Rapoport-Zink space

## Idea

A Rapoport-Zink space is a [[formal scheme]] parametrizing deformations of a [[p-divisible group]]. They may be thought of as local analogs of [[Shimura varieties]].

## Related ideas

* [[Lubin-Tate space]]

## References

* {#RapoportZink96} M. Rapoport, Th. Zink, _Period spaces for $p$-divisible groups_, Annals of Mathematics Studies, vol. 141, Princeton University Press, Princeton, NJ, 1996.

Review of the first three sections of ([Rapoport-Zink 96](#RapoportZink96)) is in

* _Rapoport-Zink spaces_, lecture notes 2013 [pdf](http://www.math.uni-bielefeld.de/~verhoek/files/notes-pdiv-2.pdf)

Further lecture notes include

* _KIAS Intensive Lecture Series on Shimura varieties and Rapoport-Zink spaces_ ([web](http://home.kias.re.kr/MKG/h/ILSSRZ2014/))

Further developments include

* {#ScholzeWeinstein12} [[Peter Scholze]], [[Jared Weinstein]], _Moduli of p-divisible groups_ ([arXiv:1211.6357](http://arxiv.org/abs/1211.6357))

Comments on the relation to [[shtukas]] in view of the [[function field analogy]] are in

* {#Hartl06} [[Urs Hartl]], _A Dictionary between Fontaine-Theory and its Analogue in Equal Characteristic_ ([arXiv:math/0607182](http://arxiv.org/abs/math/0607182))

[[!redirects Rapoport-Zink spaces]]


Rarita-Schwinger field

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### Context
#### Physics
+-- {: .hide}
[[!include physicscontents]]
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#### Higher spin geometry
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[[!include higher spin geometry - contents]]
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#### Fields and quanta
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[[!include fields and quanta - table]]
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#Contents#
* table of contents
{:toc}

## Idea

In [[physics]] the _Rarita-Schwinger field_ is a type of [[field (physics)|field]] of [[spin]] $= 3/2$.

The hypothetical [[gravitino]] in [[supergravity]] is closely related.

## Related concepts

* [[Velo-Zwanziger problem]]

## References

* Wikipedia, _[Rarita-Schwinger equation](http://en.wikipedia.org/wiki/Rarita%E2%80%93Schwinger_equation)_

A good account is in

* SuchIdeas, _[Rarita Schwinger (Spin 3/2) Fields](http://suchideas.com/articles/physics/quantum/rarita-schwinger/)_
[[!redirects Rarita-Schwinger fields]]

[[!redirects Rarita-Schwinger equation]]


Rasmus Bentmann


Rasmus Møgelberg

Rasmus Møgelberg is professor for [[computer science]] in Copenhagen.

* [webpage](http://www.itu.dk/people/mogel/)

## Publications

* [[Magnus Baunsgaard Kristensen]], [[Rasmus Ejlers Møgelberg]], [[Andrea Vezzosi]], *Greatest HITs: Higher inductive types in coinductive definitions via induction under clocks* ([arXiv:2102.01969](https://arxiv.org/abs/2102.01969))

category: people

[[!redirects Rasmus Mogelberg]]
[[!redirects Rasmus Ejlers Møgelberg]]
[[!redirects Rasmus Ejlers Mogelberg]]


ratio

## Related concepts

* [[fraction]]

* [[rational number]]

## References

' Wikipedia, _[Ratio](https://en.wikipedia.org/wiki/Ratio)_

[[!redirects ratios]]


rational 2d conformal field theory

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#### Functorial Quantum field theory
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

_Rational 2d conformal field theories_ are that class of relatively simple [[2d CFTs]], such as the [[WZW model]], which are characterized by the fact that their spaces of [[conformal blocks]] are [[finite dimensional vector spaces]]. Locally such 2d CFTs are given by a [[rational vertex operator algebra]].

Interpreted as [[perturbative string theory vacua]], rational [[2d CFTs]] correspond to ([[non-geometric string vacuum|non-geometric]]) [[target spaces]] which are [[compact topological spaces|compact]] (such as [[fibers]] of [[Kaluza-Klein compactifications]], but not [[Minkowski spacetime]], for instance). See e.g. [Schomerus 05](#Schomerus05) for contrast.

## Properties

### Classification

The [[FRS-theorem on rational 2d CFT]] provides a complete classification of rational 2d CFT via the [[CS-WZW correspondence]] and the [[Reshetikhin-Turaev construction]] of [[3d Chern-Simons theory]].

## Examples

* [[WZW model]]

* [[Gepner model]]

## Related concepts

* [[boundary CFT]]

* [[minimal model CFT]]

* [[logarithmic CFT]]

## References

See the references at _[[FRS-theorem on rational 2d CFT]]_.

Lecture notes on [[modular tensor categories]] and rational CFT:

* {#Runkel} [[Ingo Runkel]], *Algebra in Braided Tensor Categories and Conformal Field Theory* ([pdf](https://www.math.uni-hamburg.de/home/runkel/PDF/alg.pdf), [[Runkel-AlgebraInBTCAndCFT.pdf:file]])

The contrast with non-rational CFTs:

* {#Schomerus05} [[Volker Schomerus]], _Non-compact String Backgrounds and Non-rational CFT_, Phys.Rept.431:39-86, 2006 ([arXiv:hep-th/0509155](https://arxiv.org/abs/hep-th/0509155))

See also:

* Zhihao Duan, Kimyeong Lee, Kaiwen Sun, *Hecke Relations, Cosets and the Classification of 2d RCFTs* $[$[arXiv:2206.07478](https://arxiv.org/abs/2206.07478)$]$

Classification:

* Sunil Mukhi, Brandon C. Rayhaun, *Classification of Unitary RCFTs with Two Primaries and Central Charge Less Than 25* [[arXiv:2208.05486](https://arxiv.org/abs/2208.05486)]

Constructing [[correlators]] in [[rational conformal field theory]] via [[string net models]]:

* [[Jürgen Fuchs]], [[Christoph Schweigert]], [[Yang Yang]], *String-Net Construction of RCFT Correlators*, Springer Briefs in Mathematical Physics **45**, Springer (2023) [[doi:10.1007/978-3-031-14682-4](https://doi.org/10.1007/978-3-031-14682-4), [arXiv:2112.12708](https://arxiv.org/abs/2112.12708)]

[[!redirects rational 2d conformal field theories]]

[[!redirects rational conformal field theory]]
[[!redirects rational conformal field theories]]

[[!redirects rational CFT]]
[[!redirects rational CFTs]]

[[!redirects RCFT]]
[[!redirects RCFTs]]

[[!redirects rational 2d CFT]]
[[!redirects rational 2d CFTs]]


Rational and p-adic Homotopy Theory

This page collects some links related to the article:

* [[Jacob Lurie]],

  DAG XIII: **Rational and p-Adic Homotopy Theory**

  2011

  [pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XIII.pdf), [[Lurie_RationalHomotopyTheory.pdf:file]]

on [[rational homotopy theory]] and [[p-adic homotopy theory]] seen within the theory of algebraic [[∞-stacks]] and [[derived algebraic geometry]].

#Contents#
* table of contents
{:toc}

## Rational homotopy theory

* [[rational homotopy theory]]

* [[fundamental theorem of dg-algebraic rational homotopy theory]]

  (Prop. 1.3.3, 1.3.4)

## $E_\infty$-Algebras in positive characteristic

### 2.1 Norm maps

### 2.2 Power operations on $E_\infty$-Algebras

* [[power operation]]

* [[Frobenius homomorphism]]

### 2.3 Finitely constructible sheaves

* [[finitely constructible object]]

### 2.4 A Universal coefficient theorem

### 2.5 Compactness of relative cochain algebras

### 2.6 Affine behaviour of $p$-constructible morphisms

category: reference

[[!redirects Rational and p-Adic Homotopy Theory]]


rational Cherednik algebra

## Idea

Rational Cherednik algebras are certain degenerations of [[double affine Hecke algebra]]s.

## Literature

* [[Yuri Berest|Yu. Berest]], [[P. Etingof]], [[V. Ginzburg]], _Finite-dimensional representations of rational Cherednik algebras_, Int. Math. Res. Not. 2003, no. 19, 1053--1088.
* V. Ginzburg, N. Guay, E. Opdam, R. Rouquier, On the category $\mathcal{O}$ for rational Cherednik algebras, Invent. Math. __154__ (3) (2003) 617--651
* Kevin McGerty, _Microlocal KZ functors and rational Cherednik algebras_, [arxiv:1006.1599](https://arxiv.org/abs/1006.1599)
* [[Pavel Etingof]], [[Victor Ginzburg]], _Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphism_, Invent. Math. 147 (2) (2002) 243–348, [MR2003b:16021](https://www.ams.org/mathscinet-getitem?mr=1881922), [doi](https://doi.org/10.1007/s002220100171)

* Y. Bazlov, A. Berenstein, E. Jones-Healey, A. McGaw, _Twists of rational Cherednik algebras_, [arXiv:2205.11971](https://arxiv.org/abs/2205.11971)

There is also a braided/$q$-deformation:

* Yuri Bazlov, [[Arkady Berenstein]], _Noncommutative Dunkl operators and braided Cherednik algebras_, Selecta Math. (N.S.) __14__ (2009), no. 3-4, 325--372, [pdf](https://pages.uoregon.edu/arkadiy/doubles.pdf), [MR2010k:16044](https://mathscinet.ams.org/mathscinet-getitem?mr=2511188) [doi](https://doi.org/10.1007/s00029-009-0525-x)

category: algebra
[[!redirects rational Cherednik algebra]]


rational cohomology

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#### Cohomology
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[[!include cohomology - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

By _rational cohomology_ one usually means [[ordinary cohomology]] with [[rational number]] [[coefficients]], denoted $H^\bullet\big(-, \mathbb{Q}\big)$.

Hence, with the pertinent conditions on the [[domain]] [[space]] $X$ satisfied, its rational cohomology $H^\bullet\big(-, \mathbb{Q}\big)$ is what is computed by the [[Cech cohomology]] or [[singular cohomology]] or [[sheaf cohomology]] of $X$ with coefficients in $\mathbb{Q}$.

## Properties

### Universal coefficient theorem

+-- {: .num_example #UniversalCoefficientTheoremInRationalCohomology}
###### Example
**([[universal coefficient theorem]] in [[rational cohomology]])**

For [[rational numbers]]-[[coefficients]] $\mathbb{Q}$, the
[[Ext groups]] $Ext^1(-;\mathbb{Q})$ vanish, and hence the [[universal coefficient theorem]] identifies [[rational cohomology]] [[cohomology groups|groups]] with the [[dual vector space]] of the [[rational vector space]] of rational [[homology groups]]:

$$
  H^\bullet
  \big(
    -; \, \mathbb{Q}
  \big)
  \;\;
  \simeq
  \;\;
  Hom_{\mathbb{Z}}
  \Big(
    H_\bullet\big(-;\,\mathbb{Q} \big);
    \,
    \mathbb{Q}
  \Big)
  \,.
$$

=--

(e.g. [Moerman 15, Cor. 1.2.1](rational+homotopy+theory#Moerman15))

### PL de Rham theorem

* [[PL de Rham theorem]]

## Related concepts

* [[rational vector space]]

* [[integral cohomology]], [[real cohomology]], [[complex cohomology]]

* [[rational Cohomotopy]]

* [[rational homotopy theory]]

* [[Chern character]], [[Chern-Dold character]]

* [[rational equivariant K-theory]]

* [[equivariant Chern character]]


rational Cohomotopy

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#Contents#
* table of contents
{:toc}

## Idea

_Rational Cohomotopy theory_ $\pi^\bullet_{\mathbb{Q}}$ is the approximation in [[rational homotopy theory]] of [[Cohomotopy cohomology theory]], i.e. the ([[non-abelian cohomology|non-abelian]]) [[generalized cohomology theory]] whose [[cocycle spaces]] are [[rationalizations]] of the [[cocycle spaces]] of plain [[Cohomotopy cohomology theory]], hence which are [[spaces of maps]] into [[rational n-spheres]] $S^n_{\mathbb{Q}}$:

$$
  \pi^n_{\mathbb{Q}}(-)
  \;=\;
  \pi_0 Maps\big(-, S^n\big)_{\mathbb{Q}}
  \;\simeq\;
  \pi_0 Maps\big(-, S^n_{\mathbb{Q}}\big)
  \;.
$$

## Properties

### Sullivan models for cocycle spaces

Discussion of [[rational cohomology]] and of [[Sullivan models]] for [[cocycle spaces]] in rational Cohomotopy.
+-- {: .num_prop #RationalHomotopyTypeOfMapsNSphereToNsphere}
###### Proposition
**([[rational homotopy type]] of [[space of maps]] from [[n-sphere]] to itself)**

Let $n \in \mathbb{N}$ be a [[natural number]] and $f\colon S^n \to S^n$ a [[continuous function]] from the [[n-sphere]] to itself. Then the [[connected component]] $Maps_f\big( S^n, S^n\big)$ of the [[space of maps]] which contains this map has the following [[rational homotopy theory|rational]] [[homotopy type]]:

\[
  \label{RationalHomotopyTypeOfMappingSpacesSnToSn}
  Maps_f\big( S^n, S^n\big)
  \;\simeq_{\mathbb{Q}}\;
  \left\{
    \array{
      S^n \times S^{n-1} &\vert& n\,\text{even}\,, deg(f) = 0
      \\
      S^{2n-1} &\vert& n \, \text{even}\,, deg(f) \neq 0
      \\
      S^n &\vert& n\, \text{odd}
    }
  \right.
\]

where $deg(f)$ is the [[degree of a continuous function|degree]] of $f$.

Moreover, under the canonical morphism expressing the canonical [[action]] of  the [[special orthogonal group]] $SO(n+1)$ on $S^n = S\big( \mathbb{R}^{n+1}\big)$ (regarded as the [[unit sphere]] in $(n+1)$-[[dimension|dimensional]] [[Cartesian space]]) we have that on [[ordinary homology]]

$$
  \array{
    H_\bullet\Big(
      SO\big( n+ 1 \big)
    \Big)
    &\longrightarrow&
    H_\bullet\Big(
      Maps_{f = id}\big( S^n, S^n  \big)
    \Big)
  }
$$

the generator in $\left\{ \array{ H_{2n+1}\big( SO(n+1), \mathbb{Q} \big) \simeq \mathbb{Q} &\vert& n\, \text{even} \\ H_{n}\big( SO(n+1), \mathbb{Q} \big) \simeq \mathbb{Q} &\vert& n\, \text{odd}   }  \right.$ maps to the [[fundamental class]] of the respective [[spheres]] in (eq:RationalHomotopyTypeOfMappingSpacesSnToSn), all other generators mapping to [[zero]].

=--

([Møller-Raussen 85, Example 2.5](#MollerRaussen85), [Cohen-Voronov 05, Lemma 5.3.5](#CohenVoronov05))

See at _[[Sullivan model of a spherical fibration]]_ for more on this.

+-- {: .num_prop #RationalCohomologyOfIteratedLoopSpaceOf2kSphere}
###### Proposition
**([[rational cohomology]] of [[iterated loop space]] of the [[n-sphere|2k-sphere]])**

Let

$$
  1 \leq D \lt n = 2k \in \mathbb{N}

$$

(hence two [[positive number|positive]] [[natural numbers]], one of them required to be [[even number|even]] and the other required to be smaller than the first) and consider the [[iterated loop space|D-fold loop space]] $\Omega^D S^n$ of the [[n-sphere]].

Its [[rational cohomology|rational]] [[cohomology ring]] is the [[free construction|free]] [[graded-commutative algebra]] over $\mathbb{Q}$ on one [[generators and relations|generator]] $e_{n-D}$ of degree $n - D$ and one generator $a_{2n - D - 1}$ of degree $2n - D - 1$:

$$
  H^\bullet
  \big(
    \Omega^D S^n
    ,
    \mathbb{Q}
  \big)
  \;\simeq\;
  \mathbb{Q}\big[ \omega_{n - D}, \omega_{2n - 1 - D} \big]
  \,.
$$

=--

(by [this Prop.](Sullivan+model+of+free+loop+space#SullivanModelsOfMapsFromSkToSnFornLargerk) at _[[Sullivan model of based loop space]]_; see also  [Kallel-Sjerve 99, Prop. 4.10](#KallelSjerve99))

For the edge case $\Omega^D S^D$ the above formula does not apply, since $\Omega^{D-1} S^D$ is not [[simply connected topological space|simply connected]] (its [[fundamental group]] is $\pi_1\big( \Omega^{D-1}S^D \big) = \pi_0 \big(\Omega^D S^D\big) = \pi_D(S^D) = \mathbb{Z}$, the 0th [[stable homotopy group of spheres]]).

But:

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

The rational model for $\Omega^D S^D$ follows from Prop. \ref{RationalHomotopyTypeOfMapsNSphereToNsphere} by realizing the pointed mapping space as the [[homotopy fiber]] of the [[evaluation map]] from the free mapping space:

$$
  \array{
    \mathllap{
      \Omega^D S^D
      \simeq
    \;}
    Maps^{\ast/\!}\big( S^D, S^D\big)
    \\
    \big\downarrow^{\mathrlap{fib(ev_\ast)}}
    \\
    Maps(S^D, S^D)
    \\
    \big\downarrow^{\mathrlap{ev_\ast}}
    \\
    S^D
  }
$$

This yields for instance the following examples.

In odd dimensions:

\begin{xymatrix}
    \mathrm{Maps}^{\ast/\!}
    \big(
      S^3, S^3
    \big)
    \ar[d]^-{ \mathrm{fib}_{(\mathrm{ev}_\ast)} }
    \ar@{}[r]|-{ \simeq_{\mathbb{Q}} }
    &
    \underset{
      n \in \mathbb{Z}
    }{\sqcup}
    \ast
    \ar@{^{(}->}[d]
    \\
    \mathrm{Maps}
    \big(
      S^3, S^3
    \big)
    \ar[d]^-{ \mathrm{ev}_\ast }
    \ar@{}[r]|-{ \simeq_{\mathbb{Q}} }
    &
    \underset{
      n  \in \mathbb{Z}
    }{\sqcup}
    S^3
    \ar[d]^-{ (\mathrm{id}_{S^3})_{n \in \mathbb{N}} }
    \\
    S^3
    \ar@{=}[r]
    &
    S^3
\end{xymatrix}

In even dimensions:

(In the following $h_{\mathbb{K}}$ denotes the [[Hopf fibration]] of the [[division algebra]] $\mathbb{K}$, hence $h_{\mathbb{C}}$ denotes the [[complex Hopf fibration]] and $h_{\mathbb{H}}$ the [[quaternionic Hopf fibration]].)

\begin{xymatrix}
    \mathrm{Maps}^{\ast/\!}
    \big(
      S^2, S^2
    \big)
    \ar[d]^-{ \mathrm{fib}_{(\mathrm{ev}_\ast)} }
    \ar@{}[r]|-{ \simeq_{\mathbb{Q}} }
    &
    \underset{
      n \in \mathbb{Z}
    }{\sqcup}
    S^1
    \ar@{^{(}->}[d]
    \\
    \mathrm{Maps}
    \big(
      S^2, S^2
    \big)
    \ar[d]^-{ \mathrm{ev}_\ast }
    \ar@{}[r]|-{ \simeq_{\mathbb{Q}} }
    &
    \big(
      S^2 \times S^1
    \big)
    \sqcup
    \big(
      \underset{
        n \neq 0  \in \mathbb{Z}
      }{\sqcup}
      S^3
    \big)
    \ar[d]^-{
      \big(
        p_1, (h_{\mathbb{C}})_{n \neq 0 \in \mathbb{N}}
      \big)
    }
    \\
    S^2
    \ar@{=}[r]
    &
    S^2
\end{xymatrix}

\begin{xymatrix}
    \mathrm{Maps}^{\ast/\!}
    \big(
      S^4, S^4
    \big)
    \ar[d]^-{ \mathrm{fib}_{(\mathrm{ev}_\ast)} }
    \ar@{}[r]|-{ \simeq_{\mathbb{Q}} }
    &
    \underset{
      n \in \mathbb{Z}
    }{\sqcup}
    S^3
    \ar@{^{(}->}[d]
    \\
    \mathrm{Maps}
    \big(
      S^4, S^4
    \big)
    \ar[d]^-{ \mathrm{ev}_\ast }
    \ar@{}[r]|-{ \simeq_{\mathbb{Q}} }
    &
    \big(
      S^4 \times S^3
    \big)
    \sqcup
    \big(
      \underset{
        n \neq 0  \in \mathbb{Z}
      }{\sqcup}
      S^7
    \big)
    \ar[d]^-{
      \big(
        p_1, (h_{\mathbb{H}})_{n \neq 0 \in \mathbb{N}}
      \big)
    }
    \\
    S^4
    \ar@{=}[r]
    &
    S^4
\end{xymatrix}

=--

## Examples

We discuss examples of cocycle spaces for rational 4-Cohomotopy, hence with [[coefficients]] in the [[rational n-sphere|rational]] [[4-sphere]], whose [[Sullivan model]] is

$$
  CE
  \big(
    S^4
  \big)
  \;=\;
  \left(
  \begin{aligned}
  d\,g_4 & = 0
  \\
  d\,g_7 & = -\tfrac{1}{2} g_4 \wedge g_4
  \end{aligned}
  \right)
  \,.
$$

### The cocycle space for $\pi^4\big( S^1\big)_{\mathbb{Q}}$

The [[Sullivan model]] for the [[space of maps]] $Maps(S^1, S^4)$ from the [[1-sphere]] to the [[4-sphere]] is

$$
  CE
  \Big(
    \mathfrak{l}
    Maps
    \big(
      S^1, S^4
    \big)
  \Big)
  \;=\;
  \left(
  \begin{aligned}
    d\,h_3 & = 0
    \\
    d\, \omega_4 & = 0
    \\
    d\, \omega_6 & = h_3 \wedge \omega_4
    \\
    d\, h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4
    \\
  \end{aligned}
  \right)
$$

By [this Prop.](#Sullivan+model+of+free+loop+space#SullivanModelForTheFreeLoopSpace), see [FSS 16, Section 3](#FSS16).

### The cocycle space for $\pi^4\big( S^3\big)_{\mathbb{Q}}$

The [[Sullivan model]] for the [[space of maps]] $Maps(S^3, S^4)$ from the [[3-sphere]] to the [[4-sphere]] is

$$
  CE
  \Big(
    \mathfrak{l}
    Maps
    \big(
      S^3, S^4
    \big)
  \Big)
  \;=\;
  \left(
  \begin{aligned}
    d\, b_1 & = 0
    \\
    d\, \omega_4 & = 0
    \\
    d\, v_{{}_{4}} & = \omega_4 \wedge b_1
    \\
    d\, \omega_7 & = - \tfrac{1}{2} \, \omega_4 \wedge \omega_4
  \end{aligned}
  \right)
$$

By [Møller-Raussen 85, Prop. 2.3](#MollerRaussen85).

## Related concepts

[[!include flavours of cohomotopy -- table]]

See also

* [[rational cohomology]]

## References

### Cocycle spaces

Discussion of [[cocycle spaces]] in [[rational Cohomotopy]] (see also at _[[rational model for mapping spaces]]_):

* {#KallelSjerve99} [[Sadok Kallel]], [[Denis Sjerve]], _On Brace Products and the Structure of Fibrations with Section_, 1999 ([pdf](https://www.math.ubc.ca/~sjer/brace.pdf), [[KallelSjerv99.pdf:file]])

* {#MollerRaussen85} [[Jesper Møller]], [[Martin Raussen]], _Rational Homotopy of Spaces of Maps Into Spheres and Complex Projective Spaces_, Transactions of the American Mathematical Society Vol. 292, No. 2 (Dec., 1985), pp. 721-732 ([jstor:2000242](https://www.jstor.org/stable/2000242))

* J.-B. Gatsinzi, _Rational Gottlieb Group of Function Spacesof Maps into an Even Sphere_, International Journal of Algebra, Vol. 6, 2012, no. 9, 427 - 432 ([pdf](http://www.m-hikari.com/ija/ija-2012/ija-9-12-2012/gatsinziIJA9-12-2012.pdf))

### On super-spaces

The observation that the [[equations of motion]] of the [[supergravity C-field]] and its dual in [[D=11 N=1 supergravity]] characterize it as a [[cocycle]] in rational 4-Cohomotopy:

* {#Sati13} [[Hisham Sati]], Section 2.5 of: _Framed M-branes, corners, and topological invariants_, J. Math. Phys. 59 (2018), 062304 ([arXiv:1310.1060](http://arxiv.org/abs/1310.1060))

Rational Cohomotopy of [[super-spaces]] (see also at _[[geometry of physics -- fundamental super p-branes]]_):

* {#FSS16} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]: _[[schreiber:Rational sphere valued supercocycles in M-theory|Rational sphere valued supercocycles in M-theory and type IIA string theory]]_, Journal of Geometry and Physics, Volume 114, Pages 91-108 (2017) ([arXiv:1606.03206](https://arxiv.org/abs/1606.03206))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]: _[[schreiber:The WZW term of the M5-brane|The WZW term of the M5-brane and differential cohomotopy]]_, J. Math. Phys. 56, 102301 (2015) ([arXiv:1506.07557](https://arxiv.org/abs/1506.07557))

  (**[arXiv:1506.07557](http://arxiv.org/abs/1506.07557)**)

Review in:

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]: _[[schreiber:The rational higher structure of M-theory]]_, in: _Proceedings of the [LMS-EPSRC Durham Symposium](http://www.maths.dur.ac.uk/lms/):_ _[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html), August 2018_m Fortschritte der Physik, 2019 ([arXiv:1903.02834](https://arxiv.org/abs/1903.02834))

### Rational equivariant Cohomotopy

Discussion of [[rational equivariant homotopy theory|rational]] [[equivariant Cohomotopy]]:

* [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]_, Comm. Math. Phys. 2019 ([arXiv:1805.05987](https://arxiv.org/abs/1805.05987))

### Rational twisted Cohomotopy

Discussion of rational [[twisted Cohomotopy]]:

* {#FSS19b} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], Section 3 of _[[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation]]_ ([arXiv:1904.10207](https://arxiv.org/abs/1904.10207))

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization|Twisted Cohomotopy implies level quantization of the full 6d Wess-Zumino-term of the M5-brane]]_ ([arXiv:1906.07417](https://arxiv.org/abs/1906.07417))

[[!redirects rational Cohomotopy theory]]
[[!redirects rational Cohomotopy cohomology theory]]

[[!redirects rational cohomotopy]]
[[!redirects rational cohomotopy theory]]
[[!redirects rational cohomotopy cohomology theory]]


rational dagger category

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## Definition

A **rational dagger category** is a [[dagger category]] with a [[zero object]] and dagger [[biproducts]] such that every [[morphism]] has a [[kernel]] that is a dagger section and every binary [[diagonal morphism]] is a [[normal monomorphism]].

## Related concepts

* [[dagger category]]

* [[category of Hilbert spaces]]

## References

* [[Matthew Di Meglio]], *Rational dagger categories* [[arXiv:2312.02883](https://arxiv.org/abs/2312.02883)]

[[!redirects rational dagger categories]]


rational equivalence

Let $X$ be a [[noetherian scheme]].  One defines a notion of [[rational equivalence]] on $X$ which, roughly speaking, identifies [[algebraic cycles]] that are connected by a family of cycles parametrized by the [[projective line]].

A cycle $C \in Z_k(X)$ is **rationally equivalent to zero** if there are [[closed subspace|closed]] [[integral scheme|integral]] subschemes $Z_\alpha$ of [[dimension]] $k + 1$ and invertible [[rational functions]] $r_\alpha \in R(Z_\alpha)^*$ such that

$$
  C = \sum_\alpha j_{\alpha,*}(div(r_\alpha)) \,,
$$

where

* $div$ denote the associate [[Weil divisors]];

* $j_{\alpha,*}$ denotes the [[direct image]] along the inclusion $Z_\alpha \hookrightarrow X$.

Rational equivalence generalizes [[linear equivalence]] of [[Weil divisors]].  It is an example of an [[adequate equivalence relation]].

## References

* [[Stacks Project]], [02RV](http://stacks.math.columbia.edu/tag/02RV)

[[!redirects rationally equivalent]]


rational equivariant K-theory

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=--
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* table of contents
{:toc}

## Idea

While the [[rationalization]] of [[complex topological K-theory]] has lost most interesting information (being just a [[direct sum]] of [[ordinary cohomology|ordinary]] [[rational cohomology]] in every even degree) the [[rationalization]] of its enhancement to [[equivariant cohomology]] -- [[equivariant K-theory]] -- is still a comparatively rich object, controlled by the rational [[representation theory]] of the given equivariance group.

Rational equivariant K-theory appears in a variety of different but equivalent concrete incarnations, such as [[Bredon cohomology]] with [[coefficients]] in the [[representation ring]]/[[class functions]] or as [[Chen-Ruan cohomology]] (here shown over the [[complex numbers]], see [Lück-Oliver 01](#LueckOliver01) for discussion over the [[rational numbers]]):

[[!include incarnations of rational equivariant topological K-theory -- table]]

As any [[Chern-Dold character]], the [[equivariant Chern character]] on [[equivariant K-theory]] takes values in its [[rationalization]]. Hence, together with the above variety of presentations of rational equivariant K-theory, there is a corresponding variety of realizations of the [[equivariant Chern character]].

## Related concepts

* [[equivariant Chern character]]

* [[rational cohomology]], [[rational Cohomotopy]]

* [[Bredon cohomology]]

* [[rational equivariant stable homotopy theory]]

## References

See also the references at _[[equivariant Chern character]]_.

* {#ConnesBaum89} [[Alain Connes]], [[Paul Baum]], _Chern character for discrete groups_, A Fête of Topology, Papers Dedicated to Itiro Tamura 1988, Pages 163-232 ([doi:10.1016/B978-0-12-480440-1.50015-0](https://doi.org/10.1016/B978-0-12-480440-1.50015-0))

* {#LueckOliver01} [[Wolfgang Lück]], [[Bob Oliver]], Section 1 of: _Chern characters for the equivariant K-theory of proper G-CW-complexes_, In: Aguadé J., Broto C., Casacuberta C. (eds.) _Cohomological Methods in Homotopy Theory_ Progress in Mathematics, vol 196. Birkhäuser 2001 ([doi:10.1007/978-3-0348-8312-2_15](https://doi.org/10.1007/978-3-0348-8312-2_15))

* {#MislinValette03} [[Guido Mislin]], [[Alain Valette]], Theorem 6.1 in: _Proper Group Actions and the Baum-Connes Conjecture_, Advanced Courses in Mathematics CRM Barcelona, Springer 2003 ([doi:10.1007/978-3-0348-8089-3](https://link.springer.com/book/10.1007/978-3-0348-8089-3))

With focus on [[commutative ring]]-structures:

* [[Anna Marie Bohmann]], [[Christy Hazel]], [[Jocelyne Ishak]], [[Magdalena Kędziorek]], [[Clover May]], *Naive-commutative structure on rational equivariant K-theory for abelian groups* ([arXiv:2002.01556](https://arxiv.org/abs/2002.01556))

* [[Anna Marie Bohmann]], [[Christy Hazel]], [[Jocelyne Ishak]], [[Magdalena Kędziorek]], [[Clover May]], *Genuine-commutative ring structure on rational equivariant K-theory for finite abelian groups* ([arXiv:2104.01079](https://arxiv.org/abs/2104.01079))

[[!redirects equivariant rational K-theory]]


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#Contents#
* table of contents
{:toc}

## Idea

_Rational [[equivariant stable homotopy theory]]_ is the study of [[equivariant spectra]] just on the level of their [[rationalization]], hence concerning only their non-[[torsion subgroup|torsion]] [[homotopy groups]]. This is the [[equivariant cohomology|equivariant]] and [[stable homotopy theory|stable]] version of [[rational homotopy theory]].

A key general statement of the theory is that rationally the [[homotopy theory]] of $G$-equivariant spectra is equivalently given by [[homological algebra]] of [[Mackey functors]] (even for non-[[finite group|finite]] $G$). At the level of [[equivalence of categories|equivalences]] of [[homotopy categories]] this was established by [[John Greenlees]], at the level of [[zig-zags]] of [[Quillen equivalences]] of [[model categories]] this was established Greenlees and by his students, [[David Barnes]] and [[Magdalena Kedziorek]].

## Properties

### Greenlees-May splitting into equivariant Eilenberg-MacLane spectra
 {#SplittingIntoMackeyFunctors}

Let $G$ be a [[finite group]]. For $X$ a [[G-spectrum]], write $\pi_\bullet(X) \in \mathcal{M}[G]$ for its [[Mackey functor]], the one which sends $G/H$ to the $H$-[[equivariant homotopy groups]] of $X$.

Every rational $G$-equivariant spectrum $E$ is the direct sum of the [[Eilenberg-MacLane spectra]] (Mackey functors) on its [[equivariant homotopy groups]]:

$$
  E \simeq \prod_n \Sigma^n H\pi_n(E)
  \,.
$$

([Greenlees-May 95, theorem A.1](#GreenleesMay95), [Greenlees, theorem 5.1](#Greenlees))

For $X,Y$ two $G$-spectra, there is a canonical morphism

$$
  [X,Y]_G \longrightarrow \underset{n}{\prod} Hom_{\mathcal{M}[G]}(\pi_n(X),\pi_n(Y))
  \,.
$$

When $Y$ is rational, then this is an [[isomorphism]] ([Greenlees-May 95, theorem A.4](#GreenleesMay95)).

### Rational tom Dieck splitting

for the moment see at _[[tom Dieck splitting]]_

## Examples

Just as for the plain [[sphere spectrum]], the [[equivariant homotopy groups]] of the [[equivariant sphere spectrum]] in ordinary [[integer]] degrees $n$ are all [[torsion subgroup|torsion]], except at $n = 0$:

$$
  \pi_n^H(\mathbb{S})\otimes \mathbb{Q} =
  \left\{
    \array{
       \cdots & for \; n = 0
       \\
       0 & otherwise
    }
  \right.
$$

([Greenlees-May 95, prop. A.3](#GreenleesMay95))

But in some [[RO(G)-degrees]] there may appear further non-torsion groups, see at _[[equivariant sphere spectrum]]_ the section  [Examples](equivariant+sphere+spectrum#Examples).

## Related concepts

* [[rational homotopy theory]]

* [[rational equivariant homotopy theory]]

  * [[rational topological G-space]]

  * [[model structure on equivariant dgc-algebras]]

* [[rational equivariant stable homotopy theory]]

* [[rational stable homotopy theory]]

* [[rational parameterized stable homotopy theory]]

## References

General:

* {#GreenleesMay95} [[John Greenlees]], [[Peter May]], appendix A of _Generalized Tate cohomology_, Mem. Amer. Math. Soc. 113 (1995) no 543 ([pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/GM-Tate-543.pdf))

* {#Greenlees} [[John Greenlees]], p. 3 of _Triangulated categories of rational equivariant cohomology theories_ ([pdf](https://pdfs.semanticscholar.org/eda8/ee6effb2ae487085a2e7615a4a7e1964c493.pdf), [[GreenleesRationalEquivariant.pdf:file]])

* [[David Barnes]], _Rational Equivariant Spectra_ ([arXiv:0802.0954](http://arxiv.org/abs/0802.0954))

* [[David Barnes]], _Classifying Rational $G$-Spectra for Finite $G$_ ([arXiv:0812.0317](http://arxiv.org/abs/0812.0317))

* [[David Barnes]], [[Magdalena Kedziorek]], _An introduction to algebraic models for rational G-spectra_ ([arXiv:2004.01566](https://arxiv.org/abs/2004.01566))

For $G = O(2)$ or [[SO(2)|$SO(2)$]] and so also for $G = $ [[dihedral group]] and [[cyclic group]]:

* {#Greenlees96} [[John Greenlees]], _Rational $O(2)$-equivariant cohomology theories_. In _Stable and unstable homotopy_ (Toronto, ON, 1996), volume 19 of Fields Inst. Commun., pages 103–110. Amer. Math. Soc., Providence, RI, 1998. ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.6588))

* {#Greenlees99} [[John Greenlees]], _Rational $S^1$-equivariant stable homotopy theory_, Memoirs of the AMS, 1999

* [[Brooke Shipley]], _An algebraic model for rational $S^1$-equivariant stable homotopy theory_ ([pdf](http://homepages.math.uic.edu/~bshipley/qt15.pdf))

* [[David Barnes]], _Classifying Dihedral $O(2)$-Equivariant Spectra_ ([arXiv:0804.3357](http://arxiv.org/abs/0804.3357))

* [[David Barnes]], _Rational $Z_p$-Equivariant Spectra_, Algebr. Geom. Topol. 11 (2011) 2107-2135 ([arXiv:1011.5785](http://arxiv.org/abs/1011.5785))

* [[David Barnes]], _Rational $O(2)$-Equivariant Spectra_ ([arXiv:1201.6610](http://arxiv.org/abs/1201.6610))

* [[David Barnes]], _A monoidal algebraic model for rational $SO(2)$-spectra_ ([arXiv:1412.1700](http://arxiv.org/abs/1412.1700))

* [[David Barnes]], [[John Greenlees]], [[Magdalena Kedziorek]], [[Brooke Shipley]], _Rational $SO(2)$-Equivariant Spectra_ ([arXiv:1511.03291](http://arxiv.org/abs/1511.03291))

For $G = (S^1)^{\times_n}$ a [[torus]]:

* [[John Greenlees]], [[Brooke Shipley]], _An algebraic model for rational torus-equivariant spectra_ ([arXiv:1101.2511](http://arxiv.org/abs/1101.2511))

For $G = SO(3)$ and hence also for the [[finite groups]] of [[ADE classification|ADE type]]:

* [[John Greenlees]], _Rational SO(3)-Equivariant Cohomology Theories_, in _Homotopy methods in algebraic topology_ (Boulder, CO, 1999), Contemp. Math. **271**,  Amer. Math. Soc. (2001) 99 ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.36.5444), [GBooks](https://books.google.de/books?id=X2YbCAAAQBAJ&lpg=PR7&ots=tdiPe0oWYn&dq=%22Homotopy%20methods%20in%20algebraic%20topology%22&lr&pg=PR7#v=onepage&q=%22Homotopy%20methods%20in%20algebraic%20topology%22&f=false))

* [[Magdalena Kedziorek]], _Algebraic models for rational G-spectra_, thesis 2014 ([web](http://etheses.whiterose.ac.uk/7699/))

* [[Magdalena Kedziorek]], _An algebraic model for rational $SO(3)$–spectra_, Algebraic & Geometric Topology 17 (2017) 3095–3136 ([arXiv:1611.08415](https://arxiv.org/abs/1611.08415), [doi:10.2140/agt.2017.17.3095](https://doi.org/10.2140/agt.2017.17.3095))

Discussion of [[commutative ring]]-structures with an eye towards [[rational equivariant K-theory]]:

* [[Anna Marie Bohmann]], [[Christy Hazel]], [[Jocelyne Ishak]], [[Magdalena Kędziorek]], [[Clover May]], *Naive-commutative structure on rational equivariant K-theory for abelian groups* ([arXiv:2002.01556](https://arxiv.org/abs/2002.01556))

* [[Anna Marie Bohmann]], [[Christy Hazel]], [[Jocelyne Ishak]], [[Magdalena Kędziorek]], [[Clover May]], *Genuine-commutative ring structure on rational equivariant K-theory for finite abelian groups* ([arXiv:2104.01079](https://arxiv.org/abs/2104.01079))

[[!redirects equivariant rational stable homotopy theory]]

[[!redirects rational stable equivariant homotopy theory]]


rational fiber lemma

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## Idea

The _fibration lemma_ ([Bousfield-Kan 72, Chapter II](#BousfieldKan72)) in ([[rational parametrized homotopy theory|parametrized]]) [[rational homotopy theory]] states sufficient conditions under which [[rationalization]] preserves [[homotopy fibers]].

## Properties

+-- {: .num_prop #RationalFibrationLemma}
###### Proposition
**(rational fibration lemma)**
Let

$$
  \array{
    F &\overset{fib(p)}{\longrightarrow}& A
    \\
    && \big\downarrow{}^{\mathrlap{p}}
    \\
    && B
  }
$$

be a [[Serre fibration]] of [[connected topological space|connected]] [[topological spaces]], with [[fiber]] $F$ (over any base point) also connected.

If, for $k$ a [[ground field]] of [[characteristic zero]]:

1. the [[fundamental group]] $\pi_1(B)$ [[nilpotent module|acts nilpotently]] on the [[rational cohomology|k-rational]] [[homology groups]] $H_\bullet(F,k)$

   (e.g. if $B$ is [[simply connected topological space|simply connected]], or if the fibration is a [[principal bundle]]),

1. at least one of $A$, $F$ is rationally [[of finite type]],

then the [[cofiber]] of any [[relative Sullivan model]] for $p$ is a [[Sullivan model]] for $F$ (both as [[dgc-algebras]] over $k$).

=--

([Félix-Halperin-Thomas 00, Theorem 15.3](#FelixHalperinThomas00), following [Halperin 83, Section 16](#Halperin83))

Moreover, if $CE(\mathfrak{l}B)$ is a [[minimal Sullivan model]] for $B$, then the cofiber of the corresponding minimal [[relative Sullivan model]] for $p$ is the [[minimal Sullivan model]] $CE(\mathfrak{l}F)$ for $F$:

\[
  \label{SullivanModelForFiber}
  \array{
    CE(\mathfrak{l}F)
    &\overset{
      cofib
      \big(
        CE(\mathfrak{l}p)
      \big)
    }{\longleftarrow}&
    CE(\mathfrak{l}_{{}_B}A)
    \\
    && \big\uparrow{}^{\mathrlap{ CE(\mathfrak{l}p) }}
    \\
    && CE(\mathfrak{l}B)
  }
\]

([Félix-Halperin-Thomas 00, Corollary on p. 199](#FelixHalperinThomas00))

But this cofiber, being the cofiber of a [[relative Sullivan model]] and hence of a [[cofibration]] in the [[projective model structure on dgc-algebras]], is in fact the [[homotopy cofiber]], and hence is a model for the [[homotopy fiber]] of the [[rationalization|rationalized]] fibration.

Therefore (eq:SullivanModelForFiber) implies that:

On fibrations of [[connected topological space|connected]] rational [[finite type]]-spaces, where the [[fundamental group]] of the base space [[nilpotent module|acts nilpotently]] on the [[homology groups]] of the [[fiber]]: _[[rationalization]] preserves [[homotopy fibers]]_.

(This is the _fibration lemma_ orginally due to [Bousfield-Kan 72, Chapter II](#BousfieldKan72).)

## Related concepts

* [[parametrized homotopy theory]]

* [[rational parametrized homotopy theory]]

* [[rational parametrized stable homotopy theory]]

## References

* {#BousfieldKan72} [[Aldridge Bousfield]], [[Daniel Kan]], Chapter II "Fiber Lemmas" of: _Homotopy Limits, Completions and Localizations_, Springer 1972 ([doi:10.1007/978-3-540-38117-4](https://doi.org/10.1007/978-3-540-38117-4))

* {#FelixHalperinThomas00} [[Yves Félix]], [[Stephen Halperin]], [[Jean-Claude Thomas]], Sections 14 and 15 of: _Rational Homotopy Theory_, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 ([doi:10.1007/978-1-4613-0105-9](https://link.springer.com/book/10.1007/978-1-4613-0105-9))

[[!redirects rational fiber lemmas]]

[[!redirects rational fibre lemma]]
[[!redirects rational fibre lemmas]]

[[!redirects rational fibration lemma]]
[[!redirects rational fibration lemmas]]

[[!redirects fiber lemma]]
[[!redirects fiber lemmas]]

[[!redirects fibre lemma]]
[[!redirects fibre lemmas]]


rational fraction

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## Definition ##

Given an [[integral domain]] $R$, the commutative ring of **rational fractions**, **rational expressions**, or **polynomial fractions** $R(z)$ with coefficients in $R$ is the [[field of fractions]] of the [[polynomial ring]] $R[z]$.

## See also ##

* [[polynomial ring]]

* [[field of fractions]]

* [[rational function]]

* [[partial fraction decomposition]]

## References

* Wikipedia, [Rational fraction](https://en.wikipedia.org/wiki/Rational_fraction)

* Wikipedia, [Rational expression](https://en.wikipedia.org/wiki/Rational_expression)

* [[Frank Quinn]], *Proof Projects for Teachers* ([pdf](https://personal.math.vt.edu/fquinn/education/pfs4teachers0.pdf))

[[!redirects rational fractions]]

[[!redirects rational expression]]
[[!redirects rational expressions]]

[[!redirects polynomial fraction]]
[[!redirects polynomial fractions]]


rational function

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## Definition

### As partial functions on a field

If $k$ is a [[field]], given the [[polynomial ring]] $k[x]$, there is a canonical [[ring homomorphism]] $j:k[x] \to (k \to k)$ which sends [[constant function|constant]] [[polynomials]] in $k[x]$ to [[constant functions]] in $k \to k$, and the [[generator]] $x \in k[x]$ to the [[identity function]] $id_k$, where $k \to k$ is the [[function algebra]] on $k$.

There is a function
$$i:k(x) \to \{A \in Ob(Part(k)) \vert Hom(A, k)\}$$
from the field of [[rational expression]] to the set of all [[partial functions]] in the [[category of partial functions]] $Part(k)$, where the [[function algebra]] $k \to k$ is the [[endomorphism]] $k$-algebra on the [[improper subset]] $k$. The function $i$ is defined as follows: given two polynomials $p, q \in k[x]$ where $q \neq 0$, for all $x \in \{a \in k | q(a) \neq 0\}$
$$i\left(\frac{p}{q}\right)(x) \coloneqq \frac{j(p)(x)}{j(q)(x)}$$
where for $x \in k$ and $y \in k$
$$\frac{x}{y}$$
is the [[division]] of $x$ by $y$ in $k$, and for $r \in k(x)$ and $s \in k(x)$,
$$\frac{r}{s}$$
is the [[division]] of $r$ by $s$ in $k(x)$. A **rational function** $f$ is an [[element]] of the [[image]] of $i$, $f \in \im(i)$.

### As functions on the projective line

It is perhaps more illuminating to think of this partial function (with domain $D$) as coming from a (total) function $[p/q]: \mathbb{P}^1(k) \to \mathbb{P}^1(k)$ on the [[projective line]], where we have inclusion functions $i: k \to \mathbb{P}^1(k)$ and the partial function is given by the pair of projection maps in the [[pullback]]

$$\array{
D & \to & k \\
\downarrow & & \downarrow_\mathrlap{i} \\
k & \underset{[p/q] \circ i}{\to} & \mathbb{P}^1(k)
}$$

It should also be noticed that such endofunctions $[p/q]$ on $\mathbb{P}^1(k)$ are closed under composition (except that special provision must be made for the constant function valued at $\infty$, which corresponds to the "fraction" $1/0$).

## Properties

### Rational functions on a field do not form a field

This comes from the fact that the reciprocal function is not a multiplicative inverse of the identity function $f(x) = \frac{x}{x} \neq 1$, due to the fact that $f(x)$ is undefined at $0$, and thus has a different domain than the constant function $1$, which is the multiplicative unit.

### As continuous functions

[[rational functions are continuous]] on their domain of definition

## Related concepts

* [[polynomial]]

* [[power series]]

* [[rational map]]

* [[homotopy of rational maps]]

* [[category of partial functions]]

* [[rational fraction]]

* [[real rational function]]

* [[rational function on an affine variety]]

## References

* Wikipedia, [Rational function](https://en.wikipedia.org/wiki/Rational_function)

On the [[homotopy type of spaces of rational maps]] from the [[Riemann sphere]] to itself (related to the [[moduli space of monopoles]] in $\mathbb{R}^3$ and to the [[configuration space of points]] in $\mathbb{R}^2$):

* [[Graeme Segal]], _The topology of spaces of rational functions_, Acta Math. Volume 143 (1979), 39-72 ([euclid:1485890033](https://projecteuclid.org/euclid.acta/1485890033))

[[!redirects rational functions]]

[[!redirects ring of rational functions]]
[[!redirects rings of rational functions]]


rational function on an affine variety

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## Definition

Let $X$ be an [[affine variety]] over a field $k$ with the ring of [[regular function]]s $\mathcal{O}(X)$. A __rational function__ is any element of the [[field of fractions]] of $\mathcal{O}(X)$, that is the [[function field]] of the variety.

## Function field analogy

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

## See also

* [[rational function]]

* [[rational expression]]

* [[function field]]


rational homotopy equivalence

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## Idea

A **rational homotopy equivalence** is the notion of equivalence of [[topological space]]s as used in [[rational homotopy theory]]. Where a [[weak homotopy equivalence]] in ordinary [[homotopy theory]] identifies spaces under morphisms that induce isomorphisms on all [[homotopy group]]s, rational homotopy equivalences identify spaces under morphisms that induce isomorphisms on all _rationalized_ homotopy groups.

## Definition

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

For $X$ and $Y$ be [[simply connected space|simply connected]] [[topological space]]s and $f : X \to Y$ a continuous map between them, $f$ is called a **rational homotopy equivalence** if the following equivalent conditions are satisfied:

1. it induces an isomorphism on rationalized homotopy groups: $\pi_*(f) \otimes \mathbb{Q} : \pi_*(X) \otimes \mathbb{Q} \stackrel{\simeq}{\to} \pi_*(X) \otimes \mathbb{Q}$;

1. it induces an isomorphism on [[ordinary homology]] groups with rational [[coefficients]]: $H_*(f,\mathbb{Q}) : H_*(X,\mathbb{Q}) \stackrel{\simeq}{\to} H_*(X,\mathbb{Q})$;

1. it induces an isomorphism on rational [[cohomology]] groups: $H^*(f,\mathbb{Q}) : H^*(X,\mathbb{Q}) \stackrel{\simeq}{\to} H^*(X,\mathbb{Q})$;

1. it induces a [[weak homotopy equivalence]] on the  [[rationalization]]s $X_{ra}$ and $Y_{ra}$ : $f_{ra} : X_{ra} \stackrel{\simeq_{whe}}{\to} Y_{ra}$.

=--

That (the first two of) these conditions are equivalent is due to [Serre 53](#Serre53)

## References

The concept is due to

* {#Serre53} [[Jean-Pierre Serre]], _Groupes d'homotopy et classes de groupes abelians_, Ann. of Math. 58 (1953) 258-294

Review includes

* [[Kathryn Hess]], around definition 1.17 in _Rational homotopy theory: a brief introduction_ ([arXiv:math.AT/0604626](http://arxiv.org/abs/math.AT/0604626))