [[Edward Witten]], _Quantum Yang-Mills theory_ (2000) [[[JaffeWitten-QuantumYangMills.pdf:file]]]

## Related entries

* [[constructive quantum field theory]]

category: people


Arthur L. Stone

> Not to be confused with [[Arthur Harold Stone]], a general topologist.

* [Mathematics Genealogy page](https://genealogy.math.ndsu.nodak.edu/id.php?id=12504)

## Selected writings

On [[adjunctions]], their induced ([[comonad|co]]-)[[monads]] and their decomposition, such as of [[idempotent adjunctions]]:

* {#MacDonaldStone82} [[John L. MacDonald]], [[Arthur L. Stone]],  *The tower and regular decomposition*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 23 (1982) no. 2, pp. 197-213 ([numdam:CTGDC_1982__23_2_197_0](http://www.numdam.org/item/?id=CTGDC_1982__23_2_197_0))

category: people

[[!redirects Arthur Stone]]
[[!redirects Arthur Lewis Stone]]


Arthur Ogus

* [webpage](http://math.berkeley.edu/~ogus/)

## related $n$Lab entries

* [[logarithmic geometry]]

category: people

[[!redirects Ogus]]


Arthur Parzygnat

* [webpage](https://arthurparzygnat.com/)

## Selected writings

On [[higher parallel transport]] for [[connections on a 2-bundle]]:

* {#Parzygnat18} [[Arthur Parzygnat]], _Two-dimensional algebra in lattice gauge theory_, Journal of Mathematical Physics 60, 043506 (2019) ([arXiv:1802.01139](https://arxiv.org/abs/1802.01139), [doi:10.1063/1.5078532](https://doi.org/10.1063/1.5078532))

## Related $n$Lab entries

* [[Gelfand-Naimark-Segal construction]]

* [[higher parallel transport]], [[connection on a 2-bundle]]

* [[tensor network]]

category: people

[[!redirects Arthur J. Parzygnat]]


Arthur Prior

A logician and philosopher, noted for his treatment of tense logic, a form of [[temporal logic]].

* [SEP entry](http://plato.stanford.edu/entries/prior/)


Arthur Sale

[[!redirects Arhtur Sale]]

* [webpage](http://www.utas.edu.au/information-and-communication-technology/people/individual-details/_nocache?user=ahjs)

## related $n$Lab entries

* [[computer science]]

* [[type theory]]

category: people


Arthur Sard

Arthur Sard was a mathematician at Queens College.

He got his PhD in 1936 from Harvard University, advised by [[Marston Morse]],
in which he proved the [[Morse–Sard theorem]], named after him and [[Anthony P. Morse]] (no relation to Marston Morse).

* [Wikipedia entry](https://en.wikipedia.org/wiki/Arthur_Sard)

## Selected writings

On the [[Morse–Sard theorem]]:

* [[Arthur Sard]], _The measure of the critical values of differentiable maps_, Bulletin of the American Mathematical Society, 48:12 (1942), 883–890, [doi](https://doi.org/10.1090/S0002-9904-1942-07811-6).

category: people


Arthur Schopenhauer

* [Wikipedia entry](https://en.wikipedia.org/wiki/Arthur_Schopenhauer)

## related $n$Lab entries

* [[philosophy]]

* [[first law of thought]]

category: people


Arthur Wasserman

* [webpage](https://lsa.umich.edu/math/people/emeritus-faculty/awass.html)

## Selected writings

On [[equivariant differential topology]]:

* {#Wasserman69} [[Arthur Wasserman]], section 3 of: _Equivariant differential topology_, Topology Vol. 8, pp. 127-150, 1969 (doi:10.1016/0040-9383(69)90005-6[pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/wasserman.pdf))

## Related $n$Lab entries

* [[differential topology]], [[equivariant homotopy theory]]

* [[equivariant bordism]]

* [[equivariant cohomotopy]]

category: people


Arthur Wightman

* [Wikipedia entry](http://en.wikipedia.org/wiki/Arthur_Wightman)

* [a biography at princeton.edu](https://www.princeton.edu/physics/arthur-wightman)

## Selected writings

On [[non-perturbative quantum field theory|non-perturbative]]/[[AQFT|algebraic]] [[quantum field theory]] and its key [[theorems]] ([[spin-statistics theorem]], [[PCT theorem]], ...):

* [[Raymond F. Streater]], [[Arthur S. Wightman]], *PCT, Spin and Statistics, and All That*, Princeton University Press (1989, 2000) [[ISBN:9780691070629](https://press.princeton.edu/books/paperback/9780691070629/pct-spin-and-statistics-and-all-that), [jstor:j.ctt1cx3vcq](https://www.jstor.org/stable/j.ctt1cx3vcq)]

## Related entries

* [[Wightman axioms]], [[AQFT]]

category: people

[[!redirects Arthur S. Wightman]]


Arthur-Selberg trace formula

A generalization of the [[Selberg trace formula]], developed by [[James Arthur]].

## References

* Steve Gelbart, _Lectures on the Arthur--Selberg Trace Formula_, ([pdf](http://arxiv.org/abs/math/9505206))

* Wikipedia, _[Arthur-Selberg trace formula](http://en.wikipedia.org/wiki/Arthur%E2%80%93Selberg_trace_formula)_


Artin gluing

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

**Artin gluing** is a fundamental construction in [[locale]] theory and [[topos]] theory. The original example is the way in which a [[topological space]] or [[locale]] $X$ may be glued together from an [[open subspace]] $i \colon U \hookrightarrow X$ and its [[closed subspace|closed]] [[complement]] $j \colon K \hookrightarrow X$. The analogous construction for toposes gives the [[sheaf topos]] $Sh(X)$ via a gluing together of $Sh(U)$ and $Sh(K)$, and applies more generally to give a sense of how to put two toposes together so that one becomes an [[open geometric morphism|open]] [[subtopos]] and the other a [[closed subtopos]] of the gluing.

## The topological case
 {#TheTopologicalCase}

Let us consider first the case of topological spaces. Let

* $X$ be a [[topological space]],

* $i \colon U \hookrightarrow X$ an [[open subspace]], and

* $j \colon K \hookrightarrow X$ the [[complement|complementary]] [[closed subspace]].

Let $O(X)$ denote the topology of $X$. There is an [[injective]] [[map]]

$$
  \array{
    \langle i^\ast, j^\ast \rangle
      &\colon\;&
    O(X) &\longrightarrow& O(U) \times O(K)
    \\
    &&
    V
      &\mapsto&
    (U \cap V, K \cap V)
  }
$$

that is a [[homomorphism]] of [[frame|frames]]. The general problem is to characterize the [[image]] of this map: in terms of structure pertaining to $O(U)$ and $O(K)$, which pairs $(W, W')$ of relatively open sets in $U$ and $K$ "glue together" to form an open set $W \cup W'$ in $X$?

Let $int_X \colon P(X) \to P(X)$ denote the [[interior]] operation, assigning to a [[subset]] of $X$ its interior. This is a [[left exact functor|left exact]] [[comonad]] on $P(X)$. (Indeed, topologies on the set $X$ are in [[natural bijection]] with [[left exact functor|left exact]] [[comonads]] on $P(X)$.) Our problem is to understand when the inclusion

$$
  W \cup W'
   \;\xhookrightarrow{\phantom{--}}\;
  int_X(W \cup W')
$$

obtains. Since $W \in O(U)$ is already open when considered as a subset of $X$, this condition boils down to the condition that

\[
  \label{ConditionOne}
  W'
   \;\xhookrightarrow{\phantom{--}}\;
  int_X(W \cup W')
\]

+-- {: .un_thm}
######Proposition

A necessary and sufficient condition for (eq:ConditionOne) is that the inclusion $W' \hookrightarrow int_X(W \cup K)$ obtains.

=--

+-- {: .proof}
######Proof
The necessity is clear since $W' \subseteq K$. The sufficiency is equivalent to having an inclusion

$$
  W' \cap int_X(W \cup K)
   \;\xhookrightarrow{\phantom{--}}\;
  int_X(W \cup W')
  \,.
$$

Since $W'$ is relatively open in the subspace $K$, we may write $W' = K \cap V$ for some $V \in O(X)$, and so we must check that there is an inclusion
$$
  (K \cap V) \cap int_X(W \cup K)
   \;\xhookrightarrow{\phantom{--}}\;
  int_X(W \cup (K \cap V))
$$

or in other words -- using distributivity and the fact that $int_X$ preserves intersections -- an inclusion

$$
  K \cap V \cap int_X(W \cup K)
   \;\xhookrightarrow{\phantom{--}}\;
  int_X(W \cup K) \cap int_X(W \cup V)
  \,.
$$

But this is clear, since we have

$$
  K \cap V \cap int_X(W \cup K)
   \;\xhookrightarrow{\phantom{--}}\;
  int_X(W \cup K)
$$

and

$$
  K \cap V \cap int_X(W \cup K)
   \;\xhookrightarrow{\phantom{--}}\;
  V
    \;\xhookrightarrow{\phantom{--}}\;
  W \cup V
    \;=\;
  int_X(W \cup V)
  \,,
$$

where to derive the last equation, we use the fact that $W \in O(U)$ and $V$ are open in $X$.
=--

+-- {: .un_thm}
######Proposition
The operation
$$
  O(U) \ni W
    \;\mapsto\;
  int_X(W \cup K)
    \;=\;
  int_X(W \cup \neg U) \in O(X)
$$
is the [[right adjoint]] $i_\ast$ to $i^\ast \colon O(X) \to O(U)$.
=--

This is well-known.

+-- {: .proof}
######Proof
For $V \in O(X)$ we have

$$
  \frac
    {V \subseteq int_X(W \cup \neg U) \qquad \text{in} \: O(X)}
    {V \subseteq W \cup \neg U \qquad \text{in} \: P(X)}
  \,,
$$

but the last condition is equivalent to having $U \cap V \subseteq W$ in $P(X)$, or to $i^\ast(V) = U \cap V \subseteq W$ in $O(X)$.
=--

Summarizing, the gluing condition (1) above (for $W' \in O(K)$, $W \in O(U)$) translates into saying that there is an inclusion

$$
  W'
   \;\xhookrightarrow{\phantom{--}}\;
  j^\ast i_\ast W
  \,.
$$

where $i^\ast, j^\ast$ are [[restriction]] maps and $i^\ast \dashv i_\ast$.

For future reference, observe that the operator $j^\ast i_\ast \colon O(U) \to O(K)$ is [[left exact functor|left exact]].

We can turn all this around. Suppose $U$ and $K$ are topological spaces, and suppose $f \colon O(U) \to O(K)$ is left exact. Then we can manufacture a topological space $X$ which contains $U$ as an open subspace and $K$ as its closed complement, and (letting $i$, $j$ being the inclusions as above) such that $f = j^\ast i_\ast$. The open sets of $X$ may be identified with pairs $(W, W') \in O(U) \times O(K)$ such that $W' \subseteq f(W)$; here we are thinking of $(W, W')$ as a stand-in for $W \cup W'$. In particular, open sets $W$ of $U$ give open sets $(W, \varnothing)$ of $X$, while open sets $W'$ of $K$ also give open sets $U \cup W'$ of $X$.

## The localic case

The development given above generalizes readily to the context of [[locales]].
Thus, let $X$ be a locale, with corresponding [[frame]] denoted by $O(X)$. Each element $U \in O(X)$ gives rise to two distinct frames:

* The frame whose elements are [[algebra for a monad|algebras]] ([[fixed point|fixed points]]) of the left exact [[idempotent monad]] $U \vee - \colon O(X) \to O(X)$. The corresponding locale is the [[closed sublocale]] $\neg U$ (more exactly, the frame surjection $O(X) \to Alg(U \vee -)$ is identified with a sublocale $\neg U \to X$).

* The frame whose elements are algebras of the left exact idempotent monad $U \Rightarrow - \colon O(X) \to O(X)$. (NB: for topological spaces, this is $U \Rightarrow V = int_X(V \cup \neg U)$.  This is isomorphic as a frame (but not as a subset of $O(X)$) to the [[principal ideal]] of $O(X)$ generated by $U$, which is more obviously the topology of $U$.) The sublocale corresponding to the frame surjection $O(X) \to Alg(U \Rightarrow -)$ is the [[open sublocale]] corresponding to $U$.

Put $K = \neg U$, and let $i^\ast: O(X) \to O(U)$, $j^\ast: O(X) \to O(K)$ be the frame maps corresponding to the open and closed sublocales attached to $U$, with [[right adjoints]] $i_\ast$, $j_\ast$. Again we have a left exact functor

$$f = j^\ast i_\ast \colon O(U) \to O(K).$$

Observe that this gives rise to a left exact comonad

$$O(U) \times O(K) \to O(U) \times O(K): (W, W') \mapsto (W, W' \wedge f(W)) \qquad (2)$$

whose coalgebras are pairs $(W, W')$ such that $W' \leq f(W)$. The coalgebra category forms a frame.

+-- {: .un_thm}
######Theorem
The frame map $\langle i^\ast, j^\ast \rangle \colon O(X) \to O(U) \times O(K)$ is identified with the comonadic functor attached to the comonad (2).  In particular, $O(X)$ can be recovered from $O(U)$, $O(K)$, and the comonad (2).
=--

Since $O(U + K) \cong O(U) \times O(K)$, we can think of the frame map $\langle i^\ast, j^\ast \rangle$ as giving a localic surjection $U + K \to X$.

Again, we can turn all this around and say that given any two locales $U$, $K$ and a left exact functor

$$f \colon O(U) \to O(K),$$

we can manufacture a locale $X$ whose frame $O(X)$ is the category of coalgebras for the comonad

$$1_{O(U)} \times \wedge \circ (f \times 1_{O(K)}) \colon O(U) \times O(K) \to O(U) \times O(K) \qquad (3)$$

so that $U$ is naturally identified with an open sublocale of $X$, $K$ with the corresponding closed sublocale, and with a localic surjection $U + K \to X$. This is the (Artin) **gluing construction** for $f$.

## The toposic case

Now suppose given [[topos|toposes]] $E$, $E'$ and a [[left exact functor]] $\Phi \colon E \to E'$. There is an induced left exact [[comonad]]

$$E \times E' \stackrel{\delta \times 1}{\to} E \times E \times E' \stackrel{1 \times \Phi \times 1}{\to} E \times E' \times E' \stackrel{1 \times product}{\to} E \times E' \qquad (3)$$

whose category of coalgebras is again (by a basic theorem of topos theory; see for instance [here](http://ncatlab.org/toddtrimble/published/Three+topos+theorems+in+one)) a topos, called the **Artin gluing** construction for $\Phi$, denoted $\mathbf{Gl}(\Phi)$.

Objects of $\mathbf{Gl}(\Phi)$ are triples $(e, e', f \colon e' \to \Phi(e))$. A morphism from $(e_0, e_0^', f_0)$ to $(e_1, e_1^', f_1)$ consists of a pair of maps $g \colon e_0 \to e_1$, $g'\colon e_0^' \to e_1^'$ which respects the maps $f_0, f_1$ :

$$
 \array{
    e_0^' &\overset{f_0}{\to} & \Phi (e_0)
    \\
    g'\downarrow &&\downarrow \Phi(g)
    \\
    e_1^' &\underset{f_1}{\to} & \Phi(e_1)
  }
$$

In other words, the Artin gluing is just the [[comma category]] $E' \downarrow \Phi$.  (In fact, this comma category is a topos whenever $\Phi$ preserves pullbacks.)

On the other hand, if $E$ is a topos and $U\in E$ is a [[subterminal object]], then it generates two [[subtoposes]] that are [[complement|complements]] in the [[lattice of subtoposes]], namely, an [[open subtopos]] whose [[reflector]] is $(-)^U$, and a [[closed subtopos]] whose reflector is the [[pushout]] $A\mapsto A +_{A\times U} U$.  If $E=Sh(X)$ is the topos of sheaves on a locale, then $U$ corresponds to an element of $O(X)$, hence an open sublocale with complement $K$ (say), and the open subtopos can be identified with $Sh(U)$ and the closed one with $Sh(K)$.

Returning to the general case, let us denote the [[geometric embedding]] of the open subtopos by $i\colon E_U \hookrightarrow E$ and that of the closed subtopos by $j\colon E_{\neg U}\hookrightarrow E$.  Then we have a composite functor, sometimes called the **fringe functor**,
$$ E_U \xrightarrow{i_*} E \xrightarrow{j^*} E_{\neg U} $$
which is left exact.

+-- {: .un_thm}
######Theorem
Let $U$ be a subterminal object of a topos $E$, as above.  Then the left exact left adjoint
$$\langle i^\ast, j^\ast \rangle \colon E \to E_U \times E_{\neg U}$$
is canonically identified with the comonadic gluing construction $\mathbf{Gl}(j^\ast i_\ast) \to E_U \times E_{\neg U}$.  In particular, $E$ can be recovered from $E_U$, $E_{\neg U}$, and the functor $j^* i_*$.
=--

For a proof, see A4.5.6 in the [[Elephant]].

Once again, the import of this theorem may be turned around. If $f \colon E \to F$ is any left exact functor, then the projection

$$\mathbf{Gl}(f) \to E \times F \stackrel{proj}{\to} E$$

is easily identified with a [[logical functor]] $\mathbf{Gl}(f) \to \mathbf{Gl}(f)/X$ where $X$ is the [[subterminal object]] $(1, 0, 0 \to f(1))$. This realizes $E$ as an [[open subtopos]] of $\mathbf{Gl}(f)$.
On the other hand, for the same subterminal object $X \hookrightarrow 1$, the corresponding classifying map

$$[X] \colon 1 \to \Omega$$

induces a [[Lawvere-Tierney topology]] $j$ given by

$$\Omega \cong 1 \times \Omega \stackrel{[X] \times 1}{\to} \Omega \times \Omega \stackrel{\wedge}{\to} \Omega.$$

Then, the category of sheaves $Sh(j)$, or more exactly the left exact left adjoint $\mathbf{Gl}(f) \to Sh(j)$ to the category of sheaves, is naturally identified with the projection

$$\mathbf{Gl}(f) \to E \times F \stackrel{proj}{\to} F,$$

thus realizing $F$ as equivalent to the [[closed subtopos]] ([[Elephant]], A.4.5, pp. 205-206) attached to the subterminal object $X$.

### Some details and further adjunctions

The sheaves in $\mathbf{Gl}(f)$ corresponding to the open resp. closed subtoposes can be described explicitly. Recall that the objects of $\mathbf{Gl}(f)$ have the form $(X, Y, u:Y\to f(X))$: then the open copy of $E$ corresponds to the subcategory on those objects $(X, Y, u:Y\to f(X))$ with $u$ an isomorphism in $F$ and the closed copy of $F$  to the subcategory with objects $(X, Y, u:Y\to f(X))$ such that $X\simeq 1$ in $E$.

The [[open subtopos]] corresponding to $E$ is [[dense subtopos|dense]] in $\mathbf{Gl}(f)$ precisely if $f:E\to F$ preserves the initial object since $(0,0,0\to f(0))$ is the initial object in $\mathbf{Gl}(f)$ and $0\to f(0)$ is an isomorphism precisely if $f$ preserves $0$.

To summarize: given a left exact $f\colon E\to F$ we get an open inclusion of $E$ with a further left adjoint:

$$ i_\ast \colon E\to \mathbf{Gl}(f) \qquad X\mapsto (X,f(X),id_{f(X)})$$

$$ i^\ast\colon \mathbf{Gl}(f)\to E \qquad (X,Y,u)\mapsto X$$

$$ i_{!} \colon E\to \mathbf{Gl}(f) \qquad X\mapsto (X,0,0\to f(X)) \quad ,$$

and a closed inclusion of $F$ into $\mathbf{Gl}(f)$ with

$$ j_\ast \colon F\to \mathbf{Gl}(f) \qquad X\mapsto (1,X,X\to 1)$$

$$ j^\ast\colon\mathbf{Gl}(f)\to F \qquad (X,Y,u)\mapsto Y$$

that will lack the left adjoint $j_!$ in general. The situation when $j_!$ exists is characterized by the following observation:

+-- {: .un_thm}
######Proposition
The closed inclusion $j$ is essential i.e. $j^\ast$ has a left adjoint $j_!$ precisely if the fringe functor $f$ has a left adjoint $l$.
=--

+-- {: .proof}
######Proof
Suppose $j_!$ exists. The fringe functor $f$ is up to natural isomorphism just $j^\ast i_\ast$ and $i^\ast j_!\dashv j^\ast i_\ast$ since adjoints compose.

Conversely, suppose that $l\dashv f$ with $\eta\colon id\to f{l}$ the corresponding [[unit of an adjunction|unit]]. Define

$$ j_{!} \colon F\to \mathbf{Gl}(f) \qquad Y\mapsto (l(Y),Y,\eta_Y\colon Y\to f{l}(Y)) \quad .$$

Now given morphisms $\alpha\colon Y_1\to Y$ and $u\colon Y\to f(X)$ in $F$ by general properties of a unit there is precisely one morphism $\overline{u\circ\alpha}\colon l(Y_1)\to X$ corresponding to $u\circ\alpha$ under the adjunction such that the following diagram commutes:

$$
 \array{
    Y_1 &\overset{\eta_{Y_1}}{\to} & f{l}(Y_1)
    \\
    \alpha\downarrow &&\downarrow f(\overline{u\circ\alpha})
    \\
    Y &\underset{u}{\to} & f(X)
  }
$$

This establishes a bijective correspondence between $Y_1\to j^\ast(X,Y,u)$ and $j_!(Y_1)\to (X,Y,u)$ which is natural since $\eta$ is.

=--

In particular, the left adjoint $j_!$ exists if the fringe functor $f$ is the [[direct image]] of a geometric morphism, or the [[inverse image]] of an [[essential geometric morphism]].
$$j_!\dashv j^\ast\dashv j_\ast \colon \mathbf{Gl}(f)\to F\quad .$$

The existence of a _right_ adjoint for the fringe functor: $f\dashv r\colon F\to E$, on the other hand, corresponds to the existence of an 'amazing' right adjoint for the open subtopos inclusion: $i_\ast\dashv i^!:\mathbf{Gl}(f)\to E$.

One direction follows again from the composition of adjoints: $j^\ast i_\ast\dashv i^! j_\ast$ , whereas for the other direction we define:

$$
 i^!\colon \mathbf{Gl}(f)\to E \qquad (X,Y,u)\mapsto r(Y) \times_{rf(X)} X\quad .
$$

When $f$ is fully faithful, the unit $X \to rf(X)$ is an iso, and so we can instead use
$$
 i^!\colon \mathbf{Gl}(f)\to E \qquad (X,Y,u)\mapsto r(Y)\quad .
$$

Note that in this case $E$ is dense and we get a 'co-cohesive' adjoint string

$$i_!\dashv i^\ast\dashv i_\ast \dashv i^!\colon \mathbf{Gl}(f)\to E$$

where $i_!$ and $i_\ast$ are fully faithful.

### Examples

Since $i:\ast\hookrightarrow E$ is left exact where $\ast$ is the degenerate topos with one identity morphism, _every_ topos $E$ is trivially a result of Artin gluing: $E\simeq E\downarrow i$.

Of course, more interesting examples of the gluing construction abound as well. Here are a few:

* Let $E$ be an (elementary, not necessarily Grothendieck) topos, and let $\hom(1, -): E \to Set$ represent the terminal object $1$ -- this of course is left exact. The gluing construction $\mathbf{Gl}(\hom(1, -))$ is called the **scone** (Sierpinski cone), or the [[Freyd cover]], of $E$.

* If $E$ is a Grothendieck topos and $\Delta \colon Set \to E$ is the (essentially unique) left exact left adjoint, then we have a gluing construction $E \downarrow \Delta$. This gluing may be regarded as the result of _attaching a generic open point_ to $E$.

* A concrete instance of the constructions in both the preceding examples is the [[Sierpinski topos]] $Set^{\to}$ corresponding e.g. to $Set\downarrow id_{Set}$: its objects are functions $X\to Y$ between sets $X,Y$ and the closed copy of $Set$ sits on the objects of the form $X\to 1$ and the open copy on the objects $X\overset{\simeq}{\to}Y$.

* Since a topos $\mathcal{E}$ is finitely bicomplete, the product functor $\sqcap:\mathcal{E}\times\mathcal{E}\to\mathcal{E}$ with $(X,Y)\mapsto X\times Y$ is part of an adjoint string $\sqcup\dashv\triangle\dashv\sqcap$ involving the [[diagonal functor]] and the coproduct functor. Since $\sqcap$ is left exact, Artin gluing applies. In the case $\mathcal{E}=Set$ , $\mathbb{Gl}(\sqcap)$ yields the [[hypergraph|topos of hypergraphs]]; this example is discussed in detail at [[hypergraph]]. These cases are somewhat unusual in that the fringe functor here has a left adjoint which itself has a further left adjoint.

### Remarks

* Artin gluing for toposes carries over in some slight extra generality, replacing left exact functors $f$ by _pullback-preserving functors_.

* Artin gluing applies also to other [[doctrine|doctrines]]: [[regular category|regular categories]], [[pretopos|pretoposes]], [[quasitopos|quasitoposes]], etc. See ([Carboni-Johnstone](#CJ)) and ([Johnstone-Lack-Sobocinski](#JLS07)).

## Related entries

* [[Freyd cover]]

* [[comma category]]

* [[recollement]]

## References

* [[M. Artin]], [[A. Grothendieck]], [[J. L. Verdier]], _Théorie des Topos et Cohomologie Etale des Schémas ([[SGA4]])_, LNM **269** Springer Heidelberg 1972. (exposé IV, sect. 9; in particular: sect. 9.5) {#SGA4}

* [[Gavin Wraith]], _Artin Glueing_ , JPAA **4** (1974) pp.345-358 [link](http://www.sciencedirect.com/science/article/pii/0022404974900140).

* [[Aurelio Carboni]], [[Peter Johnstone]], _Connected limits, familial representability and Artin glueing_ , Mathematical Structures in Computer Science **5** (1995) pp.441-459. ([pdf](http://journals.cambridge.org/article_S0960129500001183))
{#CJ}

* [[Aurelio Carboni]], [[Peter Johnstone]], _Corrigenda to 'Connected limits...'_ , Mathematical Structures in Computer Science **14** (2004)  pp.185-187.

* Peter F. Faul, Graham R. Manuell, _Artin Glueings of Frames as Semidirect Products_ , arXiv:1907.05104 (2019). ([abstract](https://arxiv.org/abs/1907.05104))

* Peter F. Faul, Graham Manuell, José Siqueira, _Artin glueings of toposes as adjoint split extensions_ , arXiv:2012.04963 (2020). ([abstract](https://arxiv.org/abs/2012.04963))

* [[Matthias Hutzler]], _Syntactic presentations for glued toposes and for crystalline toposes_, Phd. diss. Universität Augsburg 2021. ([arXiv:2206.11244](https://arxiv.org/abs/2206.11244))

* [[Mamuka Jibladze]], _Lower Bagdomain as a Glueing_ , Proc. A. Razmadze Math. Inst. **118** (1998) pp.33-41. ([pdf](http://www.rmi.ge/~jib/pubs/baglue.pdf))

* [[Peter Johnstone]], _Topos Theory_ , Academic Press New York 1977 (Dover reprint 2014). (section 4.2, pp.107-112)

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_ , Oxford UP 2002. (sec. A2.1.12, pp.82-84; A4.5.6, p.208)
  {#Johnstone}

* {#JLS07}[[Peter Johnstone]], [[Steve Lack]], [[Pawel Sobocinski]], _Quasitoposes, Quasiadhesive Categories and Artin Glueing_ , pp.312-326 in LNCS **4624** Springer Heidelberg 2007. ([preprint](http://users.ecs.soton.ac.uk/ps/papers/calco07.pdf))

* [[Anders Kock|A. Kock]], T. Plewe, _Glueing analysis for complemented subtoposes_ , TAC **2** (1996) pp.100-112. ([pdf](http://www.tac.mta.ca/tac/volumes/1996/n9/n9.pdf))

* J. C. Mitchell, A. Scedrov, _Notes on sconing and relators_ , Springer LNCS **702** (1993) pp.352-378. ([ps-draft](ftp://ftp.cis.upenn.edu/pub/papers/scedrov/rel.ps.Z)) [PDF](https://pdfs.semanticscholar.org/ba4e/b38b05470481dc3c912fb37f0598bbda94fe.pdf)

* [[Susan Niefield]], _The glueing construction and double categories_ , JPAA **216** no.8/9 (2012) pp.1827-1836.

[[!redirects Artin-Wraith gluing]]
[[!redirects Artin-Wraith glueing]]
[[!redirects Artin glueing]]


Artin L-function

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## Idea

An _Artin L-function_ $L_\sigma$ ([Artin 23](#Artin23)) is an [[L-function]] associated with a [[number field]] $K$ and induced from the choice of an $n$-dimensional [[Galois representation]], hence a [[linear representation]]

$$
  \sigma\;\colon\; Gal(L/K) \longrightarrow GL_n(\mathbb{C})
$$

of the [[Galois group]] for some finite [[Galois extension]] $L$ of $K$: it is the product ("[[Euler product]]") over all [[prime ideals]] $\mathfrak{p}$ in the [[ring of integers]] of $K$, of,  essentially, the [[characteristic polynomials]] of the [[Frobenius homomorphism]] $Frob_p$ regarded (see [here](Frobenius+morphism#AsElementsOfGaloisGroup)) as elements of [[Galois group]]

$$
  L_{K,\sigma} \colon s
  \mapsto
  \underset{\mathfrak{p}}{\prod}
  det
  \left(
    id - (N (\mathfrak{p}))^{-s}
    \sigma(Frob_{\mathfrak{p}})
  \right)^{-1}
  \,
$$

([e.g. Gelbhart 84, II.C.2](#Gelbhart84), [Snyder 02, def. 2.1.3](#Snyder02)).

> discussion of ramified primes needs to be added

For $\sigma = 1$ the [[trivial representation]] then the Artin L-function reduces to the [[Dedekind zeta function]] (see [below](#RelationToDedekindZeta)). So conversely one may think of Artin L-functions as being Dedekind zeta functions which are "twisted" by a [[Galois representation]]. (Notice that Galois representations are the analog in [[arithmetic geometry]] of [[flat connections]]/[[local systems of coefficients]]).

For $\sigma$ any 1-dimensional [[Galois representation]] (hence the case $n = 1$) then there is a _[[Dirichlet character]]_ $\chi$ such that the Artin L-function $L_\sigma$ is equal to the [[Dirichlet L-function]] $L_\chi$ -- this relation is part of _[[Artin reciprocity]]_.

For $\sigma$ any $n$-dimensional representation for $n \geq 1$ then the [[conjecture]] of _[[Langlands correspondence]]_ is that for each $n$-dimensional [[Galois representation]] $\sigma$ there is an [[automorphic representation]] $\pi$ such that the Artin L-function $L_\sigma$ equals the [[automorphic L-function]] $L_\pi$ (e.g [Gelbhart 84, pages 5-6](#Gelbhart84)).

## Properties

### For irreducible representations -- Artin's conjecture

_Artin's conjecture_ is the statement that for a nontrivial _[[irreducible representation]]_ $\sigma$ the Artin L-function $L_{K,\sigma}$ is not just a [[meromorphic function]] on the complex plane, but in fact an [[entire holomorphic function]].

e.g. ([Ram Murty 94, p. 3](#RamMurty94))

> or rather with at most a  pole at $s = 1$ [Murty-Murty 12, page 29 in chapter 2](#MurtyMurty12)

### For induced representations
 {#ForInducedRepresentations}

Let $H \hookrightarrow Gal(L/K)$ be [[subgroup]] of the Galois group $G \coloneqq Gal(L/K)$ and write $L^H \hookrightarrow L$ for the subfield of elements fixed by $H$.
Let $\sigma$ be a representation of $H = Gal(L/L^H)$ and write
$Ind_H^G\sigma$ for the [[induced representation]] of $G$. Then the corresponding Artin L-functions are equal:

$$
  L_{K,{Ind_H^{G}\sigma}} = L_{L^H, \sigma}
  \,.
$$

(e.g. ([Murty-Murty 12, equation (2) in chapter 2](#MurtyMurty12))).

### Relation to the Dedekind zeta function
 {#RelationToDedekindZeta}

For $\sigma = 1$ the [[trivial representation]] then $\sigma(Frob_{\mathfrak{p}}) = id$ identically, and hence in this case the definition of the Artin L-function becomes verbatim that of the [[Dedekind zeta function]] $\zeta_K$:

$$
  L_{L,1} = \zeta_L
  \,.
$$

If $L/K$ is a [[Galois extension]], the by the behaviour of Artin L-functions for induced representation as [above](#ForInducedRepresentations) this is also the Artin L-function of $K$ itself for the [[regular representation]] of $Gal(L/K)$

$$
  \zeta_L
  =
  L_{L,1}
  =
  L_{K,{Ind_1^{Gal(L/K)}1}}
$$

(e.g. ([Murty-Murty 12, below (2) in chapter 2](#MurtyMurty12)))

### Analogy with Selberg/Ruelle zeta-functions
 {#AnalogyWithSelbergZeta}

The [[Frobenius morphism]] $Frob_p$ giving an element in the [[Galois group]] means that one may think of it as an element of the [[fundamental group]] of the given [[arithmetic curve]] (see at _[[algebraic fundamental group]]_). There is a direct analogy between Frobenius elements at prime numbers in arithmetic geometry and parallel transport along [[prime geodesics]] in hyperbolic geometry ([Brown 09, p. 6](#Brown09)).

Under this interpretation, a [[Galois connection]] corresponds to a [[flat connection]] ([[local system of coefficients]]) on an arithmetic curve, and its Artin L-function is a product of [[characteristic polynomials]] of the [[monodromies]]/[[holonomies]] of that flat connection.

Now, in [[differential geometry]], given a suitable odd-dimensional [[hyperbolic manifold]] equipped with an actual [[flat bundle]] over it, then associated with it is the _[[Selberg zeta function]]_ and _[[Ruelle zeta function]]_. Both are (by definition in the latter case and by theorems in the former) [[Euler products]] of [[characteristic polynomials]] of [[monodromies]]/[[holonomies]]. See at _[Selberg zeta function -- Analogy with Artin L-function](Selberg+zeta+function#AnalogyWithArtinLFunction)_ and at _[Ruelle zeta function -- Analogy with Artin L-function](Ruelle+zeta+function#AnalogyToTheArtinLFunction)_ for more on this.

See also ([Brown 09, page 6](#Brown09), [Morishita 12, remark 12.7](#Morishita12)).

(The definition also has some similarity to that of the [[Alexander polynomial]], see at _[[arithmetic topology]]_.)

## Related concepts

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

## References

The original article is

* {#Artin23} [[Emil Artin]],  _Über eine neue Art von L Reihen_. Hamb. Math. Abh. 3. (1923) Reprinted in his collected works, ISBN 0-387-90686-X. English translation in ([Snyder 02, section A](#Snyder02))

Reviews include

* Wikipedia, _[Artin L-function](http://en.wikipedia.org/wiki/Artin_L-function)_

* {#MurtyMurty12} M. Ram Murty, V. Kumar Murty, _Non-vanishing of L-functions and applications_, Modern Birkhäuser classics 2012 ([chapter 2 pdf](http://www.beck-shop.de/fachbuch/leseprobe/9783034802734_Excerpt_001.pdf))

* {#Snyder02} [[Noah Snyder]], _Artin L-Functions: A Historical Approach_, 2002 ([pdf](http://www.math.columbia.edu/~nsnyder/thesismain.pdf))

and in the context of the [[Langlands program]]

* {#Gelbhart84} [[Stephen Gelbart]], _An elementary introduction to the Langlands program_,  Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177–219 ([web](http://www.ams.org/journals/bull/1984-10-02/S0273-0979-1984-15237-6/))

Further development includes

* {#RamMurty94} M. Ram Murty, _Selberg's conjectures and Artin -functions_, Bull. Amer. Math. Soc. 31 (1994), 1-14 ([web](http://www.ams.org/journals/bull/1994-31-01/S0273-0979-1994-00479-3/home.html))

The analogy with the [[Selberg zeta function]] is discussed in

* {#Brown09} Darin Brown, _Lifting properties of prime geodesics_, Rocky Mountain J. Math. Volume 39, Number 2 (2009), 437-454 ([euclid](http://projecteuclid.org/euclid.rmjm/1239113439))

* {#Morishita12} [[Masanori Morishita]], section 12.1 of _Knots and Primes: An Introduction to Arithmetic Topology_, 2012 ([web](https://books.google.co.uk/books?id=DOnkGOTnI78C&pg=PA156#v=onepage&q&f=false))

The analogies between Alexander polynomial and L-functions and touched upon in

* Ken-ichi Sugiyama, _The properties of an L-function from a geometric point of view_, 2007 [pdf](http://geoquant2007.mi.ras.ru/sugiyama.pdf); _A topological $\mathrm{L}$ -function for a threefold_, 2004 [pdf](http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1376-12.pdf)

[[!redirects Artin L-functions]]


Artin reciprocity law

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## Idea

[[Emil Artin]]'s reciprocity law is a [[reciprocity law]] in [[class field theory]] for [[global fields]]. It states, in consequence, that for each 1-dimensional [[Galois representation]] $\sigma$ there exists a [[Dirichlet character]] $\chi$ such that the [[Artin L-function]] $L_\sigma$ equals the [[Dirichlet L-function]] $L_\chi$. (The generalization of this reciprocal correspondence to higher dimensional Galois representations is the content of the conjectured _[[Langlands correspondence]]_).

For $K$ a [[global field]] there is a canonical map

$$
  \mathbb{I}_K \longrightarrow Gal(K^{ab}/K)
$$

from its [[group of ideles]] to the [[abelianization]] of its [[Galois group]], given by

$$
  (\cdots, a_v, \cdots) \mapsto \prod_v Frob_v^{ord_v(a_v)}
  \,.
$$

For $K$ a [[number field]], _Artin's reciprocity law_ says that this map is surjective, that it factors through the [[idele class group]] $K^\times \backslash \mathbb{I}_K$ and moreover that further quotienting this by the connected component $\mathcal{O}$ of 1 in the idele class group yields an [[isomorphism]]

$$
  K^\times \backslash \mathbb{I}_K / \mathcal{O}
  \stackrel{\simeq}{\longrightarrow}
  Gal(K^{ab}/K)
  \,.
$$

For $K = \mathbb{Q}$ this is also the statement of the [[Kronecker-Weber theorem]], and together this is a starting point of the [[Langlands correspondence]] [[conjecture]], see there for more.

For $K$ a [[function field]] the map is no longer surjective, but yields on the quotient by the [[restricted product]]  $\prod_v \mathcal{O}_v^\times$ an injection with dense image

$$
  K^\times \backslash \mathbb{I}_K / \prod_v \mathcal{O}_v^\times
  \hookrightarrow
  Gal(K^{ab}/K)
  \,.
$$

([e.g. Toth 11, p. 3](#Toth11))

Notice that the double quotients appearing here are by the [[Weil uniformization theorem]] analogous to [[moduli stacks of bundles]] on the [[arithmetic curve]] on which $K$ is the field of [[rational functions]]. Under this [[function field analogy]] the analog of Artin's reciprocity law plays a central role in the [[geometric Langlands correspondence]].

## Properties

### Function field analogy

[[!include function field analogy -- table]]

## Related concepts

* [[Artin L-function]]

## References

* [[Serge Lang]], _Algebraic number theory_, GTM 110, Springer 1970, 1986, 1994, 2000

* {#Snyder02} [[Noah Snyder]], section 2.3 of _Artin L-Functions: A Historical Approach_, 2002 ([pdf](https://web.archive.org/web/20120204005924/http://www.math.columbia.edu/~nsnyder/thesismain.pdf))

* Wikipedia: en:[Artin reciprocity law](http://en.wikipedia.org/wiki/Artin_reciprocity_law), de:[Artinsches_Reziprozitaetsgesetz](http://de.wikipedia.org/wiki/Artinsches_Reziprozit%C3%A4tsgesetz)

A discussion with an eye towards [[geometric class field theory]] is in

* {#Toth11} Peter Toth, _Geometric abelian class field theory_, 2011 ([web](http://dspace.library.uu.nl/handle/1874/206061))

[[!redirects Artin's reciprocity law]]

[[!redirects Artin reciprocity]]


Artin representability theorem

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## Idea

A list of sufficient conditions for a [[presheaf]] on the [[opposite category]] of  [[commutative rings]] to be [[representable functor|represented]] by a suitable [[algebraic space]]/[[scheme]]. Recalled e.g. as [Lurie Rep, theorem 1 ](#LurieRep).

## Related concepts

* [[Grothendieck existence]]

* [[Artin-Lurie representability theorem]]

## References

* {#Artin69} [[Michael Artin]], _Algebraization of formal moduli. I_, Global Analysis (Papers in Honor of K. Kodaira), Princeton Univ. Press, Princeton, N.J., 1969, pp. 21-71. [MR0260746](http://www.ams.org/mathscinet-getitem?mr=0260746) (41:5369)

* {#LurieRep} [[Jacob Lurie]], _[[Representability theorems]]_

Generalization to [[supergeometry]]:

* Nadia Ott, *Artin's theorems in supergeometry* ([arXiv:2110.12816](https://arxiv.org/abs/2110.12816))

[[!redirects Artin's representability theorem]]


Artin stack

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## Idea

An _Artin stack_ is a [[geometric stack]] over the [[étale site]]: the [[stack]] quotient of a [[scheme]] whose [[automorphism group]]s are [[algebraic group]]s.

This is the general notion of [[algebraic stack]]. But often the latter term is used in a more restrictive sense to mean [[Deligne-Mumford stack]].

## Properties

* [[Artin representability theorem]]

## References

The original article:

* [[Michael Artin]], *Versal deformations and algebraic stacks*, Invent Math **27** (1974) 165–189 [[doi:10.1007/BF01390174](https://doi.org/10.1007/BF01390174), [eudml:142310](https://eudml.org/doc/142310),  [pdf](http://math.uchicago.edu/~drinfeld/Artin_on_stacks.pdf)]

Review in the general context of [[algebraic stacks]]:

* {#LaumontMoret-Bailly} [[Gérard Laumon]], [[Laurent Moret-Bailly]], _Champs algébriques_, Ergebn. der Mathematik und ihrer Grenzgebiete **39**, Springer (2000) [[doi:10.1007/978-3-540-24899-6](https://doi.org/10.1007/978-3-540-24899-6)]

* [[Bertrand Toen]], _[[Master course on algebraic stacks]]_

and in the broader context of [[derived algebraic geometry]]:

* [[Dennis Gaitsgory]], [[Nick Rozenblyum]], Section I.2.4 in: *[[A study in derived algebraic geometry]]*

[[!redirects Artin stacks]]


Artin-Lam induction exponent

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## Definition

\begin{definition}
  For $G$ a [[finite group]], its *Artin-Lam induction exponent* is the [[minimum]] $e(G)$ among [[positive numbers|positive]] [[natural numbers]] such that the $e(G)$-dimensional [[trivial representation]] is in the [[ideal]] of the [[rational number|rational]] [[representation ring]] of $G$ which is generated by $G$-representations that are [[induced representation|induced]] by [[cyclic group|cyclic]] [[subgroups]] of $G$.
\end{definition}

> This definition according to the abstract of [Madsen, Thomas & Wall 1983](#MadsenThomasWall83). Other authors take the smallest $e(G)$ such that $e(G) \cdot \chi$ is in this ideal for *all* rational representations $\chi$ (e.g. [Yamauchi 70](#Yamauchi70)), in which case $e(G)$ is the [[exponent of a group|exponent]] of the additive group underlying the quotient by the ideal, whence the terminology.

## Related concepts

* [[free group actions on n-spheres]]

## References

The original article:

* {#Lam68} [[Tsit-Yuen Lam]], *Artin exponent of finite groups*, Journal of Algebra **9** (1968) 94-119 (doi:10.1016/0021-8693(68)90007-0, [pdf](https://core.ac.uk/download/pdf/82635891.pdf))

Further discussion:

* {#Yamauchi70} Kenichi Yamauchi, *On the 2-part of Artin Exponent of Finite Groups*, Science Reports of the Tokyo Kyoiku Daigaku, Section A
Vol. 10, No. 263/274 (1970), 234-240 ([jstor:43698746](https://www.jstor.org/stable/43698746))

* K. K. Nwabueze, *Some definition of the Artin exponent of finite groups* ([arXiv:math/9611212](https://arxiv.org/abs/math/9611212))

* S. Jafari and H. Sharifi, *On the Artin exponent of some rational groups*, Communications in Algebra, 46:4, 1519-152 ([doi:10.1080/00927872.2017.1347665](https://doi.org/10.1080/00927872.2017.1347665))

Textbook account:

* Charles Curtis, Irving Reiner, *Methods of representation theory -- With applications to finite groups and orders -- Vol. I*, Wiley 1981 ([pdf](https://webusers.imj-prg.fr/~olivier.dudas/src/parisa1.pdf))

Application to discussion of [[free group actions on n-spheres]]:

* {#MadsenThomasWall83} [[Ib Madsen]], [[Charles B. Thomas]], [[C. T. C. Wall]], *Topological spherical space form problem III: Dimensional bounds and smoothing*,  Pacific J. Math. 106(1): 135-143 (1983) ([pjm:1102721110](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-106/issue-1/Topological-spherical-space-form-problem-III-Dimensional-bounds-and-smoothing/pjm/1102721110.full))


Artin-Lurie representability theorem

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## Idea

The generalization of the [[Artin representability theorem]] from [[algebraic geometry]] to [[spectral algebraic geometry]].

## Statement

Write $CAlg^{cn}$ for the [[(∞,1)-category]] of [[connective]] [[E-∞ rings]], and [[∞Grpd]] for that of [[∞-groupoids]].

+-- {: .num_theorem}
###### Theorem

Necessary and sufficient conditions for an [[(∞,1)-presheaf]]

$$
  \mathcal{F} \;\colon\; CAlg^{cn}\longrightarrow \infty Grpd
$$

over some $Spec R$, on the [[opposite (∞,1)-category]] of [[connective]] [[E-∞ rings]] to be [[representable functor|represented]] by a  [[spectral Deligne-Mumford stack|spectral Deligne-Mumford n-stack]] locally of almost finite presentation over $R$:

1. For every discrete commutative ring, $\mathcal{F}(A)$ is [[n-truncated object in an (infinity,1)-topos|n-truncated]].

1. $\mathcal{F}$ is an [[∞-stack]] for the [[étale (∞,1)-site]].

1. $\mathcal{F}$ is nilcomplete, integrable, and an [[infinitesimally cohesive (∞,1)-presheaf on E-∞ rings]].

1. $\mathcal{F}$ admits a [[connective]] [[cotangent complex]].

1. the natural transformation to $Spec R$ is locally almost of finite presentation.

=--

([Lurie Rep, theorem 2](#LurieRep))

+-- {: .num_remark}
###### Remark

The condition that $\mathcal{F}$ be [[infinitesimally cohesive (∞,1)-presheaf on E-∞ rings|infinitesimally cohesive]] implies that the [[Lie differentiation]] around any point, given by restriction to [[local Artin rings]] ([[formal duals]] of [[infinitesimally thickened points]]), is a [[formal moduli problem]], hence equivalently an [[L-∞ algebra]].

=--

## Applications

The motivating example of the Artin-Lurie representability theorem is the re-proof of the _[[Goerss-Hopkins-Miller theorem]]_. See there for more.

## Related concepts

* [[Grothendieck existence]]

* [[Artin representability theorem]]

## References

* {#LurieRep} [[Jacob Lurie]], _[[Representability theorems]]_


Artin-Mazur codiagonal

[[!redirects Artin-Mazur  codiagonal ]]

The Artin-Mazur codiagonal of a bisimplicial set is another name often given to the _[[total simplicial set]]_ of a bisimplicial set, and more discussion of it is found at [[bisimplicial set]].

(We will tend not to use this term because of possible confusion with other uses of [[codiagonal]].)

##References

The construction seems first to be used in

* [[M. Artin]], [[B. Mazur]], _On the Van Kampen theorem_ , Topology 5 (1966) 179–189.


Artin-Mazur formal group

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## Idea

Every [[variety]] in [[positive characteristic]] has a [[formal group]] attached to it, called the _Artin-Mazur formal group_. This group is often related to arithmetic properties of the variety such as being ordinary or supersingular.

The Artin-Mazur formal group in dimension $n$ is a [[formal group]] version of the [[Picard infinity-group|Picard n-group]] of flat/holomorphic [[circle n-bundles]] on the given variety. Therefore for $n = 1$ one also speaks of the _[[formal Picard group]]_ and for $n = 2$ of the _[[formal Brauer group]]_.

## Definition
 {#Definition}

### Deformations of higher line bundles (of $H^n(-,\mathbb{G}_m)$-cohomology)

Let $X$ be a [[smooth scheme|smooth]] [[proper scheme|proper]] $n$ [[dimension|dimensional]] [[variety]] over an [[algebraically closed field]] $k$ of [[positive number|positive]] [[characteristic]] $p$.

Writing $\mathbb{G}_m$ for the [[multiplicative group]] and $H_{et}^\bullet(-,-)$ for [[etale cohomology]], then $H_{et}^n(X,\mathbb{G}_m)$ classifies $\mathbb{G}_m$-[[principal infinity-bundle|principal n-bundles]] ([[line n-bundles]], [[bundle 2-gerbe|bundle (n-1)-gerbes]]) on $X$.  Notice that, by the discussion at _[Brauer group -- relation to étale cohomology](Brauer%20group#RelationToEtaleCohomology)_, for $n = 1$ this is the [[Picard group]] while for $n = 2$ this contains (as a [[torsion subgroup]]) the [[Brauer group]] of $X$.

Accordingly, for each [[Artin algebra]] regarded as an [[infinitesimally thickened point]] $S \in ArtAlg_k^{op}$ the [[cohomology group]] $H_{et}^n(X\times_{Spec(k)} S,\mathbb{G}_m)$ is that of [[equivalence classes]] of $\mathbb{G}_n$-[[principal infinity-bundle|principal n-bundles]] on a formal thickening of $X$.

The defining inclusion $\ast \to S$ of the unique global point induces a restriction map $H^n_{et}(X\times_{Spec(k)} S, \mathbb{G}_m)\to H^n_{et}(X, \mathbb{G}_m))$ which restricts an $n$-bundle on the formal thickening to just $X$ itself. The [[kernel]] of this map hence may be thought of as the group of $S$-parameterized infinitesimal [[deformations]] of the trivial $\mathbb{G}_m$-$n$-bundle on $X$.

(For $n = 1$ this is an [[infinitesimal neighbourhood]] of the neutral element in the [[Picard scheme]] $Pic_X$, for higher $n$ one will need to genuinely speak about [[Picard stacks]] and higher stacks.)

As $S$ varies, these groups of deformations naturally form a [[presheaf]] on "[[infinitesimally thickened points]]" ([[Isbell duality|formal duals]] to [[Artin  algebras]]).

+-- {: .num_defn}
###### Definition

For $X$ an [[algebraic variety]] as above, write

$$
  \Phi_X^n \;\colon\; ArtAlg_k \to Grp
$$

$$
  \Phi_X^n(S) \coloneqq \mathrm{ker}(H^n_{et}(X\times_{Spec(k)} S, \mathbb{G}_m)\to H^n_{et}(X, \mathbb{G}_m))
  \,.
$$

=--

([Artin-Mazur 77, II.1 "Main examples"](#ArtinMazur77))

The fundamental result of ([Artin-Mazur 77, II](#ArtinMazur77)) is that under the above hypotheses this presheaf is [[prorepresentable functor|pro-representable]] by a [[formal group]], which we may hence also denote by $\Phi_X^n$. This is called the **Artin-Mazur formal group** of $X$ in degree $n$.

More in detail:

+-- {: .num_prop #SufficientConditionsForRepresentability}
###### Proposition

Let $X$ be an [[algebraic variety]] [[proper morphism|proper]] over an [[algebraically closed field]] $k$ of [[positive number|positive]] [[characteristic]].

A sufficient condition for $\Phi_X^k$ to be [[prorepresentable functor|pro-representable]] by a [[formal group]] is that $\Phi_X^{k-1}$ is [[formally smooth morphism|formally smooth]].

In particular if $dim H^{k-1}(X,\mathcal{O}_X) = 0$ then $\Phi^{k-1}(X)$ vanishes, hence is trivially formally smooth, hence $\Phi^k(X)$ is representable

=--

The first statement appears as ([Artin-Mazur 77, corollary (2.12)](#ArtinMazur77)). The second as ([Artin-Mazur 77, corollary (4.2)](#ArtinMazur77)).

+-- {: .num_remark #Dimension}
###### Remark

The [[dimension]] of $\Phi^k_X$ is

$$
  dim(\Phi^k_X) = dim H^k(X,\mathcal{O}_X)
  \,.
$$

=--

([Artin-Mazur 77, II.4](#ArtinMazur77)).

### Deformations of higher line bundles with connection (of Deligne cohomology)
 {#DeformationsOfDeligneCohomology}

In ([Artin-Mazur 77, section III](#ArtinMazur77)) is also discussed the formal [[deformation theory]] of [[line n-bundles with connection]] (classified by [[ordinary differential cohomology]], being [[hypercohomology]] with [[coefficients]] in the [[Deligne complex]]). Under suitable conditions this yields a [[formal group]], too.

Notice that by the discussion at _[intermediate Jacobian -- Characterization as Hodge-trivial Deligne cohomology](http://ncatlab.org/nlab/show/intermediate%20Jacobian#CharacterizationAsHodgeTrivialDeligneCohomology)_ the formal deformation theory of Deligne cohomology yields the formal completion of [[intermediate Jacobians]] (all in suitable degree).

## Examples

### General

+-- {: .num_remark}
###### Remark

* For a [[curve]] $X$ (i.e. $dim(X)= 1$), the Artin-Mazur group is often called the **[[formal Picard group]]** $\widehat{\mathrm{Pic}}$.

* For a [[surface]] $X$ (i.e. $dim(X) =2$), the Artin-Mazur group is called the **[[formal Brauer group]]** $\widehat{Br}$.

=--

### Of Calabi-Yau varieties
 {#OfCalabiYauVarieties}

+-- {: .num_example #AMfgOfStrictCalabiYau}
###### Example

Let $X$ be a strict [[Calabi-Yau variety]] in [[positive characteristic]] of  [[dimension]] $n$ (strict meaning that the [[Hodge numbers]] $h^{0,r} = 0$ vanish for $0 \lt r \lt n$, i.e. over the [[complex numbers]] that the [[holonomy group]] exhausts $SU(n)$, this is for instance the case of relevance for [[supersymmetry]], see at _[[supersymmetry and Calabi-Yau manifolds]]_).

By prop. \ref{SufficientConditionsForRepresentability} this means that
the Artin-Mazur formal group $\Phi^n_X$ exists.
Since moreover $h^{0,n} = 1$ it follows by remark \ref{Dimension} that it is of [[dimension]] 1

=--

For discussion of $\Phi_X^n$ for [[Calabi-Yau varieties]] $X$ of [[dimension]] $n$ and in [[positive number|positive]] [[characteristic]] see ([Geer-Katsura 03](#GeerKatsura03)).

## Related concepts

* [[formal group]]

* [[height of a variety]]

[[!include moduli of higher lines -- table]]

## References

The original article is

* {#ArtinMazur77} [[Michael Artin]], [[Barry Mazur]], _Formal Groups Arising from Algebraic Varieties_, Annales scientifiques de l'École Normale Supérieure, Sér. 4, 10 no. 1 (1977), p. 87-131  ([numdam:ASENS_1977_4_10_1_87_0](http://www.numdam.org/item?id=ASENS_1977_4_10_1_87_0), [MR56:15663](http://www.ams.org/mathscinet-getitem?mr=56:15663))

Further developments are in

* {#Stienstra87} [[Jan Stienstra]], _Formal group laws arising from algebraic varieties_, American Journal of Mathematics, Vol. 109, No.5 (1987), 907-925 ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/stienstra1.pdf))

Lecture notes touching on the cases $n = 1$ and $n = 2$ include

* {#Liedtke14} Christian Liedtke, example 6.13 in _Lectures on Supersingular K3 Surfaces and the Crystalline Torelli Theorem_ ([arXiv.1403.2538](http://arxiv.org/abs/1403.2538))

Discussion of Artin-Mazur formal groups for all $n$ and of [[Calabi-Yau varieties]] of [[positive number|positive]] [[characteristic]] in [[dimension]] $n$ is in

* {#GeerKatsura03} [[Gerard van der Geer]], T. Katsura, _On the height of Calabi-Yau varieties in positive characteristic_, Documenta Math. 8. 97-113 (2003) ([arXiv:math/0302023](http://arxiv.org/abs/math/0302023))

[[!redirects Artin-Mazur formal group]]
[[!redirects Artin-Mazur formal groups]]
[[!redirects Artin–Mazur formal group]]
[[!redirects Artin–Mazur formal groups]]
[[!redirects Artin--Mazur formal group]]
[[!redirects Artin--Mazur formal groups]]


Artin-Schreier sequence

#Contents#
* table of contents
{:toc}

## Definition

Let $X$ be a [[reduced scheme|reduced]] [[scheme]] of [[characteristic]] the [[prime number]] $p$, hence such that for all points $x \in X$

$$
  p \cdot \mathcal{O}_{X,x} = 0
  \,.
$$

Write

$$
  F - id \coloneqq (-)^p - (-) \;\colon\; (\mathbb{G}_a)_X \longrightarrow (\mathbb{G}_a)_X
$$

for the [[endomorphism]] of the [[additive group]] over the [[étale site]] $X_{et}$ of $X$ (the [[structure sheaf]] regarded as just a [[sheaf of abelian groups]]) which is the [[Frobenius endomorphism]] $F(-) \coloneqq (-)^p$ minus the identity.

+-- {: .num_prop}
###### Proposition

There is a [[short exact sequence]] of [[abelian sheaves]] over the [[étale site]]

$$
  0 \to (\mathbb{Z}/p\mathbb{Z})_X \to (\mathbb{G}_a)_X \stackrel{F-id}{\to} (\mathbb{G}_a)_X \to 0
  \,.
$$

=--

This is called the _Artin-Schreier sequence_ (e.g. [Tamme, section II 4.2](#Tamme), [Milne, example 7.9](#Milne)).

+-- {: .proof}
###### Proof

By the discussion at [category of sheaves -- Epi-/Mono-morphisms](category+of+sheaves#EpiMonoIsomorphisms) we need to show that the left morphism is an injection over any [[étale morphism]] $U_Y \to X$, and that for every element $s \in \mathcal{O}_X$ there exists an [[étale site]] [[covering]] $\{U_i \to X\}$ such that $(-)^p- (-)$ restricts on this to a morphism which hits the restriction of that element.

The first statement is clear, since $s = s^p$ says that $s$ is a constant section, hence in the image of the [[constant sheaf]]
$\mathbb{Z}/p\mathbb{Z}$  and hence for each connected $U_Y \to X$ the left morphism is the inclusion

$$
  \mathbb{Z}/p\mathbb{Z} \hookrightarrow \mathcal{O}_{X'}
$$

induced by including the unit [[section]] $e_{X'}$ and its multiples $r e_{X'}$ for $0 \leq r \lt p$. (This uses the "[[freshman's dream]]"-fact that in [[characteristic]] $p$ we have $(a + b)^p = a^p + b^p$).

This is injective by assumption that $X$ is of characteristic $p$.

To show that $(-)^p - (-)$ is an epimorphism of sheaves, it is sufficient to find for each element $s \in \mathcal{O}_X = A$ an [[étale site|étale cover]] $Spec(B) \to Spec(A)$ such that its restriction along this cover is in the image of $(-)^p - (-) \colon B \to B$. The choice

$$
  B \coloneqq A[t]/(t- t^p - s)
$$

by construction has the desired property concerning $s$, the preimage of $s$ is the equivalence class of $t$.

To see that with this choice $Spec(B) \to Spec(A)$ is indeed an [[étale morphism of schemes]] it is sufficient to observe that it is a [[morphism of finite presentation]] and a [[formally étale morphism]]. The first is true by construction. For the second observe that for a ring homomorphism $B \to T$ the generator $t$ cannot go to a nilpotent element since otherwise $s$ would have to be nilpotent. This implies [[formally étale morphism|formal étaleness]] analogous to  the discussion at [étale morphism of schemes -- Open immersion is Etale](etale+morphism+of+schemes#OpenImmersionIsEtale).

=--

## Related concepts

* [[étale cohomology]]

* [[Kummer sequence]]

* [[Kummer-Artin-Schreier-Witt exact sequence]]

* [[exponential exact sequence]]

## References

* [[Günter Tamme]], section II 4.2 _[[Introduction to Étale Cohomology]]_

* [[James Milne]], example 7.9 of _[[Lectures on Étale Cohomology]]_

[[!redirects Artin-Schreier sequences]]

[[!redirects Artin-Schreier exact sequence]]
[[!redirects Artin-Schreier exact sequences]]


Artin-Tate lemma

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## Statement

\begin{proposition}\label{TheLemma}
**([Artin & Tate 1951](#ArtinTate51))**
\linebreak
Given a [[Noetherian ring|Noetherian]] [[commutative ring]] $A$, a [[commutative algebra|commutative]] [[associative algebra|algebra]] $B$ over $A$, and a commutative algebra  $C$ over $B$ (and thus a commutative algebra over $A$), if $C$ is a [[finitely generated algebra]] over $A$ and $C$ is a [[finitely generated module]] over $B$, then $B$ is a finitely generated algebra over $A$. (If $A$ is a commutative subring of commutative ring $B$, then $B$ is a commutative algebra over $A$.)
\end{proposition}

## See also

* [[Noetherian ring]]

* [[commutative algebra]]

## References

The original article:

* {#ArtinTate51} [[Emil Artin]], [[John T. Tate]], *A note on finite ring extensions*, J. Math. Soc Japan, Volume **3** (1951) 74–77 $[$[doi:10.2969/jmsj/00310074](https://doi.org/10.2969/jmsj/00310074), [pdf](https://projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-3/issue-1/A-Note-on-Finite-Ring-Extensions/10.2969/jmsj/00310074.pdf)$]$

Review:

* [[Michael Atiyah]], [[Ian G. Macdonald]], _Introduction to commutative algebra_, (1969, 1994) $[$[pdf](http://math.univ-lyon1.fr/~mathieu/CoursM2-2020/AMD-ComAlg.pdf), [ISBN:9780201407518](https://www.routledge.com/Introduction-To-Commutative-Algebra/Atiyah/p/book/9780201407518)$]$

* David Eisenbud, *Commutative Algebra with a View Toward Algebraic Geometry*, Graduate Texts in Mathematics, 150, Springer-Verlag, 1995, ISBN:0-387-94268-8


Artinian bimodule

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## Idea

An *Artinian bimodule* is a bimodule which satisfies the [[descending chain condition]] on its subbimodules.

## Definition

Given [[rings]] $R$ and $S$ and an $R$-$S$-[[bimodule]] $B$, let $\mathrm{Mono}(B)$ be the [[category]] whose [[objects]] are $R$-$S$-[[subbimodules]] of $B$ and whose [[morphisms]] are $R$-$S$-[[bimodule monomorphisms]]. A descending chain of $R$-$S$-subbimodules is an [[inverse sequence]] of $R$-$S$-subbimodules in $\mathrm{Mono}(B)$, a [[sequence]] of $R$-$S$-subbimodules $A:\mathbb{N} \to \mathrm{Mono}(B)$ with the following [[dependent type|dependent sequence]] of $R$-$S$-bimodule monomorphisms: for [[natural number]] $n \in \mathbb{N}$, a dependent $R$-$S$-bimodule monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$.

An $R$-$S$-[[bimodule]] $B$ is **Artinian** if it satisfies the [[descending chain condition]] on its subbimodules: for every descending chain of $R$-$S$-subbimodules $(A, i_n)$ of $B$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the $R$-$S$-bimodule monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$ is an $R$-$S$-[[bimodule isomorphism]].

## Examples

A [[ring]] $R$ is [[Artinian ring|Artinian]] if it is Artinian as a $R$-$R$-[[bimodule]] with respect to its canonical bimodule structure, with its [[left action]] $\alpha_L:R \times R \to R$ and [[right action]] $\alpha_R:R \times R \to R$ defined as its multiplicative binary operation and its [[biaction]] $\alpha:R \times R \times R \to R$ defined as its ternary product:
$$\alpha_L(a, b) \coloneqq a \cdot b$$
$$\alpha_R(a, b) \coloneqq a \cdot b$$
$$\alpha(a, b, c) \coloneqq a \cdot b \cdot c$$

## See also

* [[bimodule]]

* [[Noetherian bimodule]]

* [[Artinian ring]]

[[!redirects Artinian bimodules]]


Artinian local ring

[[!redirects Artin ring]]

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* table of contents
{:toc}

## Definition

A **Artinian local ring** or **local Artinian ring** or **local Artin ring** or **Artin local ring** or **Weil ring** is a [[local ring]] which satisfies one of the equivalent conditions

* The local ring is an [[Artinian ring]]; it satisfies the [[descending chain condition]]
* The [[Jacobson radical]] of the local ring is a [[nilradical]]

These are also called **Artinian local algebras**, **local Artinian algebras**, **Artin local algebras**, or **local Artin algebras** or **Weil algebras**, since Artinian local rings are [[commutative algebras]] over the residue field $\mathbb{K}$.

In [[synthetic differential geometry]], the term Weil algebra is used for real Artinian local algebras, see _[[infinitesimally thickened point]]_ for more details.

## Properties

Every Artinian local $\mathbb{K}$-algebra $A$ has a [[maximal ideal]] $\mathfrak{m}_A$, whose [[residue field]] $A / \mathfrak{m}_A$ is $\mathbb{K}$ itself. As a $\mathbb{K}$ [[vector space]] one has a splitting $A=\mathbb{K}\oplus \mathfrak{m}_A$. Moreover, the descending chain condition implies that $(\mathfrak{m}_A)^n=0$ for some $n\gg 0$, a consequence of [[Nakayama lemma]]. This implies that the maximal ideal is a [[nilradical]].

Given a field $K$ and a Artinian local $K$-algebra $A$, let $I$ be the [[nilradical]] of $A$. There is a function $v:I \to \mathbb{N}$ which takes a [[nilpotent element]] $r \in I$ to the least natural number $n$ such that $r^{v(r) + 1} = 0$, such that given [[nilpotent elements]] $r \in I$ and $s \in I$ and non-zero scalars $a \in K$ and $b \in K$,

* $v(a r + b s) = v(r) + v(s)$
* $v(r s) = \min(v(r), v(s))$

### Spectra

Passing from commutative rings to their spectra (in the sense of algebraic geometry), Artinian local algebras correspond to infinitesimal pointed spaces. As such, they appear as bases of deformations in [[infinitesimal object|infinitesimal]] [[deformation theory]]. For instance $Spec(\mathbb{K}[\epsilon]/(\epsilon^2))$ is the base space for 1-dimensional first order deformations. Similarly, $Spec(\mathbb{K}[\epsilon]/(\epsilon^{n+1}))$ is the base space for 1-dimensional $n$-th order deformations.

An Artinian local algebra has a unique [[prime ideal]], which means that its [[spectrum]] consists of a single point, i.e., $Spec(A)$ is trivial as a topological space. It is however non-trivial as a ringed space, since its ring of functions is $A$. By this reason spectra of Artinian local algebras are occasionally called _[[infinitesimally thickened point|fat points]]_ in the literature.

## Examples

* A classical example is the ring of [[dual numbers]] $\mathbb{K}[\epsilon]/(\epsilon^2)$ over a field $\mathbb{K}$.

* Every [[prime power local ring]] is a local Artinian ring.

## See also

| [[commutative ring]] | [[reduced ring]] | [[integral domain]] |
---------------------|-----------------|-----------------|
| [[local ring]] | [[reduced local ring]] | [[local integral domain]] |
| [[Artinian ring]] | [[semisimple ring]] | [[field]] |
| [[Weil ring]] | [[field]] | [[field]] |

## References

* [[The Stacks Project]]

  * [10.100. Flatness criteria over Artinian rings](http://stacks.math.columbia.edu/tag/051E)

Local Artinian $\infty$-algebras are discussed in

* [[Jacob Lurie]], _Formal moduli problems_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-X.pdf))

[[!redirects Weil ring]]
[[!redirects Weil rings]]

[[!redirects local Artinian ring]]
[[!redirects local Artinian rings]]
[[!redirects Artinian local ring]]
[[!redirects Artinian local rings]]

[[!redirects local Artinian algebra]]
[[!redirects local Artinian algebras]]
[[!redirects Artinian local algebra]]
[[!redirects Artinian local algebras]]

[[!redirects local Artin ring]]
[[!redirects local Artin rings]]
[[!redirects Artin local ring]]
[[!redirects Artin local rings]]

[[!redirects local Artin algebra]]
[[!redirects local Artin algebras]]
[[!redirects Artin local algebra]]
[[!redirects Artin local algebras]]


artinian ring

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## Definition

### Artinian rings

Every [[ring]] $R$ has a canonical $R$-$R$-[[bimodule]] structure, with [[left action]] $\alpha_L:R \times R \to R$ and [[right action]] $\alpha_R:R \times R \to R$ defined as the multiplicative binary operation on $R$ and [[biaction]] $\alpha:R \times R \times R \to R$ defined as the ternary product on $R$:
$$\alpha_L(a, b) \coloneqq a \cdot b$$
$$\alpha_R(a, b) \coloneqq a \cdot b$$
$$\alpha(a, b, c) \coloneqq a \cdot b \cdot c$$

Let $\mathrm{TwoSidedIdeals}(R)$ be the [[category of two-sided ideals]] in $R$, whose objects are [[two-sided ideals]] $I$ in $R$, [[subbimodules|sub-$R$-$R$-bimodules]] of $R$ with respect to the canonical bimodule structure on $R$, and whose [[morphisms]] are $R$-$R$-[[bimodule monomorphisms]].

A descending chain of two-sided ideals in $R$ is an [[inverse sequence]] of two-sided ideals in $R$, a [[sequence]] of two-sided ideals $A:\mathbb{N} \to \mathrm{TwoSidedIdeals}(R)$ with the following [[dependent sequence]] of $R$-$R$-bimodule monomorphisms: for natural number $n \in \mathbb{N}$, a dependent $R$-$R$-bimodule monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$.

A [[ring]] $R$ is **Artinian** if it satisfies the [[descending chain condition]] on its two-sided ideals: for every descending chain of two-sided ideals $(A, i_n)$ in $R$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the $R$-$R$-bimodule monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$ is an $R$-$R$-[[bimodule isomorphism]].

### Left Artinian rings

Let $\mathrm{LeftIdeals}(R)$ be the category of [[left ideals]] in $R$, whose objects are [[left ideals]] $I$ in $R$, sub-left-$R$-modules of $R$ with respect to the canonical [[left module]] structure $(-)\cdot(-):R \times R \to R$ on $R$, and whose [[morphisms]] are left $R$-module monomorphisms.

A descending chain of left ideals in $R$ is an [[inverse sequence]] of left ideals in $R$, a [[sequence]] of left ideals $A:\mathbb{N} \to \mathrm{LeftIdeals}(R)$ with the following [[dependent sequence]] of left $R$-module monomorphisms: for natural number $n \in \mathbb{N}$, a dependent left $R$-module monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$.

A [[ring]] $R$ is **left Artinian** if it satisfies the [[descending chain condition]] on its left ideals: for every descending chain of left ideals $(A, i_n)$ in $R$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the left $R$-module monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$ is an left $R$-module isomorphism.

### Right Artinian rings

Let $\mathrm{RightIdeals}(R)$ be the category of [[right ideals]] in $R$, whose objects are [[right ideals]] $I$ in $R$, sub-right-$R$-modules of $R$ with respect to the canonical [[right module]] structure $(-)\cdot(-):R \times R \to R$ on $R$, and whose [[morphisms]] are right $R$-module monomorphisms.

A descending chain of right ideals in $R$ is an [[inverse sequence]] of right ideals in $R$, a [[sequence]] of right ideals $A:\mathbb{N} \to \mathrm{RightIdeals}(R)$ with the following [[dependent sequence]] of right $R$-module monomorphisms: for natural number $n \in \mathbb{N}$, a dependent right $R$-module monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$.

A [[ring]] $R$ is **right Artinian** if it satisfies the [[descending chain condition]] on its right ideals: for every descending chain of right ideals $(A, i_n)$ in $R$, there exists a natural number $m \in \mathbb{N}$ such that for all natural numbers $n \geq m$, the right $R$-module monomorphism $i_n:A_{n+1} \hookrightarrow A_{n}$ is an right $R$-module isomorphism.

## Properties

In an artinian ring $R$ the [[Jacobson radical]] $J(R)$ is [[nilpotent ideal|nilpotent]]. A left artinian ring is semiprimitive if and only if the zero ideal is the unique nilpotent ideal.

## Artinian and Noetherian rings

A dual condition is noetherian: a __[[noetherian ring]]__ is a ring satisfying the ascending chain condition on ideals. The symmetry is severely broken if one considers unital rings: for example every unital artinian ring is noetherian. For a converse there is a strong condition: a left (unital) ring $R$ is left artinian iff $R/J(R)$ is semisimple in $_R Mod$ and the Jacobson radical $J(R)$ is nilpotent. Artinian rings are intuitively much smaller than generic noetherian rings.

## See also

* [[Artinian bimodule]]

* [[artinian object]]

| [[commutative ring]] | [[reduced ring]] | [[integral domain]] |
---------------------|-----------------|-----------------|
| [[local ring]] | [[reduced local ring]] | [[local integral domain]] |
| [[Artinian ring]] | [[semisimple ring]] | [[field]] |
| [[Weil ring]] | [[field]] | [[field]] |

## References

* [[The Stacks Project]]

  * [10.100. Flatness criteria over Artinian rings](http://stacks.math.columbia.edu/tag/051E)

[[!redirects artinian ring]]
[[!redirects artinian rings]]
[[!redirects Artinian ring]]
[[!redirects Artinian rings]]

[[!redirects Artin ring]]
[[!redirects Artin rings]]

[[!redirects Artin algebra]]
[[!redirects Artin algebras]]


Artur Ekert

* [personal page](https://www.arturekert.org/)

* [Wikipedia entry](https://en.m.wikipedia.org/wiki/Artur_Ekert)

## Selected writings

Introducing the notion of [[quantum key distribution]]:

* [[Artur K. Ekert]], *Quantum cryptography based on Bell’s theorem*, Phys. Rev. Lett. **67** 661 (1991) [[doi:10.1103/PhysRevLett.67.661](https://doi.org/10.1103/PhysRevLett.67.661)]

On the [[quantum physics]] of [[quantum information]]:

* [[Dirk Bouwmeester]], [[Artur Ekert]], [[Anton Zeilinger]] (eds.), *The Physics of Quantum Information -- Quantum Cryptography, Quantum Teleportation, Quantum Computation*, Springer (2020) [[doi:10.1007/978-3-662-04209-0](https://doi.org/10.1007/978-3-662-04209-0)]

category: people

[[!redirects Artur K. Ekert]]


Arturo Prat-Waldron

* [webpage](http://math.berkeley.edu/~aprat/)

category: people


Arun Debray

* [webpage](https://web.ma.utexas.edu/users/a.debray/)

## Selected writings

On [[Whitehead-generalized cohomology theory|Whitehead-generalized]] [[differential cohomology]] via [[sheaves of spectra]]:

* {#AmabelDebrayHaine21} [[Araminta Amabel]], [[Arun Debray]], [[Peter J. Haine]] (eds.), _Differential Cohomology: Categories, Characteristic Classes, and Connections_. Based on [Fall 2019 talks at MIT's Juvitop seminar](https://math.mit.edu/juvitop/pastseminars/2019_Fall.html) by: A. Amabel, D. Chua, A. Debray, S. Devalapurkar, D. Freed, P. Haine, M. Hopkins, G. Parker, C. Reid, and A. Zhang. ([arXiv:2109.12250](https://arxiv.org/abs/2109.12250))

On [[differential cohomology]] in [[mathematical physics]]:

* [[Arun Debray]], *Differential Cohomology*, in *[[Encyclopedia of Mathematical Physics 2nd ed]]*, Elsevier (2024) [[arXiv:2312.14338](https://arxiv.org/abs/2312.14338)]

category: people


Arun Kumar Pati

* [Wikipedia entry](https://en.wikipedia.org/wiki/Arun_K._Pati)

* [Research group page](https://www.hri.res.in/~akpati/)

## Selected writings

Introducing the statement of the "[[no-deleting theorem]]" in [[quantum physics]]/[[quantum information theory]]:

* [[Arun Kumar Pati]], [[Samuel L. Braunstein]], *Impossibility of deleting an unknown quantum state*, Nature **404** (2000) 164-165 [[doi:10.1038/404130b0](https://doi.org/10.1038/404130b0), [arXiv:quant-ph/9911090](https://arxiv.org/abs/quant-ph/9911090)]

category: people

[[!redirects Arun K. Pati]]
[[!redirects Arun Pati]]


Arun Ram

__Arun Ram__ is a mathematician at the University of Melbourne (formerly at the Univ. of Wisconsin-Madison), a specialist in [[combinatorial representation theory]] and Lie theory (Lie groups and Lie algebras, quantum groups, Iwahori-Hecke algebras etc.).

* [web](http://www.ms.unimelb.edu.au/~ram)

## Selected writings

On [[quantum cohomology]] of [[flag varieties]]:

* [[Dale H. Peterson]] (notes by [[Arun Ram]]), *Quantum Cohomology of $G/H$*, MIT (1997) [[web](http://math.soimeme.org/~arunram/Resources/QuantumCohomologyOfGPL1-5.html), [pdf](http://math.soimeme.org/~arunram/Resources/PetersonGmodBcourse1997.pdf), [[Peterson-QuantumCohomology.pdf:file]]]

category: people


Arunas Liulevicius

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=6494)

## Selected writings

On the [[symplectic cobordism ring]]:

* [[Arunas Liulevicius]], _Notes on Homotopy of Thom Spectra_, American Journal of Mathematics
Vol. 86, No. 1 (1964), pp. 1-16 ([jstor:2373032](https://www.jstor.org/stable/2373032))

category: people


Arvind Rajaraman

* [webpage](http://www.physics.uci.edu/~arajaram/)

## Selected writings

On [[D-brane polarization]] in a context of [[AdS/QCD]]:

* [[Ofer Aharony]], [[Arvind Rajaraman]], _String Theory Duals for Mass-deformed $SO(N)$ and $USp(2N)$ $\mathcal{N}=4$ SYM Theories_, Phys. Rev. D62:106002, 2000 ([arXiv:hep-th/0004151](https://arxiv.org/abs/hep-th/0004151))

On [[black brane|black]] [[M2-M5 brane bound states]]:

* [[Arvind Rajaraman]], *Supergravity Solutions for Localised Brane Intersections*, JHEP 0109:018 (2001) [[arXiv:hep-th/0007241](https://arxiv.org/abs/hep-th/0007241), [doi:10.1088/1126-6708/2001/09/018](https://doi.org/10.1088/1126-6708/2001/09/018)]

## Related entries

* [[de Sitter spacetime]]

category: people