Fourier-Mukai transform

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###Context###
#### Higher linear algebra
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[[!include homotopy - contents]]
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#### Motivic cohomology
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[[!include motivic cohomology - contents]]
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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

The _Fourier-Mukai transform_ is a [[categorification|categorified]] [[integral transform]] roughly similar to the standard [[Fourier transform]].

Generally, for $X,Y$ two suitably well-behaved [[schemes]] (e.g. affine, smooth, complex) and with $D(X)$, $D(Y)$ their [[derived categories]] of [[quasicoherent sheaves]], then a _Fourier-Mukai transform_ with [[integral kernel]] $E \in D(X\times Y)$ is a [[functor]] (of [[triangulated categories]]/[[stable (infinity,1)-categories]])

$$
  \Phi \colon D(X)\longrightarrow D(Y)
$$

which is given as the composite of the ([[derived functor|derived]])  operations of

1. pull ([[inverse image]]) along the [[projection]] $p_X\colon X\times Y \to X$

1. [[tensor product]] with $E$;

1. push ([[direct image]]) along the other projection $p_Y \colon X\times Y \to Y$

i.e.

$$
  \Phi(A) \coloneqq (p_Y)_\ast (E\otimes p_X^\ast A)
$$

(where here we implicitly understand all operations as [[derived functors]]). (e.g. [Huybrechts 08, page 4](#Huybrechts08))

Hence this is a pull-tensor-push [[integral transform]] through the product [[correspondence]]

$$
  \array{
    && X \times Y
    \\
    & \swarrow && \searrow
    \\
    X && && Y
  }
$$

with twist $E$ on the correspondence space.

Such concept of [[integral transform]] is rather general and may be considered also in [[derived algebraic geometry]] (e.g. [BenZvi-Nadler-Preygel 13](#BenZviNadlerPreygel13)) and lots of other contexts.

As discussed at _[[integral transforms on sheaves]]_ this kind of [[integral transform]] is a [[categorification]] of an integral transform/[[matrix multiplication]] of functions induced by an [[integral kernel]], the role of which here is played by $E\in D(X \times Y)$.

Indeed, the central kind of result of the theory (theorem \ref{OrlovTheorem}) says that every suitable linear functor $D(X)\to D(Y)$ arises as a Fourier-Mukai transform for some $E$, a statement which is the [[categorification]] of the standard fact from [[linear algebra]] that every [[linear function]] between finite dimensional [[vector spaces]] is represented by a [[matrix]].

The original Fourier-Mukai transform proper is the special case of the above where $X$ is an [[abelian variety]], $Y = X^\vee$ its [[dual abelian variety]] and $E$ is the corresponding [[Poincaré line bundle]].

If $X$ is a [[moduli space of bundles|moduli space of line bundles]] over a suitable [[algebraic curve]], then a slight variant of the Fourier-Mukai transform is the [[geometric Langlands correspondence]] in the abelian case ([Frenkel 05, section 4.4, 4.5](#Frenkel05)).

## Definition

Let $X$ and $Y$ be [[schemes]] over a [[field]] $K$.
Let $E \in D(QCoh(O_{X \times Y}))$ be an object in the [[triangulated categories of sheaves|derived category of quasi-coherent sheaves]] over their [[product]]. (This is a [[correspondence]] between $X$ and $Y$ equipped with a [[chain complex]] $E$ of [[quasi-coherent sheaves]]).

The functor $\Phi(E) : D(QCoh(O_X)) \to D(QCoh(O_Y))$ defined by
  \[ F \mapsto \mathbf{R}q_*(\mathbf{L}p^*(F) \otimes^{\mathbf{L}} E), \]
where $p$ and $q$ are the [[projections]] from $X \times Y$ onto $X$ and $Y$, respectively, is called the **Fourier-Mukai transform of $E$**, or the **Fourier-Mukai functor induced by $E$**.

When $F : D(QCoh(O_X)) \to D(QCoh(O_Y))$ is isomorphic to $\Phi(E)$ for some $E \in D(QCoh(O_{X \times Y}))$, one also says that $F$ is **represented by $E$** or simply that $F$ is **of Fourier-Mukai type**.

## Properties

The key fact is as follows

+-- {: .num_theorem #OrlovTheorem}
###### Theorem ([[Orlov]])

Let $X$ and $Y$ be smooth projective [[varieties]] over a [[field]] $K$.  Let $F : D(X) \to D(Y)$ be a [[triangulated functor|triangulated]] [[fully faithful functor]].  Then $F$ is represented by some object $E \in D(X \times Y)$ which is unique up to isomorphism.

=--

See [Orlov 2003, 3.2.1](#OrlovSurvey) for a proof.

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

Though theorem \ref{OrlovTheorem} is stated there for $F$ admitting a [[right adjoint]], it follows from [Bondal-van den Bergh 2002](#BondalBergh2002) that every [[triangulated functor|triangulated]] [[fully faithful functor]] admits a [[right adjoint]] automatically (see e.g. [Huybrechts 08, p. 6](#Huybrechts08)).

=--

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

It was believed that theorem \ref{OrlovTheorem} should be true for _all_ [[triangulated functors]] (e.g. [Huybrechts 08, p. 5](#Huybrechts08)). However according to [(RVdB 2015)](#RVdB2015) this is not true.

=--

## Enhancements

On the level of the [[DG enhancements]], it is true for all smooth proper $K$-[[schemes]] that, in the [[homotopy category]] of [[DG categories]], _every_ functor corresponds bijectively to an isomorphism class of objects on $D(X \times Y)$.  See [(Toen 2006)](#Toen2006).

## Related concepts

* [[triangulated categories of sheaves]]

* [[Grothendieck duality]]

## References

* {#Mukai81} [[Shigeru Mukai]],  _Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves_. Nagoya Mathematical Journal 81: 153–175. (1981)

* {#BondalBergh2002} [[Alexei Bondal]], [[Michel van den Bergh]].  _Generators and representability of functors in commutative and noncommutative geometry_, 2002, [arXiv](http://arxiv.org/abs/math/0204218)

* {#OrlovSurvey} [[Dmitri Orlov]], _Derived categories of coherent sheaves and equivalences between them_, Russian Math. Surveys, 58 (2003), 3, 89-172, [translation](http://www.mi.ras.ru/~orlov/papers/Uspekhi2003.pdf).

*  Lutz Hille, Michel van den Bergh, _Fourier-Mukai transforms_ ([arXiv:0402043](http://arxiv.org/abs/math/0402043))

* {#Huybrechts08} [[Daniel Huybrechts]], _Fourier-Mukai transforms_, 2008 ([pdf](http://www.math.uni-bonn.de/people/huybrech/Garda2.pdf))

* C. Bartocci, [[Ugo Bruzzo]], D. Hernandez Ruiperez, _Fourier-Mukai and Nahm transforms in geometry and mathematical physics_, Progress in Mathematics 276, Birkhauser 2009.

* {#RVdB2015} Alice Rizzardo, [[Michel Van den Bergh]], _An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves_
([arXiv:1410.4039](http://arxiv.org/abs/1410.4039))

* {#Belmans} [[Pieter Belmans]], section 2.2 of _Grothendieck duality: lecture 3_, 2014 ([[BelmansDuality.pdf:file]])

Banerjee and Hudson have defined Fourier-Mukai functors analogously on [[algebraic cobordism]].

* Anandam Banerjee, Thomas Hudson, _Fourier-Mukai transformation on algebraic cobordism_, [pdf](https://sites.google.com/site/anandamb/research/fourier-mukai.pdf?attredirects=0).

Discussion of [[internal homs]] of [[dg-categories]] in terms of refined Fourier-Mukai transforms is in

* [[Bertrand Toën]], _The homotopy theory of dg-categories and derived Morita theory_, Invent. Math. 167 (2007), 615–667

* Alberto Canonaco, Paolo Stellari, _Internal Homs via extensions of dg functors_ ([arXiv:1312.5619](http://arxiv.org/abs/1312.5619))

Discussion in the context of [[geometric Langlands duality]] is in

* {#Frenkel05} [[Edward Frenkel]], _Lectures on the Langlands Program and Conformal Field Theory_ ([arXiv:hep-th/0512172](http://arxiv.org/abs/hep-th/0512172))

For a discussion of Fourier-Mukai transforms in the setting of $(\infty,1)$-enhancements, see

* {#BenZviNadlerPreygel13} [[David Ben-Zvi]], [[David Nadler]], Anatoly Preygel.  _Integral transforms for coherent sheaves_,  [arXiv:1312.7164](http://arxiv.org/abs/1312.7164)

[[!redirects Fourier-Mukai transforms]]
[[!redirects Fourier-Mukai functor]]
[[!redirects Fourier-Mukai functors]]
[[!redirects Fourier-Mukai transformation]]


fourth generation of fundamental particles

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### Context
#### Fields and quanta
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[[!include fields and quanta - table]]
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#Contents#
* table of contents
{:toc}

## Idea

A hypothetical fourth [[generation of fundamental particles]] ([[fermions]]), beyond the three currently observed.

Still consistent with experiment, suggested by flavour physics, and implied by [[Spin(16)]], $Spin(18)$-[[GUT]].

## References

### General

Exposition and review:

* David B. Cline, _Beyond Truth and Beauty: A Fourth Family of Particles, _Scientific American Vol. 259, No. 2 (AUGUST 1988), pp. 60-67 ([jstor:24989195](https://www.jstor.org/stable/24989195))

* Bob Holdom, WS Hou, Tobias Hurth, Michelangelo L Mangano, Saleh Sultansoy, Gokhan Ünel
_Four statements about the fourth generation_, PMC Phys A 3, 4 (2009) ([doi:10.1186/1754-0410-3-4](https://doi.org/10.1186/1754-0410-3-4))

### Experimental constraints
 {#ReferencesExperimentalConstraints}

* {#Lenz13} [[Alexander Lenz]], _Constraints on a fourth generation of fermions from Higgs Boson searches_, Adv. High Energy Phys. 2013 (2013) 910275 ([doi:10.1155/2013/910275](https://doi.org/10.1155/2013/910275))

* [[Alexander Lenz]], _How to really kill a physics model_, talk notes 2012 ([pdf](https://www.ippp.dur.ac.uk/~lenz/Talk_Bern_Lenz.pdf), [[Lenz4thGenerationConstraints.pdf:file]])

* N.S. Mankoč Borštnik, H.B.F. Nielsen, _Do Present Experiments Exclude the Fourth Family Quarks as Well as the Existence of More Than One Scalar?_, Bled Workshops Phys. 17 (2016) no. 2, 128-146 ([spire:1510166](http://inspirehep.net/record/1510166))

### Motivation from flavour physics

Motivation from [[flavour (particle physics)|flavour]]-physics ([[B-meson]] decays)

* A.K. Ciftci, R. Ciftci, S. Sultansoy, _A Search for the Fourth SM Family Fermions and $E_6$ Quarks at $\mu^+ \mu^-$ Colliders_, Phys. Rev. D65 (2002) 055001 ([arXiv:hep-ph/0106222](https://arxiv.org/abs/hep-ph/0106222))

* Amarjit Soni, Ashutosh Kumar Alok, Anjan Giri, Rukmani Mohanta, Soumitra Nandi, _The fourth family: a simple explanation for the observed pattern of anomalies in B-CP asymmetries_, Phys. Lett. B683:302-305, 2010 ([arXiv:0807.1971](https://arxiv.org/abs/0807.1971))

### In string/grand-unified theory

In [[Spin(16)]]- and [[Spin(18)]]-[[GUT]] ("SO(16)-, SO(18)-GUT"):

* {#WilczekZee82} [[Frank Wilczek]], [[Anthony Zee]], _Families from spinors_, Phys. Rev. D 25, 553 (1982) ([doi:10.1103/PhysRevD.25.55310.1103/PhysRevD.25.553](https://doi.org/10.1103/PhysRevD.25.553))

* {#SenjanovicWilczekZee84} Goran Senjanović, [[Frank Wilczek]], [[Anthony Zee]], _Reflections on mirror fermions_,  Physics Letters B Volume 141, Issues 5–6, 5 July 1984, Pages 389-394 Physics Letters B (doi:10.1016/0370-2693(84)90269-7)

* Homero Martínez, Alejandra Melfo, Fabrizio Nesti, Goran Senjanović, _Three Extra Mirror or Sequential Families: a Case for Heavy Higgs and Inert Doublet_, Phys. Rev. Lett.106:191802, 2011 ([arXiv:1101.3796](https://arxiv.org/abs/1101.3796))

* {#diLucio11} Luca di Lucio, p. 44 of: _Aspects of Symmetry Breaking in Grand Unified Theories_, 2011 ([pdf](https://www.sissa.it/tpp/phdsection/AlumniThesis/Luca%20Di%20Luzio.pdf))

* {#Guigan19} Michael McGuigan, _Dark Horse, Dark Matter: Revisiting the $SO(16) \times SO(16)'$ Nonsupersymmetric Model in the LHC and Dark Energy Era_ ([arXiv:1907.01944](https://arxiv.org/abs/1907.01944))

In [[heterotic string theory]]:

* A.A. Maslikov, I.A. Naumov, G.G. Volkov, _The Fermion Generations Problem in the Gust in the Free World-Sheet Fermion Formulation_, Int. J. Mod. Phys. A11 (1996) 1117 ([arXiv:hep-ph/9512429](https://arxiv.org/abs/hep-ph/9512429))

[[!redirects fourth generation of fermions]]

[[!redirects fourth family of fundamental particles]]
[[!redirects fourth family of fermions]]

[[!redirects 4th generation of fundamental particles]]
[[!redirects 4th generation of fermions]]

[[!redirects 4th family of fundamental particles]]
[[!redirects 4th family of fermions]]


Fox derivative

Let $F$ be a [[free group]] with basis $X = \{ x_i\}_{i\in I}$ and $\mathbb{Z}F$ the integer [[group ring]] of $F$.

 Differentiation or [[derivation on a group|derivation]], $D$, in this context is defined using a sort of nonsymmetric analogue of the Leibniz rule: it is an additive map $D:\mathbb{Z}F\to\mathbb{Z}F$ such that for all $u,v\in F$,

$$D(u v) = D(u) + u D(v). $$

The __Fox partial derivatives__ $\frac{\partial}{\partial x_i}$ are defined by the rules

$$ \frac{\partial 1}{\partial x_i} = 0$$

$$ \frac{\partial x_i}{\partial x_i} = 1$$

extended to the products $u = y_1\ldots y_n$ where $y_i = x_k$ or $y_i=x_k^{-1}$ for some $k = k(i)$ by the formula

$$
\frac{\partial u }{\partial x_i} = \sum_{s=1}^n y_1\cdots y_{s-1} \frac{\partial y_s }{\partial x_i}.
$$

This then implies that

$$ \frac{\partial x_i^{-1}}{\partial x_i} = -x_i^{-1}$$

$$ \frac{\partial x_j^{\pm 1}}{\partial x_i} = 0,\;\;i\neq j$$

Notice that the summands on the right-hand side are "of different length".

The lemma given in [[derivation on a group]] allows the following alternative form of the above definition to be given:

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

For each $x \in X$, let
$$\frac{\partial}{\partial x} : F \to \mathbb{Z}F$$
be defined by

1. for $y \in X$,

   $$\frac{\partial y}{\partial x} = 1\,\,\, if\,\,\,  x = y\,\,\, and \,\,\,=
   0  \,\,\, y \neq x;
   .$$

2. for any words, $w_1,w_2 \in F$,

   $$\frac{\partial}{\partial x}(w_1w_2) = \frac{\partial}{\partial x}w_1 + w_1\frac{\partial}{\partial x}w_2.$$

Then these uniquely determine the Fox derivative of $F$ with respect to $x$.
=--

The Fox derivatives give a way of expanding any derivation (differentiation) defined on $F$. For every differentiation

$$
D(u)=\sum_{i\in I} \frac{\partial u }{\partial x_i} D(x_i)
$$

(This is a finite sum since $u$ will only involve finitely many of the generators.)

In particular if  $\epsilon:\mathbb{Z}F\to\mathbb{Z}$ is the augmentation map given by $\epsilon:x_i\mapsto 1$, then the differentiation $u\mapsto u-\epsilon(u) 1_F$ satisfies

$$
u - \epsilon(u) 1_F = \sum_i \frac{\partial u }{\partial x_i} (x_i -1)
$$

hence it belongs to the left [[ideal]] in $\mathbb{Z}F$ which is generated by $(x_i-1)$.

This construction is important in [[combinatorial group theory]], particularly in the study of [[free product]]s of groups and the study of [[metabelian group]]s.

Given any group $G$ with a [[group presentation|presentation]] $\langle X; R\rangle = F/N$ such that $F=\langle X\rangle$ is the [[free group]] on the set of letters $X$ and $N$ the [[normal subgroup|normal closure]] of the set of relations $R$, let $\bar{G}:=G/[G,G]$, let $\phi:F\to G$, $\bar\phi:F\to \bar{G}$ be the canonical projections; denote by the same letter their linearizations for group rings $\phi:\mathbb{Z}F\to \mathbb{Z}G$ and $\bar\phi:\mathbb{Z}F\to\mathbb{Z}\bar{G}$. The __Jacobi matrix__ of the presentation is the matrix

$$
J = \left(\phi(\frac{\partial r_i}{\partial x_j})\right)
$$
and also the projected matrix $\bar{J}$ which is the image of $J$ as a matrix over $\mathbb{Z}\bar{G}$. The __determinant ideal__ $D_i$ of order $i$ of the matrix $\bar{J}$ is the ideal of $\mathbb{Z}\bar{G}$ generated by all minors (= determinants of submatrices) of size $i\times i$ in $\bar{J}$. The sequence $D_1,D_2,\ldots$ is invariant (up to some technical details), that is does not depend on the presentation. In the case when $G=\pi(S)$ where $S$ is the complement of a [[knot]], $\bar{G}$ is an infinite cyclic group. Let $t$ be its generator; then the highest nonzero determinant ideal (of $\bar{J}$) in $\mathbb{Z}\bar{G}$ is a principal ideal, hence it has a normalized (in the sense that the heighest coefficient is $1$) generator, which is a polynomial in $t$. This polynomial is an invariant of the knot, the __Alexander polynomial__ of the knot.

###References

The orginal articles include:

* R. H. Fox, _Free differential calculus I: Derivation in the free group ring_, Annals Math. (2) __57__, 547--560 (1953) [doi:10.2307/1969736](http://dx.doi.org/10.2307/1969736)

* R. H. Fox, _Free differential calculus II: The Isomorphism Problem of Groups_, Annals Math. (2) __59__ 196--210 (1954); _III:Subgroups_, Annals Math. (2)  __64__, 407--419; IV: __71__, 408--422 (1960)

with a nice introduction in

* * [[R. H. Crowell]] and [[R. H. Fox]], Introduction to Knot Theory, Springer, Graduate Texts 57, 1963.

see also

* B. Chandler, [[W. Magnus]], _The history of combinatorial group theory: a case study in the history of ideas_, Springer 1982

* [[R. Lyndon]], [[P. Schupp]], _Combinatorial group theory_, Ch. II.3, Springer 1977(Russian transl. Mir, Moskva 1980)

and more recently

* Valentino Zocca, _Fox calculus, symplectic forms and moduli spaces_, Trans. AMS__350__, 4, (1998) 1429-1466, [pdf](http://www.ams.org/journals/tran/1998-350-04/S0002-9947-98-02052-2/S0002-9947-98-02052-2.pdf)

* [en.wikipedia:Fox derivative](http://en.wikipedia.org/wiki/Fox_derivative)

Connections to double Poisson structures/brackets are discussed in

* Gwenael Massuyeau, [[Vladimir Turaev]], _Quasi-Poisson structures on representation spaces of surfaces_, [arxiv/1205.4898](http://arxiv.org/abs/1205.4898), [irma pdf](http://www-irma.u-strasbg.fr/~massuyea/papers/double_brackets.pdf)

See also

* Gwenael Massuyeau, Vladimir Turaev, _Fox pairings and generalized Dehn twists_, [arxiv/1109.5248](http://arxiv.org/abs/1109.5248)

For a pro-$l$-version of Fox calculus see

* _Pro-$l$ Fox free differential calculus_, section 8.3 of [[Masanori Morishita]], _Knots and primes: an introduction to arithmetic topology_, Springer 2012

[[!redirects Fox derivatives]]


Fox theorem

#Contents#
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{:toc}

## Idea

This is a theorem due to [[Thomas Fox]] which characterizes products in a [[cartesian monoidal category]] as an algebraic structure given by natural transformations rather than in terms of a universal poperty.

## Theorem

A [[symmetric monoidal category]] is cartesian if and only if it is [[isomorphic]] to its own category of cocommutative [[comonoids]]. Thus every object is equipped with a unique cocommutative comonoid structure, and these structures are respected by all maps.

## Related concepts

- A similar but slightly more complicated theorem seems to exist for [[symmetric powers in a symmetric monoidal (Q plus)-linear category|symmetric powers in symmetric monoidal (Q plus)-linear categories]].
- [[semicartesian monoidal category]]

## Reference

* {#Fox76} [[Thomas Fox]], *Coalgebras and cartesian categories*, Communications in Algebra **4** 7 (1976) 665-667 ([doi:10.1080/00927877608822127](https://doi.org/10.1080/00927877608822127), [pdf](https://www.tandfonline.com/doi/pdf/10.1080/00927877608822127))


fppf site

[[!redirects fppf-site]]

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#### Topos Theory
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#### Geometry
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[[!include higher geometry - contents]]
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#Contents#
* table of contents
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## Definition

Fix some [[scheme]] $S$.

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

the **fppf-site** (over $S$) is the [[site]]

* whose underlying category is the category $Aff/S$ of [[affine scheme]]s [[over category|over]] $S$;

* whose [[coverage]] has as [[covering]] families $\{f : U_i \to X\}$
  those families of morphisms that are  such that

  * each $f_i$ is a [[flat morphism]];

  * each $f_i$ is [[locally of finite presentation]];

  * we have that $X = \cup_i f_i(U_i)$.

=--

This appears as ([de Jong, def. 27.7.1, def 27.7.6, lemma 27.7.11](#deJong)).

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

The abbreviation "fppf" is for _fidèlement plat de présentation finie_ : faithfully flat and of finite presentation.

But notice that it is common practice, as in the above definition, to require only _local_ finite presentability.

=--

## Related concepts

[[fpqc-site]] $\to$ **fppf-site** $\to$ [[syntomic site]] $\to$ [[étale site]] $\to$ [[Nisnevich site]] $\to$ [[Zariski site]]

## References

Chaper 27.7 in

* [[Aise Johan de Jong]], _[[The Stacks Project]]_
{#deJong}

[[!redirects fppf topology]]
[[!redirects fppf-topology]]


fpqc site

[[!redirects fpqc-site]]

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### Context
#### Topos Theory
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[[!include topos theory - contents]]
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#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Fix some [[scheme]] $S$.

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

The **fpqc-site** (over $S$) is the [[site]]

* whose underlying category is the category $Aff/S$ of [[affine scheme]]s [[over category|over]] $S$;

* whose [[coverage]] has as [[covering]] families $\{f : U_i \to X\}$
  those families of morphisms that are  such that

  * each $f_i$ is a [[flat morphism]];

  * for every affine open $W \hookrightarrow X$ there exists $n \geq  0$, a [[function]] $a : \{1, \cdots, n\} \to I$ and affine opens
    $V_j \hookrightarrow U_{a(j)}$ with

    $$
      \cup_{j = 1}^{n} f_{a(j)}(V_j) = W
      \,.
    $$

=--

This appears as ([Stacks Project, Tag 022B](https://stacks.math.columbia.edu/tag/022B)).

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

The last condition does imply that $\cup_i f_i(U_i) = X$.

=--

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

The abbreviation "fpqc" is for _fidèlement plat quasi-compacte_ : faithfully flat and quasi-compact.

=--

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

Because the collection of fpqc covers of a scheme does not have a small collection of refinements (Stacks project, [Tag 0BBK](http://stacks.math.columbia.edu/tag/0BBK)), working with the fpqc topology can be set-theoretically tricky.  Indeed, in 1975, Waterhouse gave an example of a functor on schemes that admits no fpqc sheafification.  This contradicts many claims in the literature that fpqc sheafification and stackification is functorial (and such claims continue to be made).

=--

## Related concepts

**fpqc-site** $\to$ [[fppf-site]] $\to$ [[syntomic site]] $\to$ [[étale site]] $\to$ [[Nisnevich site]] $\to$ [[Zariski site]]

## References

Chaper 27.8 in

* _[[The Stacks Project]]_
{#Stacks}

* W. C. Waterhouse, _Basically bounded functors and flat sheaves_, Pacific Journal of Mathematics __57__ (1975), no. 2, 597--610 [MR396578](http://www.ams.org/mathscinet-getitem?mr=396578), [euclid](http://projecteuclid.org/euclid.pjm/1102906018)

[[!redirects fpqc-site]]
[[!redirects fpqc topology]]
[[!redirects fpqc-topology]]


fpqc topology

#Contents#
* table of contents
{:toc}

## Idea

The fpqc topology is a [[Grothendieck topology]] on the category of (commutative) [[affine scheme]]s.  It is one of the main [[Grothendieck topologies]] used in [[algebraic geometry]].

## Definition

Let $Aff$ be the category [[opposite category|opposite]] to the category of [[commutative algebra]]s. A family of maps $\{f_i : U_i\to X\}_{i\in I}$ in that category is a cover in the __fpqc topology__ if the induced morphism $\coprod U_i \to X$ is [[faithfully flat morphism|faithfully]] [[flat morphism|flat]] and [[quasicompact morphism|quasicompact]].  In particular, the union of images then cover $X$ as a [[topological space]].  The French for this is *fidèlement plat et quasicompact* (fpqc).

## Related concepts

[[fpqc-site]] $\to$ [[fppf-site]] $\to$ [[syntomic site]] $\to$ [[étale site]] $\to$ [[Nisnevich site]] $\to$ [[Zariski site]]

category: algebraic geometry
[[!redirects fpqc-topology]]


fractal

#Contents#
* table of contents
{:toc}

## Idea

A _fractal_ is a [[space]] of sorts which is self-similar, to some extent, under rescalings.

There is a concept of [[fractal dimension]] which measures how the amount of detail of a given space changes with scale. For ordinary [[manifolds]] this fractal dimension coincides with the usual [[dimension]], taking values in the [[natural numbers]]. For fractal spaces however the fractal dimension may be a non-negative [[rational number]], in fact a [[real number]], hence a [[fraction]], whence the name "fractal".

## Examples

* [[Cantor space]]

* [[Mandelbrot set]]

* [[Julia set]]

## Related concepts

* [[conformal invariance]]

## References

For [[category theory|category theoretic]] treatments of the self-similarity found in fractals in terms of [[terminal coalgebras]], see

* [[Tom Leinster]], _A general theory of self-similarity_, ([arXiv:1010.4474](https://arxiv.org/abs/1010.4474))

* Prasit Bhattacharya, Lawrence S. Moss, Jayampathy Ratnayake, and Robert Rose, _Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra_, ([pdf](http://www3.nd.edu/~pbhattac/papers/Sierpinski.pdf))

See also

* Wikipedia, _[Fractal](https://en.wikipedia.org/wiki/Fractal)_

[[!redirects fractals]]


fractal dimension

#Contents#
* table of contents
{:toc}

## Idea

A concept of _fractal dimension_ measures how the amount of detail of a given space changes with scale. For ordinary [[manifolds]] this fractal dimension coincides with the usual [[dimension]], taking values in tha [[natural numbers]]. For fractal spaces however the fractal dimension may be a non-negative [[rational number]], in fact a [[real number]], hence a [[fraction]], whence the name "fractal".

There are various precise realizations of the general idea. They include

* [[Hausdorff dimension]]

* ...

## References

* Wikipedia, _[Fractal dimension](https://en.wikipedia.org/wiki/Fractal_dimension)_

[[!redirects fractal dimensions]]


fraction

#Contents#
* table of contents
{:toc}

## Idea

A _fraction_ (or _quotient_) $\frac{x}{y}$ is the expression for the [[ratio]] of the quantity $x$ over $y$, hence for the [[product]] of a quantity $x$ with the multiplicative [[inverse]] of a quantity $y$.

Here $x$ is called the _numerator_ and $y$ is called the _denominator_.

## Related concepts

* [[field of fractions]]

* [[rational number]], [[rational function]]

* [[calculus of fractions]]

* [[localization of a ring]]

[[!redirects fractions]]

[[!redirects numerator]]
[[!redirects numerators]]
[[!redirects denominator]]
[[!redirects denominators]]


fractional D-brane

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#Contents#
* table of contents
{:toc}

## Idea
 {#DBranesAtOrbifoldSingularities}

For [[type II superstrings]] on a global [[orbifold]] [[spacetime]] $X\sslash G$, [[D-brane charge]] is measured by the [[equivariant K-theory]] $K_G(X)$ ([Witten 98, section 5.1](#Witten98)) or at least some [[subgroup]] or [[quotient group]] thereof ([BDHKMMS 01, around (137)](#BDHKMMS01)). Under the canonical map $K(X/G) \longrightarrow K_G(X)$ from the plain [[topological K-theory]] of the [[quotient space]] [[simple object|irreducible elements]] of [[K-theory]] in general decompose into [[direct sums]] of smaller [[simple object|irreducible elements]], hence into "smaller fractions stuck at the orbifold singularity" . The corresponding [[D-branes]] are called _fractional_ D-branes in the literature.

In particular, at least for D-branes at an [[cyclic group|A-type]] singularity $\mathbb{C}^2\sslash (\mathbb{Z}/n)$, hence for $G = \mathbb{Z}/n$ a [[cyclic group]], the [[tension]] (hence [[mass]]) of a fractional D-brane is supposed to be the [[fraction]] $\tfrac{1}{{\vert G\vert}}  = \tfrac{1}{n}$ of that of the unit [[bulk]] brane away from the singularity ([Douglas-Greene-Morrison 97, p.10](#DouglasGreeneMorrison97)).

### At global linear orbifold singularities
 {#AtGlobalLinearOrbifoldSingularities}

If here $X$ is a [[contractible space]] (at it is in the majority of examples discussed in the literature!), then its [[equivariant K-theory]] is equivalently that of the point

$$
  K_G(X)
  \;\simeq\;
  K_G(\ast)
  \;\simeq\;
  R_{\mathbb{C}}(G)
$$

which is [[isomorphism|isomorphic]] to the [[representation ring]] of $G$.

Under this identification, the (non-fractional) unit brane is identified with the [[regular representation]] $k[G/1]$ of $G$:

$$
  \array{
    &&
    K(X) &\longrightarrow & K_G(X)
    \\
    &&
    \simeq && \simeq
    \\
    \mathbb{Z}
    &\simeq&
    K(\ast) &\longrightarrow& K_G(\ast) &\simeq& R_{\mathbb{C}}(G)
    \\
    1 && & \mapsto & && k[G/1]
  }
$$

The non-fractional nature of the [[permutation representation]] is reflected in the fact that its [[character]] is still unity on the [[neutral element]] $e \in G$ and vanishes on other elements (i.e. in the non-trivial [[twisted sectors]]):

$$
  \chi_{k[G/1]}
  \;\colon\;
  g
    \mapsto
  \left\{
  \array{
    1 &\vert& g = e
    \\
    0 &\vert& \text{otherwise}
  }
  \right.
$$

But, by a standard fact from [[representation theory]], the [[regular representation]] is far from [[irreducible representation|irreducible]], instead it is a [[direct sum]] of _all_ existing [[irreducible representations]] $V_i \in R_{\mathbb{C}}(G)$ (each appearing with multiplicity equal to its [[dimension]]):

$$
  k[G/1]
  \;\simeq\;
  \underset{i}{\bigoplus}
  d_i V_i
  \,.
$$

In this sense, the unit brane on $X$ decays into fractional branes at the orbifold  singularities $X\sslash G$.

For a general representation $V \in R(G)$ the general formula (see [this Prop.](character+of+a+linear+representation#NormalizedSumOfCharacters))

$$
  dim\left( V^G \right)
  \;=\;
  \frac{1}{{\vert G\vert}}
  \underset{g \in G}{\sum}
  \chi_V(g)
$$

shows that the total charge of the brane corresponding to $V$, summed over all [[twisted sectors]], is equal to the [[dimension]] of the [[fixed point space]] $V^G$.

### In terms of twisted sector boundary states

In the [[worldsheet]]-description of [[D-branes]] via [[boundary conformal field theory]], fractional D-branes are reflected by [[boundary states]] in "[[twisted sectors]]".
(e.g.  [Diaconescu-Gomis 99](#DiaconescuGomis99), [Recknagel-Schomerus 13, around p. 173](#RecknagelSchomerus13))

### D-branes on resolutions of orbifold singularities

In parts of the string theory literature, fractional D-branes are identified in a "[[duality in string theory|dual]]" formulation of the situation:

At least for $X  \simeq \mathbb{C}^2$ and $G$ a [[finite subgroup of SU(2)]] [[action|acting]] in the canonical way, hence for $X\sslash G$ an [[ADE-singularity]], the K-theoretic [[McKay correspondence]] ([Gonzalez-Sprinberg & Verdier 83](#GSV83)) identifies the [[equivariant K-theory]] of $X$ with the plain [[K-theory]] of a nice [[blow-up]] [[resolution of singularities|resolution]] $\tilde X$:

\[
  \label{KTheoreticMcKayIso}
  \array{
     &&
     && K( X \times_{X/G} \tilde S )
     \\
     &&
     & {}^{\mathllap{ p_1^\ast }}\nearrow && \searrow^{\mathrlap{Inv \circ (p_2)_\ast}}
     \\
     R(G) \simeq K_G(\ast) &\simeq& K_G(X) && \overset{\simeq}{\longrightarrow}  && K(\tilde X)
  }
\]

Under this [[equivalence]] ([[isomorphism]] of [[K-theory]] [[cohomology group|groups]]), fractional D-branes on the [[orbifold]] are identified with [[D-branes]] on $\tilde X$ which are [[wrapped brane|wrapped]] around some of the cycles in $\tilde X$ that appear through the [[blow-up]] of the [[ADE-singularity]] (in physics jargon these are the "vanishing cycles").

### In terms the worldvolume gauge theories
 {#InTermsOfTheWorldvolumeGaugeTheories}

In terms of the [[worldvolume]] [[gauge field theory]] the equivalence (eq:KTheoreticMcKayIso) between

1. fractional D-branes stuck at [[orbifold]] [[singularities]]

1. and [[wrapped branes]] on the [[blow-up]] [[resolution of singularities|resolution]]

is supposed to be exhibited by passage from the [[Higgs branch]] to the [[Coulomb branch]]:

The first key insight is due to [Kronheimer 89](ALE+space#Kronheimer89). He showed that the (resolutions of) the orbifold quotients $\mathbb{C}^2/\Gamma$ for $\Gamma$ a [[finite subgroup of SU(2)]] are precisely the generic form of the gauge orbits of the direct product of $U(n_i)$-s acting in the evident way on the direct sum of $Hom(C^{n_i}, C^{n_j})$-s, where $i$ and $j$ range over the [[vertices]] of the [[Dynkin diagram]], and $(i,j)$ over its [[edges]].

This becomes more illuminating when interpreted in terms of [[Yang-Mills theory|Yang-Mills]] [[gauge theory]]: in a "[[quiver gauge theory]]" the [[gauge group]] is a [[direct product group]] of [[circle group]] $U(n_i)$ factors associated with [[vertices]] of a [[quiver]], and the particles which are charged under this gauge group arrange, as a linear representation, into a [[direct sum]] of $Hom(C^{n_i}, C^{n_j})$-s, for each edge of the quiver.

Pick one such particle, and follow it around as the gauge group transforms it. The space swept out is its gauge orbit, and Kronheimer says that if the quiver is a Dynkin diagram, then this gauge orbit looks like $\mathbb{C}^2/\Gamma$.

On the other extreme, gauge theories are of interest whose gauge group is not a big direct product, but is a [[simple Lie group]], in the technical sense, such as the [[special unitary group]] $SU(N)$ or the [[exceptional Lie group]] $E_8$. The mechanism that relates the two classes of examples is [[spontaneous symmetry breaking]] ("[[Higgs mechanism|Higgsing]]"): the ground state energy of the field theory may happen to be achieved by putting the fields at any one point in a higher dimensional space of field configurations, acted on by the gauge group, and fixing any one such point "spontaneously" singles out the corresponding [[stabilizer subgroup]].

Now here is the final ingredient: it is [[N=2 super Yang-Mills theories]]  ("[[Seiberg-Witten theory]]") which have a potential that is such that its vacua break a simple gauge group such as $SU(N)$ down to a Dynkin diagram quiver gauge theory. One place where this is reviewed, physics style, is  [Albertsson 03, section 2.3.4](N=2+D=4+super+Yang-Mills+theory#Albertsson03).

More precisely, these theories have two different kinds of [[vacuum|vacua]], those on the "[[Coulomb branch]]" and those on the "[[Higgs branch]]" depending on whether the scalars of the "[[vector multiplets]]" (the gauge field sector) or of the "[[hypermultiplet]]" (the matter field sector) vanish. The statement above is for the Higgs branch, but the Coulomb branch is supposed to behave "dually". (see e.g. [Diaconescu-Gomis 99](#DiaconescuGomis99))

So that then finally is the relation, in the [[ADE classification]], between the [[simple Lie groups]] and the [[finite subgroups of SU(2)]]: start with an [[N=2 super Yang-Mills theory]] with [[gauge group]] a [[simple Lie group]]. Let it spontaneously find its vacuum and consider the orbit space of the remaining spontaneously broken symmetry group. That is (a resolution of) the orbifold quotient of $\mathbb{C}^2$ by a [[finite subgroup of SU(2)]].

### Fractional M-branes

An analogous McKay correspondence for (fractional) [[M-branes]]

> graphics grabbed from [HSS18](http://ncatlab.org/schreiber/show/Equivariant+homotopy+and+super+M-branes)

is considered informally in the [[string theory]] literature (for instance in discussion of [[M-theory on G2-manifolds]])
but has not been given a correspondingly precise cohomological formulation yet.

## Properties

### RR-Charge
 {#RRCharge}

Under the identification ([above](#AtGlobalLinearOrbifoldSingularities))

$$
  KU_G^0(\ast) \simeq R_{\mathbb{C}}(G)
$$

of the fractional D-brane charges at a $G$-[[orbifold]] [[singularity]] with of the [[equivariant K-theory]] of the point and hence with the [[representation ring]] of $G$, a [[character]]

$$
  \chi_V \coloneqq tr_V(-) \;\colon\; ConjCl(G) \to \mathbb{C}
$$

of a [[representation]] $V \in R(G)$ correspondes to the [[RR-field]] [[charge]]

\[
  \label{RRChargeOfFractionalDBraneInGTwistedSector}
  Q_V(g) \;=\; \frac{ \chi_V(g) }{ {\vert G \vert} }
\]

of the corresponding fractional D-brane in the $g$-[[twisted sector]].

([Douglas-Greene-Morrison 97, (3.8)](#DouglasGreeneMorrison97), [Diaconescu-Gomis 99 (2.4)](#DiaconescuGomis99), [Billó-Craps-Roose 01, (4.65) with (4.41)](#BilloCrapsRoose01), [EGJ 05, (4.5)](#EGJ05), [Recknagel-Schomerus 13 (4.102)](#RecknagelSchomerus13))

### Tadpole cancellation

See at _[[RR-field tadpole cancellation]]_ the section _[For fractional D-branes](RR-field+tadpole+cancellation#ForFractionalDBranes)_

## Related concepts

* [NS5 half-brane](NS5-brane#NSHalfBranes)

* [[permutation brane]]

* [[fractional M2-brane]]

* [[half M5-brane]]

## References
 {#References}

The concept originates with

* {#DouglasGreeneMorrison97} [[Michael Douglas]], [[Brian Greene]], [[David Morrison]], _Orbifold Resolution by D-Branes_, Nucl.Phys.B506:84-106,1997 ([arXiv:hep-th/9704151](https://arxiv.org/abs/hep-th/9704151))

* Diaconescu, [[Michael Douglas]], [[Jaume Gomis]], _Fractional Branes and Wrapped Branes_, JHEP 9802:013,1998 ([arXiv:hep-th/9712230](http://arxiv.org/abs/hep-th/9712230))

based on the analysis of [[perturbative string theory]] on (global) [[orbifold]] [[background field|backgrounds]] in

* {#DixonHarveyVafaWitten85} [[Lance Dixon]], [[Jeff Harvey]], [[Cumrun Vafa]], [[Edward Witten]], _Strings on orbifolds_, Nuclear Physics B Volume 261, 1985, Pages 678-686 (doi:10.1016/0550-3213(85)90593-0)

* {#DixonHarveyVafaWitten86} [[Lance Dixon]], [[Jeff Harvey]], [[Cumrun Vafa]], [[Edward Witten]], _Strings on orbifolds (II)_, Nuclear Physics B Volume 274, Issue 2, 15 September 1986, Pages 285-314 (doi:10.1016/0550-3213(86)90287-7)

The proposal that D-brane charge on [[orbifolds]] is given by [[equivariant K-theory]] goes back to

* {#Witten98} [[Edward Witten]], section 5.1 of _D-Branes And K-Theory_, JHEP 9812:019,1998 ([arXiv:hep-th/9810188](http://arxiv.org/abs/hep-th/9810188))

but it was pointed out that only a subgroup or quotient group of equivariant K-theory can be physically relevant, in

* {#BDHKMMS01} [[Jan de Boer]], [[Robbert Dijkgraaf]], [[Kentaro Hori]], [[Arjan Keurentjes]], [[John Morgan]], [[David Morrison]], [[Savdeep Sethi]], around (137) of  _Triples, Fluxes, and Strings_, Adv.Theor.Math.Phys. 4 (2002) 995-1186 ([arXiv:hep-th/0103170](https://arxiv.org/abs/hep-th/0103170))

Further discussion in terms of [[equivariant K-theory]]:

* {#GarciaCompean98} Hugo Garcia-Compean, _D-branes in Orbifold Singularities and Equivariant K-Theory_, Nucl.Phys. B557 (1999) 480-504 ([arXiv:hep-th/9812226](https://arxiv.org/abs/hep-th/9812226))

Survey with an eye towards [[string phenomenology]] is in

* {#MalyshevVerlinde07} Dmitry Malyshev, [[Herman Verlinde]], _D-branes at Singularities and String Phenomenology_, Nucl.Phys.Proc.Suppl.171:139-163, 2007 ([arXiv:0711.2451](https://arxiv.org/abs/0711.2451))

The [[McKay correspondence]] as an [[integral transform]] ([[Fourier-Mukai transform]]) in ([[equivariant K-theory|equivariant]]) [[K-theory]], and hence in terms of fractional [[D-brane charge]] is due to

* {#GSV83} [[Gérard Gonzalez-Sprinberg]], [[Jean-Louis Verdier]], Construction géométrique de la correspondance de McKay, Ann. Sci. ́École Norm. Sup.16 (1983) 409–449. ([numdam](http://www.numdam.org/item?id=ASENS_1983_4_16_3_409_0))

Detailed mathematical discussion of fractional D-branes in their incarnation as [[Ext]]-groups of [[coherent sheaves]] is in

* S. Katz, [[Tony Pantev]], [[Eric Sharpe]], _D-branes, orbifolds, and Ext groups_, Nucl.Phys. B673 (2003) 263-300 ([arXiv:hep-th/0212218](https://arxiv.org/abs/hep-th/0212218))

and with relation to [[Bridgeland stability conditions]] in

* {#MalyshevVerlinde07} Dmitry Malyshev, [[Herman Verlinde]], _D-branes at Singularities and String Phenomenology_, Nucl.Phys.Proc.Suppl.171:139-163, 2007 ([arXiv:0711.2451](https://arxiv.org/abs/0711.2451))

Also on stability:

* [[Bogdan Stefanski]], _Dirichlet Branes on a Calabi-Yau Three-fold Orbifold_, Nucl.Phys.B589:292-314, 2000 ([arXiv:hep-th/0005153](https://arxiv.org/abs/hep-th/0005153))

See also

* [[David Berenstein]], Richard Corrado, [[Jacques Distler]], _Aspects of ALE Matrix Models and Twisted Matrix Strings_, Phys. Rev. D 58, 026005 (1998) ([arXiv:hep-th/9712049](https://arxiv.org/abs/hep-th/9712049))

* [[Emil Martinec]], [[Gregory Moore]], bottom of p. 3 of _On Decay of K-theory_ ([arXiv:hep-th/0212059](http://arxiv.org/abs/hep-th/0212059))

Discussion in terms of [[twisted sector]] [[boundary states]] in [[worldsheet]] [[boundary conformal field theory]] includes

* {#DiaconescuGomis99} Diaconescu, [[Jaume Gomis]], _Fractional Branes and Boundary States in Orbifold Theories_, JHEP 0010 (2000) 001 ([arXiv:hep-th/9906242](https://arxiv.org/abs/hep-th/9906242))

* [[Matthias Gaberdiel]], [[Bogdan Stefanski]], _Dirichlet Branes on Orbifolds_, Nucl.Phys.B578:58-84, 2000 ([arXiv:hep-th/9910109](https://arxiv.org/abs/hep-th/9910109))

* M. Frau, A. Liccardo, R. Musto, _The Geometry of Fractional Branes_, Nucl.Phys. B602 (2001) 39-60 ([arXiv:hep-th/0012035](https://arxiv.org/abs/hep-th/0012035))

* {#BilloCrapsRoose01} M. Billó, B. Craps, F. Roose, _Orbifold boundary states from Cardy's condition_, JHEP 0101:038, 2001 ([arXiv:hep-th/0011060](https://arxiv.org/abs/hep-th/0011060))

* {#QuirozStefanski01} N. Quiroz, [[Bogdan Stefanski]], _Dirichlet Branes on Orientifolds_, Phys.Rev. D66 (2002) 026002 ([arXiv:hep-th/0110041](https://arxiv.org/abs/hep-th/0110041))

  (for [[orientifolds]] of orbifolds)

* {#MMR06} Jaydeep Majumder, Subir Mukhopadhyay, Koushik Ray, _Fractional Branes in Non-compact Type IIA Orientifolds_, JHEP0611:008, 2006 ([arXiv:hep-th/0602135](https://arxiv.org/abs/hep-th/0602135))

* {#KrizZayasQuiroz08} [[Igor Kriz]], Leopoldo A. Pando Zayas, Norma Quiroz, _Comments on D-branes on Orbifolds and K-theory_, Int.J.Mod.Phys.A23:933-974, 2008 ([arXiv:hep-th/0703122](https://arxiv.org/abs/hep-th/0703122))

* {#RecknagelSchomerus13} [[Andreas Recknagel]], [[Volker Schomerus]], _Boundary Conformal Field Theory and the Worldsheet Approach to D-branes_, Cambridge 2013 ([spire:1308990](http://inspirehep.net/record/1308990))

Relation to [[permutation branes]]:

* {#EGJ05} Bobby Ezhuthachan, Suresh Govindarajan, T. Jayaraman, _A quantum McKay correspondence for fractional 2p-branes on LG orbifolds_, JHEP 0508 (2005) 050 ([arXiv:hep-th/0504164](https://arxiv.org/abs/hep-th/0504164))

On [[polarized D-brane|polarization]] of fractional D-branes:

* Timothy J. Hollowood, S. Prem Kumar, _World-sheet Instantons via the Myers Effect and $\mathcal{N} = 1^\ast$ Quiver Superpotentials_, JHEP 0210:077, 2002 ([arXiv:hep-th/0206051](https://arxiv.org/abs/hep-th/0206051))

[[!redirects fractional D-branes]]

[[!redirects fractional brane]]
[[!redirects fractional branes]]

[[!redirects half-brane]]
[[!redirects half-branes]]

[[!redirects half brane]]
[[!redirects half branes]]


fractional ideal

Given a commutative [[domain]] $k$, a __fractional ideal__ of $k$ in the quotient field $Q(k)$ is a $k$-submodule $M\subset Q(k)$ such that there exist an element $c\in k$ such that $c M\subset k$.

If $k$ is Noetherian domain, then $c M$ is finitely generated, and hence $M$ is also a finitely generated $k$-module.

Fractional ideals are of importance in [[algebraic number theory]].

[[!redirects fractional ideals]]


fractional integral

* wikipedia [fractional calculus](https://en.wikipedia.org/wiki/Fractional_calculus)
* S. G. Samko, A. A. Kilbas, O. I. Marichev, _Fractional integrals and derivatives_, Gordon and Breach, 1993, transl. from Russian original Nauka i Tehnika, Minsk 1987.
* A. Kilbas, H.M. Srivastava, J.J. Trujillo, _Theory and Applications of Fractional differential equations_, Elsevier, Amsterdam, 2006
* [Fractional calculus](http://www.mathpages.com/home/kmath616/kmath616.htm) at mathpages.com
* [[G. H. Hardy]], [doi](http:/dx.doi.org/10.1112/jlms/s1-20.1.48)
* Thomas Osler, _Leibniz rule for fractional derivatives generalized and an application to infinite series_, SIAM J. Appl. Math., 18 (1970), pp. 658-674 [pdf](http://www.rowan.edu/open/colleges/csm/departments/math/facultystaff/osler/LeibRuleFractDerivGen.pdf);  _Leibniz rule, the chain rule and Taylor's theorem for fractional derivatives,_ PhD thesis, Courant Institute 1970, 87 pp. [pdf](http://www.rowan.edu/open/colleges/csm/departments/math/facultystaff/osler/000%20Oslers%20Doctoral%20Thesis.pdf)
* Fractional integration by parts and hypergeometric integrals, Section 2.9 in G. E. Andrews, [[Richard Askey|R. Askey]], R. Roy, _Special functions_, Enc. of Math. and its Appl. 1999.

[[!redirects fractional derivative]]
[[!redirects fractional calculus]]
[[!redirects Riemann-Liouville operator]]


fractional M2-brane

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{:toc}

## Idea

An [[M5-brane]] [[wrapped brane|wrapped]] on a [[torsion subgroup|torsion]] 3-[[cycle]] is called a _fractional M2-brane_ in [Aharony-Bergman-Jafferis 08, (2.2)](#AharonyBergmanJafferis08).

## Related concepts

* [[3d-3d correspondence]]

* [[discrete torsion]]

* [[fractional D-brane]]

* [[Myers effect]]

## References

* [[Neil Lambert]], [[David Tong]], _Membranes on an Orbifold_, Phys.Rev.Lett.101:041602, 2008 ([arXiv:0804.1114](https://arxiv.org/abs/0804.1114))

* {#AharonyBergmanJafferis08} [[Ofer Aharony]], [[Oren Bergman]], [[Daniel Jafferis]], *Fractional M2-branes*, JHEP 0811:043 (2008) [[arXiv:0807.4924](https://arxiv.org/abs/0807.4924), [doi;10.1088/1126-6708/2008/11/043](https://doi.org/10.1088/1126-6708/2008/11/043)]

[[!redirects fractional M2-branes]]


fractional superstring

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{:toc}

## Idea

The fractional superstrings are supposed to be a family of generalizations of the [[superstring]] that exhibit [[parafermion]] statistics. They were first described in [Argyres and Henry Tye (1991)](#AH91) in an attempt to find other kinds of string theories with lower critical dimension (see [[string theory]]).

A string theory (as a [[sigma-model]]) defines in particular a two-dimensional [[worldvolume]] (i.e. worldsheet) theory with specific [[symmetries|symmetry]]: a [[conformal field theory]] in the bosonic case, and a [[SCFT|superconformal field theory]] in the super- case. The idea is to start from the worldsheet theory and impose particular transformations as symmetries of the theory, motivated by the assumption that any theory defined as such corresponds to a worldsheet theory. These symmetry transformations include generators, or currents, with fractional spins, of which the $\frac{3}{2}$-spin current of the superstring case is an example.

Some consistency checks are described in ([Henry Tye (1993)](#HenryTye93)).

The relevant geometric structure on the worlsheet appears to be an *r-spin* structure (see e.g. ([Randal-Williams (2010)](#Randal-Williams10)) ([Runkel and Szegedy (2018)](#RS18))).

## References

### General

* {#AH91} [[Philip Argyres]], and [[S.H. Henry Tye]]. *Fractional Superstrings with Space-Time Critical Dimensions Four and Six*. (1991). [arXiv:hep-th/9109001v2](https://arxiv.org/abs/hep-th/9109001v2)

* {#HenryTye93} [[S.H. Henry Tye]]. *Fractional Superstrings with Space-Time Critical Dimensions Four and Six*. (1993). [arXiv:hep-th/9311021](https://arxiv.org/abs/hep-th/9311021)

* [[Philip Argyres]]. *A three-dimensional fractional string*. (2005) [doi](https://doi.org/10.1007/3-540-59163-X_256)

On the heterotic parafermionic superstring and its potential anomalies

* Paul Frampton, and James Liu. *Heterotic Parafermionic Superstring*. (1992). [arXiv:hep-th/9210049](https://arxiv.org/abs/hep-th/9210049)

Further developments

* Ahmed El Fallah, El Hassane Saidi and Rainer Dick. *On generalized fractional superstring models and the associative division algebras*. (2000). [doi](https://doi.org/10.1088/0264-9381/17/1/305)

[[!redirects parafermionic superstring]]

On r-spin structures

* {#Randal-Williams10} [[Oscar Randal-Williams]]. *Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces*. (2010) [arXiv:1001.5366](https://arxiv.org/abs/1001.5366)

* {#RS18} [[Ingo Runkel]], Lóránt Szegedy. *Topological field theory on r-spin surfaces and the Arf invariant*. (2018). [arXiv:1802.09978](https://arxiv.org/abs/1802.09978)


fractional-level WZW model -- references

### On fractional-level WZW models as logarithmic CFTs
 {#FractionalLevelWZWModelReferences}

On [[WZW models]] at [[rational number|fractional]] [[level (Chern-Simons theory)|level]]:

* [[Sunil Mukhi]], Sudhakar Panda, *Fractional-level current algebras and the classification of characters*, Nuclear Physics B **338** 1 (1990) 263-282 $[$doi:10.1016/0550-3213(90)90632-N$]$

* {#MooreRead91} [[Gregory Moore]], [[Nicholas Read]], p. 389 of: *Nonabelions in the fractional quantum hall effect*, Nuclear Physics B **360** 2–3 (1991) 362-396 $[$doi:10.1016/0550-3213(91)90407-O, [pdf](https://www.physics.rutgers.edu/~gmoore/MooreReadNonabelions.pdf)$]$

  > (suggesting the fractional level $k = -5/4$ as related to [[Laughlin wavefunctions]] of [[anyons]])

* [[Hidetoshi Awata]], [[Yasuhiko Yamada]], *Fusion rules for the fractional level $\widehat{\mathfrak{sl}(2)}$ algebra*,     Mod. Phys. Lett. A **7** (1992) 1185-1196 $[$[spire:332974](https://inspirehep.net/literature/332974), [doi:10.1142/S0217732392003645](https://doi.org/10.1142/S0217732392003645)$]$

* P. Furlan, [[A. Ch. Ganchev]], R. Paunov, [[Valentina B. Petkova]], *Solutions of the Knizhnik-Zamolodchikov Equation with Rational Isospins and the Reduction to the Minimal Models*, Nucl. Phys. B394 (1993) 665-706 ([arXiv:hep-th/9201080](https://arxiv.org/abs/hep-th/9201080), doi:10.1016/0550-3213(93)90227-G)

* J. L. Petersen, J. Rasmussen, M. Yu, *Fusion, Crossing and Monodromy in Conformal Field Theory Based on $SL(2)$ Current Algebra with Fractional Level*, Nucl. Phys. B481 (1996) 577-624 ([arXiv:hep-th/9607129](https://arxiv.org/abs/hep-th/9607129), doi:10.1016/S0550-3213(96)00506-8)

* [[Boris Feigin]], [[Feodor Malikov]], *Modular functor and representation theory of $\widehat{\mathfrak{sl}_2}$ at a rational level*, p. 357-405 in: Loday, Stasheff, Voronov (eds.) *Operads: Proceedings of Renaissance Conferences*,  Contemporary Mathematics **202**, AMS (1997) $[$[arXiv:q-alg/9511011](https://arxiv.org/abs/q-alg/9511011), [ams:conm-202](https://bookstore.ams.org/conm-202)$]$

  > (an [[osp|$\mathfrak{osp}(1\vert2)$]]-factor appears)

with a good review in:

* [[A. Ch. Ganchev]], [[Valentina B. Petkova]], G. M. T. Watts, *A note on decoupling conditions for generic level $\widehat{s l}(3)_k$ and fusion rules*, Nucl. Phys. B **571** (2000) 457-478 $[$doi:10.1016/S0550-3213(99)00745-2$]$

On [[braided fusion categories]] formed by [[affine Lie algebra]]-[[representations]] at admissible fractional [[level (Chern-Simons theory)|level]]:

* [[Thomas Creutzig]], [[Yi-Zhi Huang]], [[Jinwei Yang]], *Braided tensor categories of admissible modules for affine Lie algebras*, Commun. Math. Phys. **362** (2018) 827–854 $[$[arXiv:1709.01865](https://arxiv.org/abs/1709.01865)$]$

On interpreting fractional level [[WZW models]] as [[logarithmic CFT|logarithmic]] [[2d CFT|CFTs]]:

* {#Gaberdiel01} [[Matthias R. Gaberdiel]], *Fusion rules and logarithmic representations of a WZW model at fractional level*, Nucl. Phys. B **618** (2001) 407-436 $[$[arXiv:hep-th/0105046](https://arxiv.org/abs/hep-th/0105046), doi:10.1016/S0550-3213(01)00490-4)$]$

* {#Gaberdiel01} [[Matthias R. Gaberdiel]], Section 5 of: *An algebraic approach to logarithmic conformal field theory*, Int. J. Mod. Phys. A **18** (2003) 4593-4638 $[$[arXiv:hep-th/0111260](https://arxiv.org/abs/hep-th/0111260), [doi:10.1142/S0217751X03016860](https://doi.org/10.1142/S0217751X03016860)$]$

* [[David Ridout]], *$\widehat{\mathfrak{sl}}(2)_{-1/2}$: A Case Study*, Nucl. Phys. B **814** (2009) 485-521 $[$[arXiv:0810.3532](https://arxiv.org/abs/0810.3532), [doi:10.1016/j.nuclphysb.2009.01.008](https://doi.org/10.1016/j.nuclphysb.2009.01.008)$]$

* [[Thomas Creutzig]], [[David Ridout]], *Modular Data and Verlinde Formulae for Fractional Level WZW Models I*, Nuclear Physics B **865** 1 (2012) 83-114 $[$[arXiv:1205.6513](https://arxiv.org/abs/1205.6513), [doi:10.1016/j.nuclphysb.2012.07.018](https://doi.org/10.1016/j.nuclphysb.2012.07.018)$]$

* [[Thomas Creutzig]], [[David Ridout]], *Modular Data and Verlinde Formulae for Fractional Level WZW Models II*, Nuclear Physics B **875** 2 (2013) 423-458 $[$[arXiv:1306.4388](https://arxiv.org/abs/1306.4388), [doi:10.1016/j.nuclphysb.2013.07.008](https://doi.org/10.1016/j.nuclphysb.2013.07.008)$]$

* [[Thomas Creutzig]], [[David Ridout]], Section 4 of: *Logarithmic conformal field theory: beyond an introduction*,  J. Phys. A: Math. Theor. **46** (2013) 494006 ([doi:10.1088/1751-8113/46/49/494006](https://iopscience.iop.org/article/10.1088/1751-8113/46/49/494006), [arXiv:1303.0847](https://arxiv.org/abs/1303.0847))

* [[Kazuya Kawasetsu]], [[David Ridout]], *Relaxed highest-weight modules I: rank 1 cases*, Commun. Math. Phys. **368** (2019) 627–663  $[$[arXiv:1803.01989](https://arxiv.org/abs/1803.01989), [doi:10.1007/s00220-019-03305-x](https://doi.org/10.1007/s00220-019-03305-x)$]$

* [[Kazuya Kawasetsu]], [[David Ridout]], *Relaxed highest-weight modules II: classifications for affine vertex algebras*, Communications in Contemporary Mathematics, **24** 05 (2022) 2150037 $[$[arXiv:1906.02935](https://arxiv.org/abs/1906.02935), [doi:10.1142/S0219199721500371](https://doi.org/10.1142/S0219199721500371)$]$

Reviewed in:

* {#Ridout10} [[David Ridout]], *Fractional Level WZW Models as Logarithmic CFTs* (2010) $[$[pdf](https://researchers.ms.unimelb.edu.au/~dridout@unimelb/seminars/100225.pdf), [[Ridout-FractionalLevelWZW2010.pdf:file]]$]$

* {#Ridout20} [[David Ridout]], *Fractional-level WZW models* (2020) $[$[pdf](https://researchers.ms.unimelb.edu.au/~dridout@unimelb/seminars/200205.pdf), [[Ridout-FractionalLevelWZW2020.pdf:file]]$]$

In particular, the logarithmic $c = -2$ model is essentially an admissible-level WZW model (namely at level $k = 0$):

* {#Nichols02} Alexander Nichols, *Extended chiral algebras in the $SU(2)_0$ WZNW model*,  JHEP 04 (2002) $[$[doi:10.1088/1126-6708/2002/04/056](https://iopscience.iop.org/article/10.1088/1126-6708/2002/04/056), [arXiv:hep-th/0112094](https://arxiv.org/abs/hep-th/0112094)$]$

with a comprehensive account in:

* {#NicholsThesis02} Alexander Nichols, *$SU(2)_k$ Logarithmic Conformal Field Theories*, PhD thesis, Oxford (2002) $[$[arXiv:hep-th/0210070](https://arxiv.org/abs/hep-th/0210070), [spire:599081](https://inspirehep.net/literature/599081)$]$

On the $c = -1$ model as the $\mathfrak{su}(2)$ [[WZW model]] at fractional level $-1/2$ and relation to the [[beta-gamma system]]:

* {#LesageMathieuRasmussenSaleur02} F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur, *The $\mathfrak{su}(2)_{-1/2}$ WZW model and the beta-gamma system*, Nucl. Phys. B **647** (2002) 363-403  $[$[arXiv:hep-th/0207201](https://arxiv.org/abs/hep-th/0207201), doi:10.1016/S0550-3213(02)00905-7a$]$

and its lift to a [[logarithmic CFT]]:

* {#LesageMathieuRasmussenSaleur04} F. Lesage, P. Mathieu, J. Rasmussen, H. Saleur, *Logarithmic lift of the $\mathfrak{su}(2)_{-1/2}$ model*, Nuclear Physics B **686** 3 (2004) 313-346 [[doi:10.1016/j.nuclphysb.2004.02.039](https://doi.org/10.1016/j.nuclphysb.2004.02.039)]

On quasi-characters at fractional level:

* [[Sachin Grover]], *Quasi-Characters in $\widehat{\mathfrak{su}}(2)$ Current Algebra at Fractional Levels*, SciPost Phys. Core **6** 068 (2023)  $[$[arXiv:2208.09037](https://arxiv.org/abs/2208.09037), [doi:10.21468/SciPostPhysCore.6.4.068](https://scipost.org/10.21468/SciPostPhysCore.6.4.068)$]$

Identification of would-be fractional level [[su(2)|$\mathfrak{su}(2)$]] [[conformal blocks]] in [[twisted equivariant K-theory]] of [[configuration spaces of points]]:

* [[Hisham Sati]], [[Urs Schreiber]], Rem. 2.3 in: *[[schreiber:Anyonic defect branes in TED-K-theory]]*, Rev. Math. Phys. **35** 06 (2023) 2350009 $[$[arXiv:2203.11838](https://arxiv.org/abs/2203.11838), [doi:10.1142/S0129055X23500095](https://doi.org/10.1142/S0129055X23500095)$]$


fracture theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
 {#Idea}

A basic fact in [[number theory]] is that the [[natural numbers]] may be decomposed into the [[rational numbers]] and the [[p-adic integers]] for all [[prime numbers]] $p$. Dually  in [[arithmetic geometry]] this says that [[Spec(Z)]] has a [[cover]] by all its [[formal disks]] and the complements of finitely many points, a fact that is crucial in the geometric interpretation of the [[function field analogy]] and which motivates for instance the [[geometric Langlands correspondence]]. (See [below](#ArithmeticFractureSquares).)

Lifting this statement to [[stable homotopy theory]] and "[[higher arithmetic geometry]]" the _arithmetic fracture theorem_ says that [[stable homotopy types]] (and suitably tame plain [[homotopy types]]) canonically decompose into their [[rationalization]] and their [[p-completion]] for all primes $p$, hence into their images in [[rational homotopy theory]] and [[p-adic homotopy theory]]. Since these images are typically simpler than the original homotopy type itself, this decomposition is a fundamental computational tool in stable homotopy theory, often known under the slogan of "working one prime at a time". (See [below](#TheArithmeticFractureSquare).)

One finds that this arithmetic fracturing in stable homotopy theory is really a statement about the [[Bousfield localization of spectra]] with respect to the [[Moore spectrum]] for $\mathbb{Q}$ and that of $\mathbb{Q}/\mathbb{Z}$. Viewed this way there is a more general fracture theorem which says that for any suitable pair $E,F$ of [[spectra]]/[[homology theories]] the [[Bousfield localization of spectra|Bousfield localization]] at their [[coproduct]] decomposes into the separate Bousfield localizations. This generalized fracture theorem appears for instance in [[chromatic homotopy theory]] for localization at [[Morava K-theory]] and [[Morava E-theory]]. (See [below](#GeneralFractureSquares).)

In [[cohesive homotopy theory]] every [[stable homotopy type]] canonically sits in a fracture square formed from the localizations exhibited by the [[shape modality]] and the [[flat modality]]. For [[differential cohesion]] over [[infinitesimal cohesion]] this is a [[higher geometry|higher geometric]] analog of the classical arithmetic fracture. (See [below](#InCohesiveHomotopyTheory).)

## Statement
 {#Statement}

### In number theory and arithmetic geometry
 {#ArithmeticFractureSquares}

The  statement in [[number theory]]/[[arithmetic geometry]] is the following:

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

The [[integers]] $\mathbb{Z}$ are the [[fiber product]] of all the [[p-adic integers]] $\underset{p\;prime}{\prod} \mathbb{Z}_p$ with the [[rational numbers]] $\mathbb{Q}$ over the [[rationalization]] of the former, hence there is a [[pullback]] [[diagram]] in [[CRing]] of the form

$$
  \array{
     && \mathbb{Q}
     \\
     & \swarrow && \nwarrow
     \\
     \mathbb{Q}\otimes_{\mathbb{Z}}\left(\underset{p\;prime}{\prod} \mathbb{Z}_p \right) && && \mathbb{Z}
     \\
     & \nwarrow && \swarrow
     \\
     && \underset{p\;prime}{\prod} \mathbb{Z}_p
  }
  \,.
$$

Equivalently this is the [[fiber product]] of the rationals with the [[integral adeles]] $\mathbb{A}_{\mathbb{Z}}$ over the [[ring of adeles]] $\mathbb{A}_{\mathbb{Q}}$

$$
  \array{
     && \mathbb{Q}
     \\
     & \swarrow && \nwarrow
     \\
     \mathbb{A}_{\mathbb{Q}} && && \mathbb{Z}
     \\
     & \nwarrow && \swarrow
     \\
     && \mathbb{A}_{\mathbb{Z}}
  }
  \,,
$$

Since the [[ring of adeles]] is the [[rationalization]] of the integral adeles $\mathbb{A}_{\mathbb{Q}} = \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{A}_{\mathbb{Z}}$, this is also (by the discussion [here](category+of+monoids#PushoutOfCommutativeMonoids)) a [[pushout]] diagram in [[CRing]], and in fact in topological [[commutative rings]] (for $\mathbb{Q}$ with the [[discrete topology]] and $\mathbb{A}_{\mathbb{Z}}$ with its profinite/[[completion of a ring|completion]] topology).

=--

An original discussion is ([Sullivan 05, prop. 1.18](#Sullivan05)). Review includes ([Riehl 14, lemma 14.4.2](#Riehl14)).

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

Under the [[function field analogy]] we may think of

* $Spec(\mathbb{Z})$ as an [[arithmetic curve]] over [[F1]];

* $\mathbb{A}_{\mathbb{Z}}$ as the [[ring of functions]] on the [[formal disks]] around all the points in this curve;

* $\mathbb{Q}$ as the ring of functions on the complement of a finite number of points in the curve;

* $\mathbb{A}_{\mathbb{Q}}$ is the ring of functions on punctured formal disks around all points, at most finitely many of which do not extend to the unpunctured disk.

Under this [[analogy]] the arithmetic fracture square of prop. \ref{ArithmeticFractureSquare} says that the curve $Spec(\mathbb{Z})$ has a [[cover]] whose patches are the complement of the curve by some points, and the formal disks around these points.

This kind of cover plays a central role in [[number theory]], see for instance the following discussions:

* _[Moduli stack of bundles over curves](moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence)_;

* _[[geometric Langlands correspondence]]_;

* _[[Weil conjecture on Tamagawa numbers]]_.

=--

### In homotopy theory
 {#InHomotopyTheory}

In [[homotopy theory]] the corresponding statement is that [[homotopy types]] may be decomposed into that of [[rational homotopy types]] and [[p-complete homotopy types]] of [[p-local homotopy types]].

+-- {: .num_remark #FractureForSpectra}
###### Proposition

Let $p$ be a [[prime number]]. Let $X$ be a [[homotopy type]]/[[∞-groupoid]] satisfying at least one of the following sufficient conditions

* $X$ is a [[connected]], [[nilpotent space]] with finitely generated [[homotopy groups]];

* $X$ is [[p-local homotopy type]];

Then $X$ is the [[homotopy fiber product]]

$$
  X \simeq X_{\mathbb{Q}} \underset{(X_p^\wedge)_{\mathbb{Q}}}{\times} X_p^\wedge
$$

of its [[rationalization]] $X_{\mathbb{Q}}$ with its [[p-completion]] $X_p^\wedge$ over the rationalization $(X_p^\wedge)_{\mathbb{Q}}$of the $p$-completions.

=--

This originates around ([Bousfield-Kan 72, VI.8.1](#BousfieldKan72)). A detailed more modern account is in ([May-Ponto, theorem 13.1.4](#MayPonto)). A quick survey is in ([Riehl 14, theorem 14.4.14](#Riehl14)).

### In stable homotopy theory
 {#InStableHomotopyTheory}

Similar statements hold in [[stable homotopy theory]] for [[spectra]]. There is a stable version of

* [The arithmetic fracture square](#TheArithmeticFractureSquare)

but more generally there are fracture squares for the [[coproduct]] homology theory $E \vee F$ whenever $F$-localization is $E$-acyclic:

* [General fracture squares](#GeneralFractureSquares)

> warning: a condition missing in the following, see the comment section of [this MO reply](https://mathoverflow.net/a/91057/381). Somebody should add the relevant clause here...

#### The arithmetic fracture square for spectra
 {#TheArithmeticFractureSquare}

For $p$ a [[prime number]] write

* $L_p$ for [[Bousfield localization of spectra]] at the [[Moore spectrum]] $S \mathbb{F}_p$, hence for [[p-completion]] $(-)_p^\wedge$;

* $L_{\mathbb{Q}}$ for the [[Bousfield localization of spectra]] at the [[Moore spectrum]]/[[Eilenberg-MacLane spectrum]] $S \mathbb{Q} \simeq H \mathbb{Q}$, hence for [[rationalization]].

+-- {: .num_prop #SullivanArithmeticFracture}
###### Proposition
**(Sullivan arithmetic square)**

For every [[spectrum]] $X$ the canonical square

$$
  \array{
    && L_{\mathbb{Q}}X
    \\
    & \swarrow && \nwarrow
    \\
    L_{\mathbb{Q}}
    \left(
      \prod_p L_p X
    \right)
    &&  &&
    X
    \\
     & \nwarrow && \swarrow
    \\
    && \prod_p L_p X
  }
$$

is a [[homotopy pushout]] (hence also a [[homotopy pullback]]).

=--

Original statements of this include ([Bousfield 79](#Bousfield79), [Sullivan 05, prop. 3.20](#Sullivan05)). Review includes ([van Koughnett 13, prop. 4.5](#VanKoughnett13), [Bauer 11, lemma 2.1](#Bauer11)).

+-- {: .num_remark }
###### Remark
**("one prime at a time")**

The impact of prop. \ref{FractureForSpectra} is that it decomposes the study of ([[stable homotopy theory|stable]]) [[homotopy theory]] into that of

1. [[rational homotopy theory]]  and

1. [[p-adic homotopy theory]] for each prime $p$.

Both the [[rationalization]] $X_{\mathbb{Q}}$ and the [[p-completion]] $X_{p}^\wedge$ are typically much easier to analyze than $X$ itself and so the fracture theorem gives a way to decompose the remaining hard part of study of [[homotopy types]] into that of $p$-local/$p$-complete spaces.
This procedure is known in homotopy theory as working "one prime at a time".

=--

More generally:

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

The product of all [[p-completions]] is equivalently the [[Bousfield localization of spectra]] at the [[wedge sum]] $\vee_p S \mathbb{F}_p$ of all [[Moore spectra]]

$$
  \prod_p L_p X
  \simeq
   L_{\vee_p S \mathbb{F}_p} X
  \,.
$$

Moreover there is a  [[Bousfield equivalence]]

$$
  S (\mathbb{Q}/\mathbb{Z}) \simeq_{Bousf} \vee_p S \mathbb{F}_p
  \,,
$$

and therefore also an equivalence

$$
  \prod_p L_p X
  \simeq
   L_{S (\mathbb{Q}/\mathbb{Z})} X
  \,.
$$

=--

The first statement originates around ([Bousfield 79, prop. 2.6](Bousfield+localization+of+spectra#Bousfield79)), review includes ([van Koughnett 13, prop. 4.4](Bousfield+localization+of+spectra#VanKoughnett13), [Bauer 11, below prop. 2.2](#Bauer11)); the second is highlighted in ([Strickland 12, MO comment](http://mathoverflow.net/a/91057/381)).

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

By the discussion at _[[Bousfield localization of spectra]]_ and at _[[localization of a space]]_, the [[rationalization]] and the [[p-completion]] maps on spectra are [[homotopy cofibers]] of $E$-acyclifications $G_E(X) \to X$, for $E = S \mathbb{Q} \simeq H \mathbb{Q}$ and $E = S \mathbb{F}_p$  the [[Moore spectra]] of $\mathbb{Q}$ and of the [[cyclic group]]/[[finite field]] $\mathbb{F}_p = \mathbb{Z}/(p)$, respectively (e.g. [Lurie 10, lecture 20](localization+of+a+space#Lurie)).

Including this into the statement of prop. \ref{FractureForSpectra} says that for spectra $X$ satisfying sufficient conditions as above, then the canonical diagram

$$
  \array{
    && X_{\mathbb{Q}} && \longleftarrow && G_{S (\mathbb{Q}/\mathbb{Z})}(X)
    \\
    & \swarrow && \nwarrow && \swarrow
    \\
    (\prod_p X_p^\wedge)_{\mathbb{Q}} && && X
    \\
    & \nwarrow && \swarrow && \nwarrow
    \\
    && \prod_p X_p^\wedge && \longleftarrow && G_{H\mathbb{Q}}(X)
  }
$$

is homotopy exact, in that

1. the square is a [[homotopy pullback]] and hence also a [[homotopy pushout]] (this is prop. \ref{ReformulationOfProdOverPComletionByLocalizationAtCoproduct});

1. the diagonals are [[homotopy cofiber sequences]] and hence also [[homotopy fiber sequences]] (by [this proposition](Bousfield+localization+of+spectra#LocalizationCofiber) at _[[Bousfield localization of spectra]]_);

1. the top and bottom outer composite sequences are [[homotopy fiber sequences]] (and hence [[homotopy cofiber sequences]]) (by applying the [[pasting law]] to the previous two items).

Together this is like two-thirds of a [[differential cohomology hexagon]] -- to the extent that $p$-[[adic completion]] is adjoint to $p$-[[torsion approximation]]. This is indeed the case, as the next proposition asserts

Notice that in view of remark \ref{GeometricMeaning} then $X_p^\wedge$ is like the restriction of $X$ from [[Spec(Z)]] to all [[formal disks]] around the points $(p)$, and hence $G_{S\mathbb{F}_p}$ is like the restriction to the "complement of all formal disks". Finally $X_{\mathbb{Q}}$ may be understood as the restriction to the [[Ran space]] of $Spec(\mathbb{Z})$ ([Gaitsgory 11](#Gaitsgory11)), roughly the colimit of the restriction of $X$ to the complement of finitely many points, as this set of points ranges through all points.

=--

In view of remark \ref{TwoThirdHexagon} we may regard the following fact as an refinement of the traditional arithmetic fracture theorem.

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

Let $A$ be an [[E-∞ ring]] and let $\mathfrak{a} \subset \pi_0 A$ be a [[generators and relations|finitely generated]] ideal in its underlying [[commutative ring]].

Then there is an [[adjoint triple]] of [[adjoint (∞,1)-functors]]

$$
  \array{

    \underoverset{
      A Mod_{\mathfrak{a}comp}^{op}}
    {A Mod_{\mathfrak{a}tors}^{op}}
    {\simeq}
    &\stackrel{\overset{\Pi_{\mathfrak{a}}}{\longleftarrow}}{\stackrel{\hookrightarrow}{\underset{\flat_{\mathfrak{a}}}{\longleftarrow}}}&
    A Mod^{op}
  }
$$

where

* $A Mod$ is the [[stable (∞,1)-category|stable]] [[(∞,1)-category of modules]], i.e. of [[∞-modules]] over $A$;

* $A Mod_{\mathfrak{a}tors}$ and $A Mod_{\mathfrak{a} comp}$ are the [[full sub-(∞,1)-categories]] of $\mathfrak{a}$-[[torsion approximation|torsion]] and of $\mathfrak{a}$-[[completion of a module|complete]] $A$-[[∞-modules]], respectively;

* $(-)^{op}$ denotes the [[opposite (∞,1)-category]];

* the [[equivalence of (∞,1)-categories]] on the left is induced by the restriction of $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$.

=--

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

This is effectively the content of ([Lurie "Proper morphisms", section 4](#LurieProper)):

* the existence of $\Pi_{\mathfrak{a}}$ is corollary 4.1.16 and remark 4.1.17

* the existence of $\flat_{\mathfrak{a}}$ is lemma 4.2.2 there;

* the equivalence of sub-$\infty$-categories is proposition 4.2.5 there.

=--

+-- {: .num_cor #FractureFromCohesion}
###### Corollary

The traditional arithmetic fracture square of prop. \ref{SullivanArithmeticFracture}, regarded as in remark \ref{TwoThirdHexagon}, is the left part of the "[differential cohomology diagram](differential+cohomology+diagram#TheHexagonDiagram)" induced by the [[adjoint modality]] $(\Pi_{\mathfrak{a}} \dashv \flat_{\mathfrak{a}} )$ of prop. \ref{CompletionTorsionAdjointModalityForModuleSpectra}, for the special case that $X = \mathbb{S}$ is the [[sphere spectrum]] and $\mathfrak{a} = (p)$ a [[prime ideal]]

$$
  \array{
    &&  \Pi_{\mathfrak{a}dR} X && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{\mathfrak{a}dR} X
    \\
    & \nearrow & & \searrow & & \nearrow && \searrow
    \\
    \Pi_{\mathfrak{a}dR} \flat  X  && \Downarrow && X && \Downarrow && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X
    \\
    & \searrow &  & \nearrow & & \searrow && \nearrow
    \\
    && \flat_{\mathfrak{a}} X && \longrightarrow && \Pi_{\mathfrak{a}} X
  }
  \,,
$$

[[!include arithmetic cohesion -- table]]

=--

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

The special case of prop. \ref{CompletionTorsionAdjointModalityForModuleSpectra} where $A$ is the [[Eilenberg-MacLane spectrum]] of a plain [[commutative ring]], and hence -- by the [[stable Dold-Kan correspondence]] -- the case where $A$-[[∞-modules]] are equivalently [[chain complexes]], has a longer tradition in the existing literature. This we highlight separately [below](#CompletionAndTorsionOnDerivedCategories).

=--

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

By the discussion at _[[completion of modules]]_ in the section _[Monoidalness](completion+of+a+module#Monoidalness)_, the [[adjoint modality]] of prop. \ref{CompletionTorsionAdjointModalityForModuleSpectra} is a [[monoidal (∞,1)-functor]] without, possibly, respect the [[tensor unit]] in $A Mod$. This means that $(\Pi_{\mathfrak{a}}\dashv \flat_{\mathfrak{a}})$ passes to "[[commutative monoid in a symmetric monoidal (∞,1)-category|commutative ∞-monoids]]-without unit" in $A Mod$, hence to ([[Isbell duality|formal duals of]]) [[nonunital E-∞ algebras]].
By [this proposition](nonunital+Ek-algebra#RelationToAugmentedEkAlgebras) ([Lurie "Algebra", prop. 5.2.3.15](#LurieAlgebra)) [[nonunital E-∞ rings]]  are [[equivalence of (∞,1)-categories|equivalent]] to [[augmented E-∞ rings]] over the [[sphere spectrum]], hence this is [[E-∞ arithmetic geometry]] under $Spec(\mathbb{S})$.

Notice that in addition $\Pi_{\mathfrak{a}}$ here should preserve [[finite products]] (because by the discussion at  [completion of a module -- monoidalness](completion%20of%20a%20module#Monoidalness) the underlying $\Pi_{\mathfrak{a}} \colon A Mod \to A Mod$ preserves all small [[(∞,1)-colimits]] and because by
[this proposition](commutative+monoid+in+a+symmetric+monoidal+%28infinity%2C1%29-category#LimitsInCRing) finite coproducts in $CRng(A Mod)$ are computed in the underlying $A Mod$.

Therefore we may think of $\Pi_{\mathfrak{a}}$ as a [[shape modality]] and of $\flat_{\mathfrak{a}}$ as a [[sharp modality]] on affine [[E-∞ geometry|E-∞]]-[[arithmetic geometry]] under $Spec(\mathbb{S})$ -- namely on [[Isbell duality|formal duals]] of [[nonunital E-∞ rings]] .

(It may be entertaining to note that on the level of [[∞-groups of units]] then [[E-∞ arithmetic geometry]] under $Spec(\mathbb{S})$ translates to [[abelian ∞-groups]] of [[twisted cohomology|twists]] over the [[sphere spectrum]] -- which has been argued to be the homotopy-theoretic incarnation of [[superalgebra]], see at _[superalgebra -- abstract idea](super+algebra#AbstractIdea)_ for more on this.)

=--

For more discussion of this see also _[[differential cohesion and idelic structure]]_.

#### The arithmetic fracture square for chain complexes
 {#CompletionAndTorsionOnDerivedCategories}

We discuss here arithmetic fracturing on [[chain complexes]] of modules. Under the [[stable Dold-Kan correspondence]] this is a special case of prop. \ref{CompletionTorsionAdjointModalityForModuleSpectra}, but this special case has a longer tradition in the literature -- going back to _Grothendieck local duality_  and sometimes called _Greenlees-May duality_ due to ([Greenlees-May 92](#GreenlessMay92)) -- and we point to these original proofs.

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

Let $A$ be a [[commutative ring]], let $\mathfrak{a} \subset A$ be be an ideal generated by a single regular element (i.e. not a zero divisor). Write $A Mod_{\infty}^{op}$ for the [[opposite (∞,1)-category]] of the [[(∞,1)-category of modules]] over $A$.

Write

* $\flat_{\mathfrak{a}}\colon A Mod_\infty^{op} \to A Mod_{\infty}^{op}$ for the [[derived functor]] of [[formal completion]] (adic completion) of modules at $\mathfrak{a}$;

  with canonical natural transformation

  $\epsilon_{\mathfrak{a}} \colon \flat_{\mathfrak{a}} \longrightarrow id$

* $\Pi_{\mathfrak{a}} \colon A Mod_\infty^{op} \to A Mod_\infty^{op}$ for the total [[derived functor]] of the $\mathfrak{a}$-[[torsion approximation]]-functor;

  with canonical natural transformation

  $\eta_{\mathfrak{a}}\colon id \longrightarrow \Pi_{\mathfrak{a}} $

Finally write

$$
  (A Mod_\infty^{op})^{\mathfrak{a}com}, (A Mod_\infty^{op})^{\mathfrak{a}tor} \hookrightarrow A Mod_\infty
$$

for the [[full (∞,1)-subcategories]] of objects $X$ for which, $\epsilon_{\mathfrak{a}}(X)$ or $\eta_{\mathfrak{a}}(X)$ is an [[equivalence in an (∞,1)-category]], respectively.

=--

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

1. The transformation $\epsilon_{\mathfrak{a}}$ exhibits $(A Mod_\infty^{op})^{\mathfrak{a}com}\hookrightarrow A Mod_\infty$ as a [[reflective (∞,1)-subcategory]], hence $\flat_{\mathfrak{a}}$ as an [[idempotent (∞,1)-monad]].

1. The transformation $\eta_{\mathfrak{a}}$ exhibits $(A Mod_\infty^{op})^{\mathfrak{a}tor}\hookrightarrow A Mod_\infty$ as a co-reflective $(\infty,1)$-category, hence $\Pi_{\mathfrak{a}}$ as an idempotent $(\infty,1)$-comonad.

1. Restricted to these sub-$(\infty,1)$-categories both $\flat_{\mathfrak{a}}$ as well as $\Pi_{\mathfrak{a}}$ become [[equivalences of (∞,1)-categories]], hence exhibiting $(\Pi_{\mathfrak{a}} \dashv \flat_{\mathfrak{a}})$ as a higher [[adjoint modality]].

=--

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

This is a paraphrase of the results in ([Dwyer-Greenlees 99](#DwyerGreenlees99)) and ([Porta-Shaul-Yekutieli 10](#PortaShaulYekutieli10)) from the language of [[derived categories]] to [[(∞,1)-category theory]].

First of all, by our simplifying assumption that $\mathfrak{a}$ is generated by a single regular element, the running assumption of "weak proregularity" in ([Porta-Shaul-Yekutieli 10, def.3.21](#PortaShaulYekutieli10)) is satisfied.

Then in view of ([Porta-Shaul-Yekutieli 10, corollary 3.31](#PortaShaulYekutieli10)) the statement of ([Porta-Shaul-Yekutieli 10, theorem 6.12](#PortaShaulYekutieli10)) is the [characterization of reflectors](reflective+sub-(infinity,1)-category#CharacterizationOfReflectors) as discussed at _[[reflective sub-(∞,1)-category]]_, and formally dually so for the coreflection. With the fully faithfulness that goes with this the equivalence of the two inclusions on the level of [[homotopy categories]] given by ([Hovey-PalieriS-trickland 97, 3.3.5](#HoveyPalieriStrickland97), [Dwyer-Greenlees 99, theorem 2.1](#DwyerGreenlees99) [Porta-Shaul-Yekutieli 10, theorem 6.11](#PortaShaulYekutieli10)) implies the canonical equivalence of the two [[sub-(∞,1)-categories]] and this means that $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$ are the [[adjoint pair]] induced from the reflection/coreflection [[adjoint triple]].

This adjoint triple is stated more explicitly in ([Dwyer-Greenlees 99, section 4](#DwyerGreenlees99)), see also ([Porta-Shaul-Yekutieli 10, end of remark 6.14](#PortaShaulYekutieli10)).

=--

Therefore arithmetic fracture squares in the homotopy theory of chain complexes are induced by this as in corollary \ref{FractureFromCohesion} above.

#### General fracture squares of spectra
 {#GeneralFractureSquares}

By prop. \ref{ReformulationOfProdOverPComletionByLocalizationAtCoproduct}
the arithmetic fracture square of prop. \ref{SullivanArithmeticFracture} is equivalently of the form

$$
  \array{
    && L_{H\mathbb{Q}}X
    \\
    & \swarrow && \nwarrow
    \\
    L_{H\mathbb{Q}}
    L_{S \mathbb{Q}/\mathbb{Z}} X
    &&  &&
    X
    \\
     & \nwarrow && \swarrow
    \\
    && L_{S \mathbb{Q}/\mathbb{Z}} X
  }
  \,.
$$

In this form the statement holds much more generally:

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

Let $E, F, X$ be [[spectra]] such that the $F$-[[Bousfield localization of spectra|localization]] of $X$ is $E$-acyclic, i.e.  $E_\bullet(L_F X) \simeq 0$, then the canonical square [[diagram]]

$$
  \array{
    && L_F X
    \\
    & \swarrow && \nwarrow
    \\
    L_F
    L_E X
    &&  &&
    L_{E\vee F} X
    \\
     & \nwarrow && \swarrow
    \\
    && L_E X
  }
$$

is a [[homotopy pullback]] (and hence by stability also a [[homotopy pushout]]).

=--

e.g. ([Bauer 11, prop. 2.2](#Bauer11))

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

The general version of the fracture statement in prop. \ref{GeneralFractureSquare} is used frequently in [[chromatic homotopy theory]] for decomposition in [[Morava K-theory]] and [[Morava E-theory]]-localizations. For example there is a **chromatic** fracture square:

$$
  \array{
    && L_{E(n-1)} X
    \\
    & \swarrow && \nwarrow
    \\
    L_{E(n-1)}
    L_{K(n)} X
    &&  &&
    L_{E(n)} X
    \\
     & \nwarrow && \swarrow
    \\
    && L_{K(n)} X
  }
$$

In particular it is used for instance in the construction of [[tmf]], see example \ref{ConstructionOfTmf} below.

=--

From another perspective:

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

Suppose that $L \colon Spectra \to Spectra$ is a [[smashing localization]] given by [[smash product]] with some [[spectrum]] $T$. Write $F$ for the [[homotopy fiber]]

$$
  F \longrightarrow \mathbb{S} \longrightarrow T
  \,.
$$

Then there is a [[fracture diagram]] of operations

$$
  \array{
    T \wedge (-) && \longleftarrow  &&  [T,-]
    \\
    & \nwarrow && \swarrow
    \\
    && \mathbb{S}
    \\
    & \swarrow & & \nwarrow
    \\
    [F,-] && \longleftarrow && F \wedge (-)
  }
$$

where $[F,-]$ and $T \wedge (-) \colon Spectra \to Spectra$ are [[idempotent (∞,1)-monads]] and $[T,-]$, $F \wedge (-)$ are idempotent $\infty$-comonads, the diagonals are [[homotopy fiber sequences]].

=--

([[Charles Rezk]], [MO comment,August 2014](http://mathoverflow.net/a/178316/381))

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

For $T = S \mathbb{Z}[p^{-1}]$ the [[Moore spectrum]] of the [[integers]] [[localization of a ring|localized away from]] $p$, then

$$
  F = \Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z})
   \to
  \mathbb{S}
   \to
  S \mathbb{Z}[p^{-1}]
$$

and hence

* $\Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}) \wedge (-)$ is $p$-[[torsion approximation]];

* $[\Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}),-]$ is $p$-[[completion of a module|completion]];

* $S \mathbb{Z}[p^{-1}] \wedge (-)$ is $p$-[[rationalization]];

* $[T,-]$ is forming $p$-[[adic residual]].

$$
  \array{
    &&  localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual
    \\
    & \nearrow & & \searrow & & \nearrow && \searrow
    \\
    && && X &&  &&
    \\
    & \searrow &  & \nearrow & & \searrow && \nearrow
    \\
    && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation
  }
  \,,
$$

=--

With ([Bousfield 79, prop.2.5](#Bousfield79))

#### For $E_\infty$-modules

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

Let $A$ be an [[E-∞ ring]] and let $\mathfrak{a} \subset \pi_0 A$ be a [[generators and relations|finitely generated]] ideal in its underlying [[commutative ring]].

Then there is an [[adjoint triple]] of [[adjoint (∞,1)-functors]]

$$
  \array{

    \underoverset{
      A Mod_{\mathfrak{a}comp}^{op}}
    {A Mod_{\mathfrak{a}tors}^{op}}
    {\simeq}
    &\stackrel{\overset{\Pi_{\mathfrak{a}}}{\longleftarrow}}{\stackrel{\hookrightarrow}{\underset{\flat_{\mathfrak{a}}}{\longleftarrow}}}&
    A Mod^{op}
  }
$$

where

* $A Mod$ is the [[stable (∞,1)-category|stable]] [[(∞,1)-category of modules]], i.e. of [[∞-modules]] over $A$;

* $A Mod_{\mathfrak{a}tor}$ and $A Mod_{\mathfrak{a} comp}$ are the [[full sub-(∞,1)-categories]] of $\mathfrak{a}$-[[torsion approximation|torsion]] and of $\mathfrak{a}$-[[completion of a module|complete]] $A$-[[∞-modules]], respectively;

* $(-)^{op}$ denotes the [[opposite (∞,1)-category]];

* the [[equivalence of (∞,1)-categories]] on the left is induced by the restriction of $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$.

=--

This is effectively the content of ([Lurie "Completions", section 4](#LurieProper)):

### In cohesive (stable) homotopy theory
 {#InCohesiveHomotopyTheory}

In [[cohesive homotopy theory]] every [[stable homotopy type]] $X$ sits in a fracture square of the form

$$
  \array{
    &&  \Pi_{dR} X &&  \longrightarrow && \flat_{dR} X
    \\
    & \nearrow & & \searrow  && \nearrow
    \\
    \Pi_{dR} \flat X  && && X
    \\
    & \searrow &  & \nearrow && \searrow
    \\
    && \flat X   && \longrightarrow && \Pi X
  }
$$

where $\flat$ is the [[flat modality]] and $\Pi_{dR}$ the [[homotopy fiber]] of the [[unit of a monad|unit]] $X\to \Pi X$ of the [[shape modality]]. This is the left part of the [[differential cohomology hexagon]] for $X$, see there for details.

Here $\Pi_{dR} X$ is such that for any other stable cohesive homotopy type $\hat E$ then functions $\Pi_{dR} X \to \hat E$ are equivalent to functions $X \to \flat_{dR} \hat E$, where $\hat E \to \flat_{dR} \hat E$ is a generalized form of rationalization in the sense discussed at _[[differential cohomology hexagon]]_. In particular if $\hat E$ is a [Hopkins-Singer-type](differential%20cohomology%20diagram#HopkinsSingerCoefficients) [[differential cohomology]] refinement of a plain [[spectrum]] $E$, then $E\to \flat_{dR} E$ is its ordinary [[rationalization]] given by the [[Chern character]] and $\hat E \to \flat_{dR} \hat E$ is the corresponding map on Chern [[curvature forms]].

Moreover, if the ambient [[cohesion]] is [[differential cohesion]] over a base of [[infinitesimal cohesion]], then the [[flat modality]] $\flat$ takes any space $X$ to the union of all its [[formal disks]]. (See at _[[differential cohesion and idelic structure]]_.) Accordingly the collection of functions $\flat X \to \hat E$ in this case behave like the product of all [[formal power series]] of $\hat E$-valued functions around all global points of $X$, analogous to remark \ref{GeometricMeaning}.

An example of this are [[synthetic differential ∞-groupoids]] regarded as cohesive over their [[formal moduli problems]], as its its complex analytic incarnation by synthetic differential [[complex analytic ∞-groupoids]]. In this context if $X = \Sigma$ is a [[complex curve]] then $\flat \Sigma$ is precisely the analog of the [[integral adeles]] as it is predicted by the [[function field analogy]].

## Examples
 {#Examples}

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

The construction of the [[tmf]]-spectrum -- the spectrum of [[global sections]] of the [[derived Deligne-Mumford stack]] of [[derived elliptic curves]] --  as described in ([Behrens 13](tmf#Behrens13)) proceeds by first applying the arithmetic fracture square of prop. \ref{ArithmeticFractureSquare}, prop. \ref{SullivanArithmeticFracture} to decompose the [[moduli stack of elliptic curves]] into rational and $p$-adic curves, and then in a second step in applying in turn the general fracture square of prop. \ref{GeneralFractureSquare} for [[Morava K-theory]] to the remaining $p$-adic pieces.

See at _[tmf -- Decomposition via arithmetic fracture squares](tmf#DecomopositionViaArithmeticSquares)_ for more on this.

=--

## Related concepts

* [[p-adic homotopy theory]], [[p-completion]]

* [[rational homotopy theory]], [[rationalization]]

## References

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], _[[Homotopy limits, completions and localizations]]_, Lecture Notes in Mathematics, Vol 304, Springer 1972

* {#Bousfield79} [[Aldridge Bousfield]], _The localization of spectra with respect to homology_ , Topology vol 18 (1979) ([pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/bousfield-topology-1979.pdf))

* {#Sullivan05} [[Dennis Sullivan]], _Geometric topology: localization, periodicity and Galois symmetry_, volume 8 of K- Monographs in Mathematics. Springer, Dordrecht, 2005. The 1970 MIT notes, Edited and with a preface
by [[Andrew Ranicki]] ([pdf](http://www.maths.ed.ac.uk/~aar/books/gtop.pdf))

* {#Bauer11} [[Tilman Bauer]], _Bousfield localization and the Hasse square_ (2011) ([pdf](https://math.mit.edu/events/talbot/2007/tmfproc/Chapter09/Bousfield.pdf), [[Bauer_BousfieldLocalization.pdf:file]]), chapter 6 in:  [[Christopher Douglas]], [[John Francis]], [[André Henriques]], [[Michael Hill]] (eds.), _Topological Modular Forms_, Mathematical Surveys and Monographs Volume 201, AMS 2014  ([ISBN:978-1-4704-1884-7](https://bookstore.ams.org/surv-201))

* {#VanKoughnett13} [[Paul VanKoughnett]], _Spectra and localization_, 2013 ([[VanKoughnettLocalization.pdf:file]])

* {#Riehl14} [[Emily Riehl]], _Categorical homotopy theory_, new mathematical monographs 24, Cambridge University Press 2014 (published version)

* {#MayPonto} [[Peter May]], [[Kate Ponto]], chapters 7 and 8 of _More concise algebraic topology: Localization, completion, and model categories_ ([pdf](http://www.maths.ed.ac.uk/~aar/papers/mayponto.pdf))

* [[Michael Shulman]], _The Propositional Fracture Theorem_, ([blog post](http://golem.ph.utexas.edu/category/2013/05/the_propositional_fracture_the.html))

Related MO-discussion:

* _[Fracture squares of Bousfield Localizations of Spectra](http://mathoverflow.net/a/91057/381)_

Discussion of rational functions as functions on the [[Ran space]] is in

* {#Gaitsgory11} [[Dennis Gaitsgory]], _Contractibility of the space of rational maps_ ([arXiv:1108.1741](http://arxiv.org/abs/1108.1741))

Discussion of $\mathfrak{a}$-[[adic completion]] and $\mathfrak{a}$-[[torsion approximation]] as derived idempotent (co-)monads on a [[derived category]] of chain complexes of modules -- _Greenlees-May duality_ -- is in

* {#PortaShaulYekutieli10} Marco Porta, [[Liran Shaul]], [[Amnon Yekutieli]], _On the Homology of Completion and Torsion_ ([arXiv:1010.4386](http://arxiv.org/abs/1010.4386))

building on

* {#GreenlessMay92} [[John Greenlees]], [[Peter May]], _Derived functors of I-adic completion and local homology_, J. Algebra 149 (1992), 438–453 ([pdf](http://math.uchicago.edu/~may/PAPERS/73.pdf))

* Leovigildo Alonso, Ana Jeremías, [[Joseph Lipman]], _Local Homology and Cohomology on Schemes_ ([arXiv:alg-geom/9503025](http://arxiv.org/abs/alg-geom/9503025))

* {#HoveyPalieriStrickland97} [[Mark Hovey]], [[John Palmieri]], [[Neil Strickland]], _Axiomatic stable homotopy theory_, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114.

* {#DwyerGreenlees99} [[William Dwyer]], [[John Greenlees]], _Complete modules and torsion modules_, Amer. J. Math. 124, No. 1, (1999)  ([pdf](https://www3.nd.edu/~wgd/Dvi/Complete.And.Torsion.pdf))

Discussion of this in [[stable homotopy theory]] and the full generality of [[higher algebra]] is in

* {#LurieProper} [[Jacob Lurie]], section 4 of _[[Proper Morphisms, Completions, and the Grothendieck Existence Theorem]]_

And in the context of commutative DG-rings in

* [[Liran Shaul]], _Completion and torsion over commutative DG rings_, [arXiv:1605.07447](http://arxiv.org/abs/1605.07447)

This and further generalization is in

* {#BarthelHeard15} [[Tobias Barthel]], [[Drew Heard]], Gabriel Valenzuela, _Local duality in algebra and topology_ ([arXiv:1511.03526](http://arxiv.org/abs/1511.03526))

Discussion in [[homotopy type theory]]:

* {#Scoccola19} Luis Scoccola, _Nilpotent Types and Fracture Squares in Homotopy Type Theory_ ([arXiv:1903.03245](https://arxiv.org/abs/1903.03245))

Exposition from the point of view of [[modal homotopy type theory|modal homotopy theory]]:

* [[Urs Schreiber]], _[[nLab:differential cohesion and idelic structure|Fractures, Ideles and the Differential Hexagon]]_, talk at _[Workshop on differential cohomologies](http://qcpages.qc.cuny.edu/~swilson/cunyworkshop14.html)_, CUNY Graduate Center, August 12-14 2014 ([video recording](http://videostreaming.gc.cuny.edu/videos/video/1806/in/channel/55/))

* [[Urs Schreiber]], _[[schreiber:Differential cohesion and Arithmetic geometry]]_, talk at [Karslruher Weihnachts-Workshop 2015](http://www.math.kit.edu/iag3/~herrlich/seite/weihnachts-workshops)

Formalization in [[cohesive homotopy type theory|cohesive]] [[modal homotopy type theory|modal homotopy type theory]] [[homotopy type theory]]:

* [[David Jaz Myers]], _Modal Fracture of Higher Groups_,  talk at _[CMU-HoTT Seminar](https://www.cmu.edu/dietrich/philosophy/hott/seminars/index.html), 2021 ([pdf](http://davidjaz.com/Talks/CMU_March_2021.pdf), [[MyersModalFracture2021.pdf:file]])

See also

* [[John Greenlees]], _Tate cohomology in axiomatic stable homotopy theory_ ([pdf](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.490.322&rep=rep1&type=pdf))

[[!redirects fracture theorems]]

[[!redirects arithmetic square]]
[[!redirects arithmetic squares]]

[[!redirects fracture square]]
[[!redirects fracture squares]]

[[!redirects arithmetic fracture square]]
[[!redirects arithmetic fracture squares]]

[[!redirects fracture diagram]]
[[!redirects fracture diagrams]]

[[!redirects Grothendieck local duality]]
[[!redirects Greenlees-May duality]]


fractured (infinity,1)-topos

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohesive $\infty$-Toposes
+--{: .hide}
[[!include cohesive infinity-toposes - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A kind of local generalization of *[[cohesive (infinity,1)-toposes|cohesive $\infty$-toposes]]*.

(...)

## Examples

* Apparently [[condensed mathematics]] takes place in fractured $\infty$-topoi, see at *[[condensed local contractibility]]*.

## References

* [[Jacob Lurie]], *Fractured $\infty$-Topoi*, Chapter 20 in: *[[Spectral Algebraic Geometry]]*

following

* [[David Carchedi]], *Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity Topoi*, Memoirs of the American Mathematical Society **264** 1282 (2020) [[doi:10.1090/memo/1282](https://doi.org/10.1090/memo/1282), [arXiv:1312.2204](https://arxiv.org/abs/1312.2204)]

The example of [[condensed local contractibility]]:

* [[Qi Zhu]], *Fractured Structure on Condensed Anima*, MSc thesis (2023) [[pdf](https://1429cecd-24a0-4223-8b7c-1ebf47aa12e2.filesusr.com/ugd/8e912a_fe1c2f8e90094d0fa065fb8129687963.pdf), [[QiZhu-FracturedCondensed.pdf:file]]]

exposition:

* [[Qi Zhu]], *Fractured structure on condensed spaces*, talk notes (2023) [[pdf](https://1429cecd-24a0-4223-8b7c-1ebf47aa12e2.filesusr.com/ugd/8e912a_8d2a90176ed14969a877e77ec1b787da.pdf), [[QiZhuFracturedCondensed.pdf:file]]]

* [[Nima Rasekh]], *What is a topological structure?*, talk notes (April 2023) [[pdf](https://guests.mpim-bonn.mpg.de/rasekh/Condensed.pdf), [[Rasekh-TopologicalStructure.pdf:file]]]

[[!redirects fractured (infinity,1)-toposes]]
[[!redirects fractured (infinity,1)-topoi]]

[[!redirects fractured (infinity,1)-toposes]]


Fraenkel-Mostowski model

[[!redirects Fraenkel--Mostowski model]]
[[!redirects Fraenkel-Mostowski model]]

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

#Contents#
* table of contents
{:toc}

## Idea
It can be shown via $\epsilon$-induction that any model of [[ZF]] has no non-trivial [[automorphisms]]. However, if we have a [[model]] $V$ of [[ZFA]] with a set of [[atoms]] $A$, then any [[permutation]] of the atoms in $A$ gives rise to a non-trivial automorphism of $V$.

We can then look at the submodel of $V$ consisting of sets that are (hereditarily) closed under "most" permutations of $V$ (in a sense made precise by a normal filter), and it turns out this gives a [[transitive model]] of ZFA. In some sense, imposing the symmetry condition above causes the atoms to be "indistinguishable", and thus often causes the [[axiom of choice]] to fail, since there is no uniform way of picking out atoms.

By tweaking the choice of permutations, we can produce models in which some weak version of choice holds (such as the [[prime ideal theorem#boolean_pit_and_the_ultrafilter_principle|boolean prime ideal theorem]]), while full (or even countable) choice fails, and thus proving their independence (in ZFA).

Two such examples are the [[basic Fraenkel model]] and the [[second Fraenkel model]] which gave the first examples of the independence of the axiom of choice from a set theory ([[David Roberts|I]] think!).

## Definition

Let $V$ be a model of $ZFA$, and let $A$ be its set of atoms. Then any permutation of $A$ induces a unique non-trivial automorphism of $V$.

We pick a subgroup $G \leq Aut(A)$ of permutations of the atoms, and let $\mathcal{F}$ be a normal [[filter]] of subgroups of $G$, ie. a collection of subgroups such that

 * $\mathcal{F}$ is non-empty.

 * If $H, K \in \mathcal{F}$, then $H \cap K \in \mathcal{F}$.

 * If $H \in \mathcal{F}$ and $K \geq H$, then $K \in \mathcal{F}$.

 * If $H \in \mathcal{F}$ and $g \in G$, then $gHg^{-1} \in \mathcal{F}$.

We say $x$ is ($\mathcal{F}$-)symmetric if its stabilizer $stab(x) \in \mathcal{F}$. We let $\tilde{V}$ be the class of hereditarily symmetric sets.

We usually pick $\mathcal{F}$ so that for each atom $a \in A$, the singleton $\{a\}$ is symmetric. Otherwise, the non-symmetric elements would never occur in the class $\tilde{V}$.

+-- {: .num_theorem}
###### Theorem
The class $\tilde{V}$ is a model of ZFA.
=--

+-- {: .proof}
###### Not a proof
See [Felgner, Chapter III.B](#Felgner).
=--

## Category-theoretic approach
From a modern, [[structural set theory|structural]] [[foundations]] point of view, we can construct these models by viewing $G \leq Aut(A)$ as a [[topological group]], where the topology is generated by the normal filter of subgroups $\mathcal{F}$. We then construct the [[category]] whose objects are [[sets]] with an action by an [[open subset|open]] [[subgroup]] $H \le G$ (where $H$ is allowed to vary), and morphisms are functions $f\colon X\to Y$ which are equivariant for an open subgroup $L \le H \cap K$, where $H$ and $K$ act on $X$ and $Y$ respectively. The resulting category is in fact a [[Boolean topos|Boolean]] [[topos]].

The construction outlined at [[pure set]] can be varied to construct well-founded pure sets using atoms [ [[David Roberts|DR]]:which I believe they are elements of the [[transitive action|transitive]] $G$-sets.]

While this approach more-or-less translates the construction of the permutation model directly, the resulting topos in general fails to be (externally) [[complete category|complete]]. In particular, we cannot construct the [[coproduct]] of an infinite family $\{X_i\}_{i \in I}$ (at least not in the obvious way), since given functions $f_i: X_i \to A$, we cannot guarantee that the combined map $(\coprod_i X_i) \to A$ has an open stabilizer, as the infinite intersection of open sets need not be open.

Instead, we can consider the [[logical morphism|logical]] [[subtopos]] consisting of sets with a full $G$-action, and morphisms are the functions that are $G$-equivariant. This gives the [[topos of continuous G-sets]].

## Related Concepts

 * [[category of G-sets]]
 * [[G-set]]
 * [[ZFA]]

## References

Fraenkel-Mostowski Models are discussed in

 * {#Felgner} Ulrich Felgner, _Models of ZF-Set Theory_, Lecture Notes in Mathematics 223 (1971)

 * Thomas J. Jech, _The Axiom of Choice_ (1973)

The following papers discuss how we can deduce properties of the permutation model from properties of the corresponding categories of $G$ sets:

 * Michael Fourman, _Sheaf models for set theory_ ([pdf](http://www.sciencedirect.com/science/article/pii/0022404980900961/pdf?md5=0de18da810657bf1ba9de7399d224ec5&pid=1-s2.0-0022404980900961-main.pdf))

 * Andreas Blass, Andre Scedrov, _Complete topoi representing models of set theory_ ([pdf](https://deepblue.lib.umich.edu/bitstream/handle/2027.42/30052/0000420.pdf?sequence=1))

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