[[!redirects basic line bundle on the 2-sphere]]

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### Context
#### Bundles
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#Contents#
* table of contents
{:toc}

## Idea

The _basic line bundle on the 2-sphere_ is the [[complex line bundle]] on the [[2-sphere]] whose [[first Chern class]] is a generator $\pm 1 \in \mathbb{Z} \,\simeq\, H^2(S^2, \mathbb{Z})$, equivalently the [[tautological line bundle]] on the [[Riemann sphere]] regarded as [[complex projective space|complex projective 1-space]].

This is the [[pullback bundle]] of the map $S^2 \to B U(1) \simeq B^2 \mathbb{Z}$ to the [[classifying space]]/[[Eilenberg-MacLane space]] which itself represents a generator of the [[homotopy group]] $\pi_2(S^2) \simeq \mathbb{Z}$.

Beware that this basic line bundle is sometimes called the "canonical line bundle on the 2-sphere", but it is  _not_ [[isomorphism|isomorphic]] to what in complex geometry is called the [[canonical bundle]] of the 2-sphere regarded as a [[Riemann surface]]. Instead it is "one half" of the latter, its [[theta characteristic]]. See also at _[[geometric quantization of the 2-sphere]]_.

The basic line bundle is the canonically [[associated bundle]] to basic [[circle principal bundle]]: the [[complex Hopf fibration]]. Another name for it is the [[tautological line bundle]] for the complex [[projective space|projective line]] $\mathbb{P}^1(\mathbb{C})$ (the [[Riemann sphere]]), namely the map $\mathbb{C}^2 \setminus \{(0, 0)\} \to \mathbb{P}^1(\mathbb{C})$ mapping $(x, y)$ to $[x; y]$.

## Definition

The [[classifying space]] for [[circle principal bundles]], or equivalently (via forming [[associated bundles]]) that of [[complex line bundles]] is $B U(1)$, which as a [[Grassmannian]] is the infinite [[complex projective space]] $\mathbb{C}P^\infty$. The [[homotopy type]] of this space is that of the [[Eilenberg-MacLane space]] $K(\mathbb{Z},2)$. This means that $K(\mathbb{Z},2)$ is in particular [[path-connected topological space|path-connected]] and has second [[homotopy group]] the [[integers]]: $\pi_2(K(\mathbb{Z},2)) \simeq \mathbb{Z}$.

It being the [[classifying space]] for complex line bundles means that

$$
  \left\{
     \array{
        \text{isomorphism classes of}
        \\
        \text{complex line bundles}
        \\
        \text{on}\,\, S^2
     }
  \right\}
  \;\simeq\;
  \left\{
    \array{
      \text{continuous functions}
      \\
      S^2 \longrightarrow K(\mathbb{Z},2)
      \\
      \text{up to homotopy}
    }
  \right\}
  \;\simeq\;
  \pi_2(K(\mathbb{Z},2))
  \;\simeq\;
  \mathbb{Z}
  \,.
$$

The ([[isomorphism class]]) of the complex line bundle which corresponds to $+1 \in \mathbb{Z}$ under this sequence of [[isomorphisms]] is called the _basic complex line bundle on the 2-sphere_.

Hence the basic complex line bundle on the 2-sphere is [[generalized the|the]] [[pullback bundle]] of the [[universal complex line bundle]] on $B U(1)$ along the map $S^2 \to B U(1)$ which represents the element $1 \in \mathbb{Z} \simeq \pi_2(B U(1))$. If the [[classifying space]] $B U(1)$ is represented by the infintie [[complex projective space]] $\mathbb{C}P^\infty$ with its canonical [[CW-complex]] structure ([this prop.](complex+projective+space#CellComplexStructureOnComplexProjectiveSpace)), then this map is represented by the canonical cell incusion $S^2 \hookrightarrow\mathbb{C}P^\infty$.

Notice that there is a non-trivial [[automorphism]] of $\mathbb{Z}$ as an [[abelian group]] given by $n \mapsto -n$. This means that there is an ambiguity in the definition of the basic line bundle on the 2-sphere.

## Properties

+-- {: .num_lemma #ClutchingConstructionOfBasicLineBundle}
###### Lemma
**([[clutching construction]] of the basic line bundle)

Under the [[clutching construction]] of [[vector bundles]] on the [[2-sphere]], the basic complex line bundle on the 2-sphere is given by the [[transition function]]

$$
  \mathbb{C} \supset \, S^1 \longrightarrow GL(1,\mathbb{C}) \, \subset \mathbb{C}
$$

from the [[Euclidean space|Euclidean]] [[circle]] $S^1 \subset \mathbb{R}^2 \simeq \mathbb{C}$ to the complex [[general linear group]] in 1-dimension, which is $GL(1,\mathbb{C}) \simeq \mathbb{C} \setminus \{0\}$ given simply by

$$
  z \mapsto z
  \,,
$$

Alternatively, due to the sign ambiguity in the definition of the basic bundle, its clutching transition function is given by

$$
  z \mapsto - z
  \,.
$$

=--

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

Under the [[clutching construction]] the [[isomorphism class]] of a complex line bundle corresponds to the [[homotopy class]] of its clutching transition function

$$
  S^1 \to GL(1, \mathbb{C}) \simeq \mathbb{C} \setminus \{0\}
$$

hence to an element of the [[fundamental group]] $\pi_1(\mathbb{C} \setminus \{0\}) \simeq \mathbb{Z}$. Hence by definition, the basic bundle has clutching transition function corresponding to $\pm 1  \in \mathbb{Z} \simeq [S^1, GL(1,\mathbb{Z})]$ and this element is represented by the function $z \mapsto \pm z$.

=--

+-- {: .num_prop #TensorRelationForBasicLineBundleOn2Sphere}
###### Proposition
**(fundamental tensor/sum relation of the basic complex line bundle)**

Under [[direct sum of vector bundles]] $\oplus_{S^2}$ and [[tensor product of vector bundles]] $\otimes_{S^2}$, the basic line bundle on the 2-sphere $H \to S^2$ satisfies the following relation

$$
  H \oplus_{S^2} H
    \;\simeq\;
  \left(
    H \otimes_{S^2} H
  \right)
    \oplus_{S^2}
  1_{S^2}
$$

(where $1_{S^2}$ denotes the [[trivial vector bundle]] [[complex line bundle]] on the 2-sphere).

=--

(e.g ([Hatcher, Example 1.13](#Hatcher)))

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

Via the [[clutching construction]] there is a single [[transition function]] of the form

$$
  S^1 \longrightarrow GL(n,\mathbb{C})
$$

that characterizes all the bundles involved. With $S^1 \hookrightarrow \mathbb{C}$ identified with the [[topological subspace]] of [[complex numbers]] of unit [[absolute value]], the standard choice for these functions is

* for the [[trivial vector bundle|trivial]] [[line bundle]] $1_{S^2}$ we may choose

  $f_1 \colon z \mapsto \left( 1 \right)$;

* for the basic line bundle we may choose (by lemma \ref{ClutchingConstructionOfBasicLineBundle})

  $f_H \colon z \mapsto \left( z\right)$

This yields

* for $H \otimes H \oplus 1_{S^2}$ the clutching function

  $z \mapsto \left( \array{ z^2 & 0 \\ 0 & 1 }\right)$

* for $H \oplus H$ the clutching function

  $z \mapsto \left( \array{ z & 0 \\ 0 & z }  \right)$.

Since the complex [[general linear group]] $Gl(n,\mathbb{C})$ is [[path-connected topological space|path-connected]] (by [this prop.](general+linear+group#ConnectednessOfGeneralLinearGroup)), there exists a [[continuous function]]

$$
  \gamma \colon [0,1] \longrightarrow GL(2,\mathbb{C})
$$

connecting the identity matrix on $\mathbb{C}^2$ with the one that swaps the two entries, i.e.

with $\gamma(0) = \left( \array{ 1 & 0 \\ 0 & 1 }  \right)$

and $\gamma(1) = \left(  \array{ 0 & 1 \\ 1 & 0 } \right)$

Therefore the function

$$
  \array{
    S^1 \times [0,1] &\overset{}{\longrightarrow}& GL(2,\mathbb{C})
    \\
    (z,t)
      &\overset{\phantom{AA}}{\longrightarrow}&
    f_{H \oplus 1}(z) \cdot \gamma(t) \cdot f_{1 \oplus H}(z) \cdot \gamma(t)
  }
$$

(with [[matrix multiplication]] on the right) is a [[left homotopy]] from $f_{H \oplus H}$ to $f_{H \otimes H \oplus 1}$.

=--

+-- {: .num_remark}
###### Remark
**([[fundamental product theorem in topological K-theory]])**

Under the map

$$
  Vect(S^2)_{/\sim} \longrightarrow K(X)
$$

that sends [[complex vector bundles]] to their class in the [[topological K-theory]] ring $K(X)$, the fundamental tensor/sum relation of prop. \ref{TensorRelationForBasicLineBundleOn2Sphere} says that the K-theory class $H$ of the basic line bundle in $K(X)$ satisfies the relation

$$
  \begin{aligned}
    (H - 1)^2
    & =
    H^2  + 1 - \underset{= H^2 + 1}{\underbrace{2 H}}
    \\
    = &   0
  \end{aligned}
$$

in $K(X)$.

(Notice that $H-1$ is the image of $[H]$ in the [[reduced K-theory]] $\tilde K(X)$ of $S^2$ under the splitting $K(X) \simeq \tilde K(X) \oplus \mathbb{Z}$ (by [this prop.](topological+K-theory#KGrupDirectSummandReducedKGroup)).)

It follows that there is a [[ring homomorphism]] of the form

$$
  \array{
    \mathbb{Z}[h]/\left( (h-1)^2 \right)
      &\overset{}{\longrightarrow}&
    K(S^2)
    \\
    h &\overset{\phantom{AAA}}{\mapsto}& H
  }
$$

from the [[polynomial ring]] in one abstract generator, [[quotient ring|quotiented]] by this relation, to the [[topological K-theory]] ring.

It turns out that this homomorphism is in fact an [[isomorphism]], hence that the relation $(H-1)^2 = 0$ from prop. \ref{TensorRelationForBasicLineBundleOn2Sphere} is the _only_ relation satisfied by the basic complex line bundle in topological K-theory.

More generally, for $X$ a [[topological space]], then there is a composite ring homomorphism

$$
  \array{
    K(X) \otimes \mathbb{Z}[h]/((h-1)^2)
      & \longrightarrow &
    K(X) \times K(S^2)
      & \longrightarrow &
    K(X \times S^2)
    \\
    (E, h)
      &\overset{\phantom{AAA} }{\mapsto}&
    (E,H)
      &\overset{\phantom{AAA}}{\mapsto}&
    (\pi_{X}^\ast E) \cdot (\pi_{S^2}^\ast H)
  }
$$

to the topological K-theory ring of the [[product topological space]] $X \times S^2$, where the second map is the [[external tensor product]] of vector bundles.

This composite is an [[isomorphism]] if $X$ is a [[compact Hausdorff space]] (for $X = \ast$ the [[point space]] this reduces to the previous statement).

This is called the _[[fundamental product theorem in topological K-theory]]_. It is the main ingredient in the [[proof]] of [[Bott periodicity]] in complex topological K-theory.

=--

## Related concepts

* [[universal complex line bundle]]

* [[tautological line bundle]], [[equivariant tautological line bundle]]

## References

* {#Hatcher} [[Allen Hatcher]], _Vector bundles and K-theory_ ([web](https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html))

* {#Wirthmuller12} [[Klaus Wirthmüller]], _Vector bundles and K-theory_, 2012 ([[wirthmueller-vector-bundles-and-k-theory.pdf:file]])

[[!redirects basic line bundles on the 2-sphere]]


basic differential form

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## Definition

### For general submersions

Given a [[submersion]] $p\colon E\to B$, one may ask: which [[differential forms]] on $E$ are [[pullback of differential forms|pullbacks]] of [[differential forms]] on $B$?

If the [[fibers]] of $p$ are [[connected topological space|connected]] (otherwise the characterization given below is valid only locally in $E$), the answer is provided by the notion of a __basic form__: a differential form $\omega$ is basic if the following two conditions are met:

* The [[tensor contraction|contraction]] of $\omega$ with any $p$-[[vertical vector field]] is [[zero]].

* The [[Lie derivative]] of $\omega$ with respect to any $p$-[[vertical vector field]] is [[zero]].

Using [[Cartan's magic formula]], in the presence of the first condition, the second condition can be replaced by the following one:

* The [[tensor contraction|contraction]] of $d\omega$ (where $d$ is the [[de Rham differential]]) with any $p$-[[vertical vector field]] is zero.

### Via pullback to smooth principal bundles

Given a [[Lie group]] $G$ and a [[smooth manifold|smooth]] $G$-[[principal bundle]] $P \overset{p}{\longrightarrow} X$ over a base [[smooth manifold]] $X$, a [[differential form]] $\omega \in \Omega^\bullet\big( P \big)$ on the total space $P$ is called _basic_ if it is the [[pullback of differential forms]] along the [[bundle]] projection $p$ of a differential form $\beta \in \Omega^\bullet\big( X \big)$ of the base manifold

$$
  \omega \;=\; p^\ast(\beta)
  \,.
$$

### In terms of Cartan calculus

Equivalently, if $\mathfrak{g} \simeq T_e G$ denotes the [[Lie algebra]] of $G$, and for $v \in \mathfrak{g}$ we write

$$
  \hat v
  \;\colon\;
  P
  \overset{ (e,v), (-,0) }{\hookrightarrow}
  T G \times T P
  \simeq
  T ( G \times P )
  \overset{ \;\;\; d \rho \;\;\; }{\longrightarrow}
  T P
$$

for the [[vector field]] on $P$ which is the [[derivative]] of the $G$-[[action]] $G \times P \overset{\rho}{\to} P$ along $v$, then differential form $\omega$ is _basic_ precisely if

1. it is annihilated by the contraction with $\hat v$

   $$
     \iota_{\hat v} \omega = 0
   $$

1. it is annihilated by the [[Lie derivative]] along $\hat v$:

   $$
     \mathcal{L}_{\hat v}\omega
       \;=\;
     [d_{dR}, \iota_{\hat v}] \omega
       \;=\;
     0
   $$

   (where the first equality holds generally by [[Cartan's magic formula]], we are displaying it just for emphasis)

for all $v \in \mathfrak{g}$.

In this form the definition of basic forms makes sense more generally whenever a [[Cartan calculus]] is given, not necessarily exhibited by smooth vector fields on actual manifolds. This more general concept of basic differential forms appears notably in the construction of the Weil model for [[equivariant de Rham cohomology]].

## Related concepts

* [[horizontal differential form]]

[[!redirects basic differential forms]]


basic Fraenkel model

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{: toc}

## Idea

The _basic Fraenkel model_ is a model of the [[set theory]] [[ZFA]] that doesn't satisfy the [[axiom of choice]]. It was one of the first examples of a [[permutation model]] of set theory. In the basic Fraenkel model, we have a countable set of atoms $A$, and we take the full permutation group $G$ of $A$. The normal filter of subgroups is generated by the stabilizers of finite subsets of $A$.

The [[second Fraenkel model]] is similar, but uses the [[countable set|countable]] group $(\mathbb{Z}/2\mathbb{Z})^\mathbb{N}$.

## Description

### Material set theory

In the language of [[material set theory]], this is just a [[Fraenkel-Mostowski model]] given by a countable set of atoms $A$ with the full permutation group $G$ acting on it. The normal filter of subgroups is generated by the [[stabilizers]] of [[finite]] subsets of $A$, ie. groups of the form
$$
  Stab (K) = \{ g \in G: \forall x \in K, gx = x\},
$$
where $K$ is a finite subset of $A$.

### Structural set theory

We can take the same group $G$ as above, with topology generated by the stabilizers $Stab (K)$. We can then form the [[topos]] of [[category of continuous G sets|continuous G-sets]], whose objects are [[sets]] with a continuous [[action]] of $G$ (with the set given the discrete topology), and morphisms are the $G$-equivariant maps. This topos is also known as the [[Schanuel topos]].

The [[internal logic]] of this topos can be identified with the standard logic of the material model of ZFA constructed above, if we interpret the universal quantifiers in ZFA suitably (eg. via [[stack semantics]], or other semantics as mentioned in [[Fraenkel-Mostowski models]])

## Properties

It is clear that the set of atoms in the Fraenkel-Mostowski model cannot be linearly ordered. So the ordering principle (that every set can be linearly ordered) fails. Consequently, the [[axiom of choice]], [[prime ideal theorem#boolean_pit_and_the_ultrafilter_principle|boolean prime ideal theorem]] etc. all fail. In fact, even countable choice fails.

We can consider the object $\mathcal{A}$ consisting of the set of atoms with the obvious action of $G$. It can be shown that this does not biject with any finite set (ie. a set that is a finite coproduct of the terminal object), but any injection $\mathcal{A} \to \mathcal{A}$ must be a surjection (this is true both as an external statement or as a statement in the [[internal logic]]). So this is an example of a [[Dedekind-finite]] but not [[finite]] set.

## Variations

We can consider the case where we have more atoms, or allow stabilizers of larger subsets. In different cases, weak versions of Choice may hold. For example, if we have $\aleph_1$ many atoms, and take the normal filter generated by stabilizers of countable subsets, then the axiom of choice holds for any well-ordered family of sets. Details can be found in the book [Consequences of the Axiom of Choice](#consequences).

## Related concepts

 * [[Fraenkel-Mostowski model]]
 * [[G-set]]
 * [[topos of G-sets]]
 * [[second Fraenkel model]]
 * [[Schanuel topos]]
 * [[ZFA]]

## References

The model is given by $\mathcal{N}1$ in the book

 * {#consequences} Paul Howard, Jean E. Rubin, _Consequences of the Axiom of Choice_, AMS Mathematical Surveys and Monographs 59

The various properties of the model are listed in detail. Variations of the model are listed as $\mathcal{N}12$ and $\mathcal{N}16$.

[[!redirects basic Fraenkel model of ZFA]]


basic gerbe

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\tableofcontents

## Idea

With some abuse of terminology, by the "basic gerbe" over a [[Lie group]] $G$ authors mean, following [Meinrenken 2003](#Meinrenken03), a $U(1)$-[[bundle gerbe]] (typically understood [[connection on a bundle gerbe|with 2-connection]]) on the [[underlying]] [[topological space]] (or underlying [[smooth manifold]]) of $G$, whose [[Dixmier-Douady class]] is a generator (whence "basic") of the [[ordinary cohomology]]-[[cohomology group|group]] $H^3(G;\mathbb{Z})$.

For that to make verbatim sense we need for instance that $H^3(G;\mathbb{Z}) \,\simeq\, \mathbb{Z}$, which is the case notably for [[simply connected topological space|simply connected]] [[compact Lie groups|compact]] [[simple Lie groups]] $G$, such as [[SU(n)]] for $n \geq 2$ or [[Spin(n)]] for $n \geq 3$ (but the notion has been considered more generally).

If this bundle gerbe is considered [[connection on a bundle gerbe|with 2-connection]], then the [[curvature]] 3-form is typically understood to be a [[de Rham cohomology|de Rham representative]] of $H^3(G;\mathbb{R})$ by a [[differential 3-form]] which is $G$-bi-invariant (i.e. invariant under [[pullback of differential forms]] along the action of $G$ on itself by multiplication.)

With due care, the basic bundle gerbe itself has a multiplicative structure covering that on $G$ and making it a [[2-group]], as such equivalent to the [[string 2-group]] ([Waldorf 2012](#Waldorf12)).

With bundle gerbes understood as a [[Kalb-Ramond field|Kalb-Ramond]] [[background field]] for a [[string]] (its background [[B-field]]), the basic gerbes define the [[worldsheet]] [[sigma-model]] known as the [[Wess-Zumino-Witten model]], which was the original motivation of [Gawędzki & Reis 2002](#GawędzkiReis02). In terms of [[higher prequantum geometry]] the bundle gerbe is the [[prequantum circle 2-bundle]] of the theory ([Fiorenza, Sati & Schreiber 2015](#FiorenzaSatiSchreiber15), [Fiorenza, Rogers & Schreiber 2016](#FiorenzaRogersSchreiber16)).

## References

The original articles:

* {#GawędzkiReis02} [[Krzysztof Gawędzki]], [[Nuno Reis]], *WZW branes and gerbes*, Rev. Math. Phys. **14** (2002) 1281-1334 [[arXiv:hep-th/0205233](https://arxiv.org/abs/hep-th/0205233), [doi:10.1142/S0129055X02001557](https://doi.org/10.1142/S0129055X02001557)]

* {#Meinrenken03} [[Eckhard Meinrenken]], _The basic gerbe over a compact simple Lie group_, Enseign. Math. (2) 49 (2003), no. 3-4, 307-333 [[arXiv:math/0209194](https://arxiv.org/abs/math/0209194), [e-periodica](https://www.e-periodica.ch/digbib/view?pid=ens-001:2003:49::562)]

* [[Krzysztof Gawędzki]], [[Nuno Reis]], _Basic gerbe over non-simply connected compact groups_, Journal of Geometry and Physics **50** 1-4 (2004) 28–55 [[arXiv:0307010](https://arxiv.org/abs/math/0307010), [doi:10.1016/j.geomphys.2003.11.004](http://dx.doi.org/10.1016/j.geomphys.2003.11.004)]

A specific construction for [[unitary groups]] and generalization to groups with [[unitary representations]] on a [[Hilbert space]]:

* [[Michael K. Murray]], [[Danny Stevenson]], _The basic bundle gerbe on unitary groups_, Journal of Geometry and Physics **58** 11 (2008) 1571-1590 [[doi:10.1016/j.geomphys.2008.07.006](https://doi.org/10.1016/j.geomphys.2008.07.006), [arXiv:0804.3464](https://arxiv.org/abs/0804.3464)]

Review:

* [[Christoph Schweigert]], [[Konrad Waldorf]], *Gerbes and Lie Groups*, in: *Developments and Trends in Infinite-Dimensional Lie Theory*, Progress in Mathematics **288** (2011) [[arXiv:0710.5467](https://arxiv.org/abs/0710.5467), [doi:10.1007/978-0-8176-4741-4_10](https://doi.org/10.1007/978-0-8176-4741-4_10)]

As a model for the [[string 2-group]]:

* {#Waldorf12} [[Konrad Waldorf]], *A Construction of String 2-Group Models using a Transgression-Regression Technique*, Contemp. Math. **584** (2012) 99-115 [[arXiv:1201.5052](http://arxiv.org/abs/1201.5052), [doi:10.1090/conm/584/11588](https://doi.org/10.1090/conm/584/11588)]

and as the [[prequantum circle 2-bundle]] of the [[WZW model]] discussed in [[smooth infinity-groupoids]]:

* {#FiorenzaSatiSchreiber15} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], §3.4.2 in: *[[schreiber:A higher stacky perspective on Chern-Simons theory]]*, in: *Mathematical Aspects of Quantum Field Theories*, Mathematical Physics Studies, Springer (2015) 153-211 [[arXiv:1301.2580](https://arxiv.org/abs/1301.2580), [doi:10.1007/978-3-319-09949-1_6](https://doi.org/10.1007/978-3-319-09949-1_6)]

* {#FiorenzaRogersSchreiber16} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], §4.1 in: _[[schreiber:Higher geometric prequantum theory|Higher $U(1)$-gerbe connections in geometric prequantization]]_, Rev. Math. Phys. **28** 06 1650012 (2016) [[arXiv:1304.0236](http://arxiv.org/abs/1304.0236), [doi:10.1142/S0129055X16500124](https://doi.org/10.1142/S0129055X16500124)]

[[!redirects basic gerbes]]

[[!redirects basic bundle gerbe]]
[[!redirects basic bundle gerbes]]


basic ideas of moduli stacks of curves and Gromov-Witten invariants > history

< [[basic ideas of moduli stacks of curves and Gromov-Witten invariants]]


basic ideas of moduli stacks of curves and Gromov-Witten theory

+-- {: .standout}

This is a sub-entry of [[Gromov-Witten invariants]]. See there for further background and context.

This entry is supposed to provide an exposition of some basic ideas underlying Gromov-Witten theory.

=--

> **raw material**: notes taken verbatim in some seminar -- needs to be polished

#Contents#

* [part I: basics of moduli stacks of curves](#intropart1)

* [part I: basics of Gromov-Witten theory](#intropart2)

## Intro Part I: basics of moduli stacks of curves {#intropart1}

**question**: what is a **moduli problem**?

* some kind of object;

* a notion of "familiy"/"deformation" of these objects

* some [[equivalence relation]] $\sim$ (possibly trivial)
  on the set of such families $S(B)$ over $B$

from this we get a [[functor]] (a [[presheaf]])

$$
  F : C^{op} \to Set
$$

$$
  B \mapsto S(B)/_\sim
$$

**terminology** an object $M$ is called a [[fine moduli space]] if the corresponding [[representable functor|represented functor]]

$$
  Y_M : C^{op} \to Set
$$

$$
  x \mapsto Hom(x,M)
$$

is isomorphic to $F$, i.e. if it represents $F$.

**examples** in the [[homotopy category]] of [[topological space]]s we have

$$
  \left\{
    complex line bundles over B
  \right\}/isom
  \leftrightarrow
  Hom_{Ho(Top)}(B, \mathbb{C}P^\infty)
$$

so $\mathbb{C}P^\infty$ is a [[classifying space]] for complex line bundle. Similarly for higher rank vector bundles and Grassmannians.

The analogue in the algebraic category is

$$
  \left\{
      line bundles L over B with generating sections s_0,...,s_n \in \Gamma(L) \right\} / isom
\leftrightarrow
Hom(B, \mathbb{P}^n).
$$

And also similarly for higher rank [[vector bundle]]s and Grassmannians.

Despite these examples, in a lot of cases the functors are not representable. We'll see some of these examples below.

Why are [[fine moduli space]]s desireable?

They allow us to study a _single_ family which tells us universal things about all families. Even if you do not care about families or deformations, moduli spaces can help, because perhaps they can tell you something about trivial families, i.e. the objects that you are studying to begin with.

**example** studying the [[cohomology ring]]s of $Gr_n(\mathbb{R}^\infty)$ or $Gr_n(\mathbb{C}^\infty)$, which are the classifying space for higher rank real and complex [[vector bundle]]s gives universal relations (or, rather, the lack thereof!) among [[Chern class]]es, etc.

Let's look at [[elliptic curve]]s (we'll work over $\mathbb{C}$).

the functor of families here is

$$
  F : Sch/\mathbb{C}^{op} \to Set
$$

$$
  B \mapsto \left\{
     \array{
        E \\ \downarrow \\ B
     }
     flat families of elliptic curves
  \right\}
$$

**Fact**: there is no [[fine moduli space]] of [[elliptic curve]]s representing this functor

**why not?**

there is something called the  [[j-invariant]] which classifies [[elliptic curve]]s up to [[isomorphism]]

let

$$
  E : y^2 = x^3 + a x + b
$$

be an [elliptic curve]] given by parameters $a,b$. Then its [[j-invariant]] is the number

$$
  j(E) = \frac{2^8 3^3 a^3}{4 a^3 + 27 b^2}
$$

so we might guess that the "$j$-line" $\mathbb{A}^1$ is a [[fine moduli space]] for [[elliptic curve]]s,

i.e. that there is a "universal family" of elliptic curves $C \to \mathbb{A}^1$ such that the fiber over $j \in \mathbb{A}^1$ is the elliptic cuvre with that $j$-invariant.

so that for any family $E \to B$ we'd have a [[pullback]]

$$
  \array{
    E &\to& C
    \\
    \downarrow && \downarrow
    \\
    B &\to& \mathbb{A}^1
  }
$$

where $B \to \mathbb{A}^1$ sends a point in $B$ to the [[j-invariant]] of the elliptic curve in the fiber over $B$.

Now consider the family

$$
  \chi = (y^2 = x(x-1)(x+\lambda)) \subset \mathbb{A}^1 \times \mathbb{A}_\lambda^1 \times \mathbb{P}^2
$$

$$
  \array{
    \chi &\to& C
    \\
    \downarrow && \downarrow
    \\
    \mathbb{A}^1_\lambda &\to& \mathbb{A}^1
  }
$$

$$
  j(\chi_\lambda) = s^8 \frac{(\lambda^2 - \lambda + 1)^2}{\lambda^2(\lambda-1)^2}
$$

now send $\mathbb{A}^1_\lambda \to \mathbb{A}^1_{\lambda}$

by dually sending $\lambda \mapsto 1-\lambda$

check that $j(\chi_\lambda) = j(\chi_{1-\lambda})$

this induces in the fibers a map

$\phi : \chi \to \chi$

$$
  \phi : (x,y,\lambda) \mapsto (1-x,\pm i y , 1-\lambda)
$$

this maps further to $(x,-y,\lambda)$. But this would have to be the identity for $\mathbb{A}^1$ to be a fine moduli space, which it is not. So $\mathbb{A}^1$ at least is not a fine moduli space, even though it might look like one.

so that gives some computational insight that something goes wrong

This argument does not yet prove that there exists no moduli space of elliptic curves. It merely proves that the "j-line" $\mathbb{A}^1$ can not be the moduli space of elliptic curves.
However, the basic argument can be adapted, if one so desires, to in fact prove that there is no moduli space of elliptic curves.

In fact, below we will see (in an exercise), that the $j$-line _is_ a [[coarse moduli space]], as explained below. Moreover, if a coarse moduli space exists, then it is unique up to canonical isomorphism. Since fine moduli spaces, if they exist, are also coarse moduli spaces, and the $j$-line is a coarse but not a fine moduli space by the above argument, it follows that no fine moduli space exists.

**Abstract argument** Yet another, more abstract way, to see that no fine moduli space can exist it to realize that since elliptic curves have nontivial automorphisms, it is possible to construct families of elliptic curves that are locally trivial families (of the form $U \times E \to U$ for a fixed elliptic curve $U$) but which are glued together from these local pieces using nontrivial automorphisms such that the resulting family $V \to X$ is not globally trivial, i.e. not globally of the form $E \to X \to X$.

With a bit of care this alone can be used to show that a fine moduli space cannot exist. This is often summarized by a slogan of the form

> **Slogan** : nontrivial automorphisms of objects prevent the family-assignment of these objects to be representable by a fine moduli space.

However, one has to be careful with interpreting this slogan correctly. Taken naively, the slogan alone would also seem to imply that, since [[vector space]]s have nontrivial automorphisms, no classifying space for "families of vector spaces", i.e. for [[vector bundle]]s does exist, while of course this does exist (recall also the example further above). But if one interprets the slogan carefully, it does yield a true statement. For more on that see the discusson at [[moduli space]].

**How to "fix" these problems**.

1. add extra structure to the objects under consideration (e.g. add marked points) to make the automorphism groups trivial.

2. instead of looking for representing [[topological space]]s, look for representing [[groupoid]]s / [[stack]]s.

3. use [[coarse moduli space]]s  $M \in Sch/\mathbb{C}$ with $\Psi_M : F \to h_M$ such that

   a) $F(Spec(\mathbb{C})) \to h_M(Spec \mathbb{C}) = hom(Spec \mathbb{C}, M)$ is a [[bijection]]

   b) given $M'$ and $\Psi_{M'} : F \to h_{M'}$ then there exists unique $M \to M'$ such that $\array{ F && \stackrel{\Psi_{M'}}{\to}&& h_{M'} \\ & {}_{\Psi_M}\searrow && \nearrow \\ && h_M} $

So a [[coarse moduli space]] is one that at least has the right underlying set of points as the _right_ [[moduli stack]] has: as long as we don't look at families but just at single things, it does give the right classification.

**exercise** show that the $j$-line $\mathbb{A}^1$ _is_, while not a [[fine moduli space]], a [[coarse moduli space]].

**exercise** Show that if a coarse moduli space exists, then it is unique up to canonical isomorphism.

**fact** there exists a [[coarse moduli space]] $M_{g,n}$ of [[Riemann surface]]s of [[genus]] $g$ with $n$ marked points and a [[fine moduli stack]] $M_{g,n}$ such that for all $g,n$ we have

$M_{g,n}$ is a [[smooth scheme|smooth]] [[Deligne-Mumford stack]] - aka [[orbifold]]

**except** for $(g,n) =$ $(0,0), (0,1), (0,2), (1,0)$ (these are the cases where the [[automorphism group]] is infinite, so in these cases we don't get a [[Deligne-Mumford stack]])

**"Issues"**:

1. $M_{g,n}$ is not proper, meaning: not compact, so we don't have, for example, [[Poincare duality]] on $M_{g,n}$ and no [[integration]] theory. Proper, or even better projective, schemes or stacks are just a lot easier to deal with.

2. Sometimes one wants to study singular curves or families with degeneracies.

Both "issues" can be "resolved" via [[Deligne-Mumford compactification]].

$$
  \bar M_{g,n}
$$

which parameterizes at-most-nodal curves, that are connected, of arithmetic [[genus]] $g$, with $n$ smooth marked points, and the group of [[automorphism]]s is finite.

$\bar M_{g,n}$ is a smooth proper [[Deligne-Mumford stack]].

smooth here means smoothness as for [[orbifold]]s.

Deligne and Mumford were able to prove many theorems about the ordinary moduli space of curves by studying instead the compactification. For example they were able to prove that $M_{g,n}$ is irreducible.

#Intro Part II: basics of Gromov-Witten theory{#intropart2}

**Gromov** was looking for invariants of [[symplectic manifold]]s: his idea was to use $J$-holomorphic curves in compact symplectic manifolds to get symplectic invariants

**[[Edward Witten]]** and other physicists studied [[worldsheet]]s of [[string theory|string]]s in some [[spacetime]] [[manifold]] (e.g a [[Calabi-Yau space|Calabi-Yau 3-fold]])

we want to consider now [[genus]] $g$ [[Riemann surface]]s with $n$ marked points mapping into some space $X$

For a fixed

$$
  \beta \in H_2(X, \mathbb{Z}),
$$

$$
  M_{g,n}(X,\beta)
$$

is the space that parameterizes maps

$$
  \Sigma \stackrel{f}{\to} X
$$

where $\Sigma$ is smooth, and has $n$ marked points, and
such that for $[\Sigma]$ the [[fundamental homology class]] of $\Sigma$ we have

$$
  f_*[\Sigma] = \beta
  \,.
$$

this is a smooth [[Deligne-Mumford stack]]. (Again we must exclude the cases of small $(g,n)$. Excluding these cases, the automorphisms of the surfaces and the automorphisms of the maps are automatically finite.)

similarly, write $\bar M_{g,n}(X,\beta)$ for the same setup but with $\Sigma$ from $\bar M_{g,n}$ as above in part 1 (the DM compactified moduli stack). EXCEPT here we do not require that $\Sigma$ (with its $n$ marked points) has finite automorphism group; we require instead that the MAP has finite automorphism group, which means.......(fill in)

This is a _proper_ (compact) [[Deligne-Mumford stack]] - but _not_ smooth (not even in the sense of smooth stacks!). This is very important. This is what makes the theory difficult/nontrivial/interesting.

we have

$$
  \bar M_{g,n}(pt,0) = \bar M_{g,n}
$$

what do [[string theory|string theorists]] want to do?

we have evaluation maps

$$
  \bar M_{g,n}(X,\beta) \stackrel{ev_i}{\to} X
$$

where $i$ labels a marked point, and we want morphisms

$$
  H^\bullet(X)^{\otimes n}
  \stackrel{ev^*_1 \wedge ev^*_2 \wedge \cdots}{\to}
  H^*(\bar M_{g,n}(X,\beta))
  \stackrel{\int_{[\bar M_{g,n}(x,\beta)]^{virtual}}}{\to}
  \mathbb{C}
$$

There should be a virtual fundamental class $[\bar M_{g,n}(x,\beta)]^{virtual}$ that makes the maps above into the [[correlation function]]s of a [[quantum field theory]] (the integral would be the [[path integral]] of the [[worldsheet]] [[sigma-model]]) with state space $H^*(X)$. We'll explain some of what this means below.

This [[virtual fundamental class]] in algebraic geometry was constructed by Li-Tian and Behrend-Fantechi; in symplectic geometry it was done by Li-Tian.

Why do we want to use a "virtual" fundamental class? Because \bar M_{g,n}(x,\beta) is not smooth, the actual fundamental class may not behave very well. **NEEDS ELUCIDATION -- it would be nice if someone could give a more complete explanation of why we need a _[[virtual fundamental class|virtual]]_ fundamental class**

the mathematical structure of GW-theory was elucidated Ruan and then by [[Maxim Kontsevich]] and Manin in 1994.

$$
  \array{
    \bar M_{g,n}(X,\beta)
    \\
    \downarrow^\phi
    \\
    \bar M_{g,n}
  }
$$

$$
  I_{g,n,\beta} :
  H^*(X)^{\otimes}
  \to
  H^*(\bar M_{g,n}(X,\beta))
  \stackrel{\phi_*}{\to}
  H^*(\bar M_{g,n})
$$

where the map $\phi_*$ is some kind of "pushforward" or "integration along the fibers"; this map uses the virtual fundamental class

there is also a map

$$
  \alpha
  :
  \bar M_{g_1,n_1+1}
  \times
  \bar M_{g_2,n_2+1}
  \to
  \bar M_{g,n}
$$

with $g = g_1 + g_2$ and $n = n_1 + n_2$ obtained by gluing the last marked point of the first curve to the first marked point of the second curve.

so consider now the combination of these two maps

$$
  \array{
    H^\bullet(X)^{\otimes}
    &\stackrel{I_{g,n,\beta} }{\to}&
    H^\bullet(\bar M_{g,n}(X,\beta))
   \stackrel{\phi_*}{\to}
   H^*(\bar M_{g,n})
   \\
   \downarrow^{- \otimes \Delta \otimes -}
   &&
   \downarrow^\alpha
   \\
   H^\bullet(X)^{\otimes n_1}
  \otimes H^\bullet(X) \otimes
   H^\bullet(X) \otimes H^\bullet(X)^{\otimes n_2}
   &&
   H^{\bullet}(\bar M_{g_1,n_1+1} \times \bar M_{g_2,n_2+1})
   \\
   \downarrow^\simeq
   && \downarrow^{\simeq}
   \\
   H^\bullet(X)^{\otimes n_1 + 1}
   \otimes
   H^\bullet(X)^{\otimes n_2 + 1}
   &\stackrel{\sum_{\beta_1+\beta_2 = \beta}I_{g_1,n_1+1,\beta_1} \otimes I_{g_2,n_2+1,\beta_2}}{\to}&
   H^{\bullet}(\bar M_{g_1,n_1+1}) \otimes H^\bullet (\bar M_{g_2,n_2+1}))
   }
  $$

where the bottom right iso uses the [[Kunneth formula|Künneth formula]]

here $\Delta_a$ is a homogeneous basis of $H^\bullet(X)$

$$
  g_{a b} = \int_X \Delta_a \wedge \Delta_b
$$

$$
  \Delta = \sum g^{a b} \Delta_a \wedge \Delta_b
$$

where

$$
  g^{a b} = (g_{a b})^{-1}
$$

so this diagram above says that this satisfies the [[sewing law]]s that defines a [[quantum field theory]].

There are various other axioms that Gromov-Witten theory must satisfy, but the sewing law above is the most important. The hard part of all of this is constructing the virtual fundamental class, and then proving that this class indeed makes the sewing law and the other axioms (which I have omitted) satisfied.

The above discussion does not yet reveal much of the rich structure of Gromov-Witten invariants. But GW invariants indeed have a very rich and beautiful mathematical structure. Indeed, if we look at just the easiest part of the theory, namely g=0, we are lead to quantum cohomology and Frobenius manifolds.

There is also the mirror symmetry conjecture, which roughly posits that the GW invariants can be found via calculations that a priori seem completely unrelated.

#References#
* M. Kontsevich, Yu. Manin, _Gromov-Witten classes, quantum cohomology, and enumerative geometry_, Comm. Math. Phys. 164 (1994), no. 3, 525--562 ([euclid](http://projecteuclid.org/euclid.cmp/1104270948)).

* [[Yuri Manin]], _Frobenius manifolds, quantum cohomology and moduli spaces_, Amer. Math. Soc., Providence, RI, 1999.

* W. Fulton, R. Pandharipande, _Notes on stable maps and quantum cohomology_, in: Algebraic Geometry-
Santa uz 1995 ed. Kollar, Lazersfeld, Morrison. Proc. Symp. Pure Math. 62, 45–96 (1997)

* J Robbin, D A Salamon, _A construction of the Deligne-Mumford orbifold_,  J. Eur. Math. Soc. (JEMS)  8  (2006),  no. 4, 611--699 ([arxiv](http://arxiv.org/abs/math/0407090); [pdf at JEMS](http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=8&iss=4&rank=3)); corrigendum  J. Eur. Math. Soc. (JEMS)  9  (2007),  no. 4, 901--905 ([pdf at JEMS](http://www.ems-ph.org/journals/show_pdf.php?issn=1435-9855&vol=9&iss=4&rank=11)).

* J Robbin, Y Ruan, D A Salamon, _The moduli space of
regular stable maps_,  Math. Z.  259  (2008),  no. 3, 525--574 ([doi](http://dx.doi.org/10.1007/s00209-007-0237-x)).

* Martin A. Guest, _From quantum cohomology to integrable systems_, Oxford Graduate Texts in Mathematics, 15. Oxford University Press, Oxford, 2008. xxx+305 pp.

* Joachim Kock, Israel Vainsencher, _An invitation to quantum cohomology. Kontsevich's formula for rational plane curves_, Progress in Mathematics, 249. Birkhäuser Boston, Inc., Boston, MA, 2007. xiv+159 pp.

* Sheldon Katz, _Enumerative geometry and string theory_, Student Math. Library __32__. IAS/Park City AMS & IAS 2006. xiv+206 pp.

* Eleny-Nicoleta Ionel, Thomas H. Parker, _Relative Gromov-Witten invariants_, Ann. of Math. (2) 157 (2003), no. 1, 45--96 ([doi](http://dx.doi.org/10.4007/annals.2003.157.45)).

[[!redirects basic ideas of moduli stacks of curves and Gromov–Witten theory]]
[[!redirects basic ideas of moduli stacks of curves and Gromov--Witten theory]]
[[!redirects basic ideas of moduli stacks of curves and Gromov-Witten invariants]]
[[!redirects basic ideas of moduli stacks of curves and Gromov–Witten invariants]]
[[!redirects basic ideas of moduli stacks of curves and Gromov--Witten invariants]]


basic localizer

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

# Basic localizers
* table of contents
{: toc}

## Idea

By a _basic localizer_ one means a [[localizer]] on the category [[Cat]] of [[categories]], hence a choice of a [[class]] of [[functors]] to be called the _[[weak equivalences]]_, subject to some conditions.

These conditions ensure in particular that a basic localizer always contains the weak equivalences of the [[Thomason model structure]] on [[Cat]] (see [Maltsiniotis 11, 1.2.1](#Maltsiniotis11)), the [[localization]] at which is equivalent to the standard [[homotopy category]]. On the other hand, the standard [[equivalences of categories]], which are the weak equivalences in the [[canonical model structure]] on [[Cat]], do not form a basic localizer.

Hence basic localizers are a tool for [[homotopy theory]] modeled on [[category theory]]. In fact, their introduction by [[Grothendieck]] was motivated from the study of [[test categories]] (see remark \ref{CitationOnTestCategories} below).

## Definition

The definition is due to [[Grothendieck]]:

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

A **basic localizer** is a [[class]] $W$ of [[morphisms]] in [[Cat]] such that

1. $W$ contains all [[identities]], satisfies the [[2-out-of-3 property]] and, for any functor $i\colon A\to B$ such that there exists a functor $r\colon B\to A$ with $r i=1_A$ and $i r$ in $W$, we have $i$ in $W$ (in the literature this is sometimes called being *weakly saturated*),

1. If $A$ has a [[terminal object]], then the functor $A\to 1$ is in $W$, and

1. Given a commutative triangle in $Cat$:
   $$\array{ A & & \overset{u}{\to} & & B\\ & _v \searrow & & \swarrow_w \\ & & C } $$
   if each induced functor $v/c \to w/c$ between [[comma categories]] is in $W$, then $u$ is also in $W$.

=--

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

The term in French is **localisateur fondamental**, which is sometimes translated as **fundamental localizer**.

=--

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

One can show that basic localizers are stable under [[retracts]]; see Proposition 4.2.4 in ([Cisinski 06](#Cisinski06)). They are also closed under small filtered colimits; see Corollaire 4.2.22 in ([Cisinski 06](#Cisinski06)).

=--

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

In [[Pursuing Stacks]] [[Grothendieck]] wrote about def. \ref{BasicLocalizer} the following:

> These conditions are enough, I quickly checked this night, in order to validify all results developed so far on [[test categories]], [[weak test categories]], [[strict test categories]], weak test functors and test functors (with values in $(Cat)$) (of notably the review in par. 44, page 79–88), provided in the case of test functors we restrict to the case of loc. cit. when each of the categories $i(a)$ has a final object. All this I believe is justification enough for the definition above.

=--

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

Grothendieck's comments above refer in fact to a weaker definition, in which in Definition \ref{BasicLocalizer}, we replace the third condition with an analogue of Quillen's Theorem A, namely: any functor $f\colon A\to B$ such that, for any object $b$ of $B$, the induced functor $A/b\to 1$ is in $W$, is in $W$. This weaker notion is sometimes called a **weak basic localizer**.

=--

## Examples

* The class of *all* functors between small categories is, of course, the maximal basic localizer.

* The class of functors inducing an [[isomorphism]] on [[connected components]] is a basic localizer.

* The class of functors whose [[nerve]] is a [[weak homotopy equivalence]] is a basic localizer. (These are the [[weak equivalences]] in the [[Thomason model structure]].)

* For any [[derivator]] $D$, the class of $D$-equivalences is a basic localizer.  This includes all the previous examples.

* The class of [[equivalences of categories]] is *not* a basic localizer (it fails the second condition). (These are the weak equivalences of the [[canonical model structure]].)

## Properties

### Asphericity and local equivalences

If $W$ is a basic localizer, we define the following related classes.  We sometimes refer to functors in $W$ as *weak equivalences*.

* A category $A$ is **($W$-)aspherical** if $A\to 1$ is in $W$.  Thus the second axiom says exactly that any category  with a terminal object is aspherical.
* A functor $u\colon A\to B$ is **($W$-)aspherical** if for all $b\in B$, the comma category $u/b$ is aspherical.
* When the hypotheses of the third axiom are satisfied, we say that $u$ is a **local weak equivalence over $C$**.  Thus the third axiom says exactly that every local weak equivalence is a weak equivalence.

+-- {: .num_example}
###### Example
If $W = \pi_0$-equivalences, then a category is aspherical iff it is connected, and a functor is aspherical iff it is [[initial functor|initial]].
=--

+-- {: .num_example}
###### Example
If $W = $ nerve equivalences, then a category is aspherical iff its nerve is contractible, and a functor is aspherical iff it is [[homotopy initial functor|homotopy initial]].
=--

We observe the following.

* A category $A$ is aspherical iff the functor $A\to 1$ is aspherical, since the only comma category involved in the latter assertion is $A$ itself.

* An aspherical functor is a weak equivalence.  For if $u\colon A\to B$ is aspherical, then consider the triangle
  $$\array{ A & & \overset{u}{\to} & & B\\ & _u \searrow & & \swarrow \\ & & B } $$
  The third axiom tells us to consider, for a given $b\in B$, the functor $u/b \to B/b$.  But $u/b$ is aspherical by assumption, while $B/b$ is aspherical by the second axiom since it has a terminal object.  Thus, by 2-out-of-3, the functor $u/b \to B/b$ is in $W$, and thus by the third axiom $u$ is in $W$.

* If $u$ has a right adjoint, then it is aspherical.  For in this case, each category $u/b$ has a terminal object, and thus is aspherical.

* If $I$ denotes the [[interval category]], then for any category $A$ the projection $A\times I\to A$ has a right adjoint, hence is aspherical and thus a weak equivalence.  By 2-out-of-3, the two injections $A \rightrightarrows A\times I$ are also weak equivalences, so $A\times I$ is a [[cylinder object]] for $W$.  It follows that if we have a natural transformation $f\to g$, then $f$ is in $W$ if and only if $g$ is.  Moreover, if $f$ is a "homotopy equivalence" in the sense that it has an "inverse" $g$ such that $f g$ and $g f$ are connected to identities by arbitrary natural zigzags, then $f$ is a weak equivalence.

* In particular, any left or right adjoint is a weak equivalence.

### Self-duality

It is a non-obvious fact that the notion of basic localizer is self-dual.

+-- {: .num_theorem}
###### Theorem
A functor $u : A \to B$ is in a basic localizer $W$ if and only if $u^{op} : A^{op} \to B^{op}$ is in $W$.
=--

+-- {: .proof}
###### Proof
See Proposition 1.1.22 in ([Maltsiniotis 05](#Maltsiniotis05)).
=--

### Cisinski's theorem

Since the definition consists merely of closure conditions, the intersection of any family of basic localizers is again a basic localizer.  It follows that there is a unique *smallest* basic localizer.  The following was conjectured by [[Grothendieck]] and proven by [[Denis-Charles Cisinski]].

+-- {: .num_theorem}
###### Theorem (Cisinski)
The class of functors whose [[nerve]] is a [[weak homotopy equivalence]] is the smallest basic localizer.
=--

+-- {: .proof}
###### Proof
See Théorème 2.2.11 in ([Cisinski 04](#Cisinski04)). A completely different proof is given in Corollaire 4.2.19 in ([Cisinski 06](#Cisinski06))
=--

Note that this is a larger class than the class of "[[homotopy equivalences]]" considered above.  For instance, the category generated by the graph
$$\bullet \leftarrow \bullet \rightarrow \bullet \leftarrow \bullet \rightarrow \bullet \leftarrow \bullet \rightarrow \cdots$$
has a contractible nerve, but its identity functor is not connected to a constant one by any natural zigzag.

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

One can in fact prove that the class of functors whose nerve is a weak homotopy equivalence is the smallest weak basic localizer, as defined in Remark \ref{weakBasicLocalizer}; see Théorème 6.1.18 in ([Cisinski 06](#Cisinski06)). This refinement is related to the fact that weak homotopy equivalences are determined by the property that they induce equivalences of categories of locally constant presheaves; see Corollary 1.17 in ([Cisinski 09](#Cisinski09)). This is also related to the fact that cohomology with coefficients in local systems determine weak homotopy equivalences; see Théorème 6.5.11 and Scholie 6.5.13 in ([Cisinski 06](#Cisinski06)).

=--

## Related concepts

* [[canonical model structure]], [[Thomason model structure]] on [[Cat]]

* [[Cisinski model structure]]

## References

* [[Denis-Charles Cisinski]], _[[joyalscatlab:Les préfaisceaux comme type d'homotopie]]_, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages ([numdam](http://www.numdam.org/item/AST_2006__308__R1_0/))
 {#Cisinski06}

* [[Denis-Charles Cisinski]], _Le localisateur fondamental minimal_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 45 (2004) no. 2, pp. 109-140, [numdam](http://www.numdam.org/item/CTGDC_2004__45_2_109_0/)
 {#Cisinski04}

* [[Denis-Charles Cisinski]], _Locally constant functors_, Math. Proc. Camb. Phil. Soc. Volume 147 (2009), pp. 593-614, [arXiv](https://arxiv.org/abs/0803.4342)
 {#Cisinski09}

* [[Georges Maltsiniotis]], _La théorie de l'homotopie de Grothendieck _, Astérisque, Volume 301, Soc. Math. France (2005), 146 pages ([numdam](http://www.numdam.org/item/AST_2005__301__R1_0/))
 {#Maltsiniotis05}

See also at _[[Cisinski model structure]]_.

* [[Georges Maltsiniotis]], _Homotopical exact squares and derivators_ ([arXiv:1101.4144](http://arxiv.org/abs/1101.4144))
 {#Maltsiniotis11}

[[!redirects basic localizor]]
[[!redirects basic localizors]]
[[!redirects basic localizers]]
[[!redirects basic localiser]]
[[!redirects basic localisers]]
[[!redirects fundamental localizer]]
[[!redirects fundamental localizers]]
[[!redirects fundamental localiser]]
[[!redirects fundamental localisers]]
[[!redirects localisateur fondamental]]


basics of etale cohomology

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

This page goes through some basics of [[étale cohomology]].

#Contents#
* table of contents
{:toc}

## Étale topos

To every [[scheme]] $X$ is assigned a [[site]] which is a geometric analog of the collection of [[étale spaces]] over a [[topological space]]. This is called the [[étale site]] $X_{et}$ of the scheme. The [[category of sheaves]] on that site is called the [[étale topos]] of the scheme. The intrinsic [[cohomology]] of that [[topos]], hence the [[abelian sheaf cohomology]] over the [[étale site]], is the _[[étale cohomology]]_ of $X$.

This section starts with looking at some basic aspects of the [[étale topos]] as such, the basic definitions and the central [[descent theorem]] for characterizing its [[sheaves]]. The [next section](#EtaleCohomology) then genuinely considers the corresponding [[abelian sheaf cohomology]].

Étale cohomology is traditionally motivated by the route by which it was historically discovered, namely as a fix for technical problems encountered with the [[Zariski topology]]. You can find this historical motivation in all textbooks and lectures, see the _[References](#References)_ below.

But [[étale cohomology]] has a more fundamental _raison d'être_ than this. As discussed at _[[étale topos]]_ it is induced in any context in which one has a "[[reduction modality]]". While fundamental, this is actually a simple point of view which leads to a simple characterization of [[étale morphisms]], and this is what we start with now.

### Étale morphisms

Of the many equivalent characterizations of [[étale morphisms]], here we will have use of the following incarnation:

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

A morphisms of [[schemes]] is an _[[étale morphism of schemes]]_ if it is

1. [[formally étale morphism of schemes|formally étale]] -- recalled in a [moment](#ExplicitDefinition);

1. [[locally of finite presentation]].

=--

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

The first condition makes an [[étale morphism of schemes]] be like an [[étale space]] over its [[codomain]]. The second essentially just says demands this has [[finite set|finite]] [[fibers]].

=--

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

For $X$ a [[scheme]], its [[étale site]] has a [[objects]] the [[étale morphisms of schemes]] into $X$, as [[morphisms]] the morphisms of schemes [[over category|over]] $X$, and as [[coverings]] the jointly surjective [[étale morphisms of schemes|étale morphisms]] over $X$.

The [[category of sheaves]] on $X_{et}$ is the _[[étale topos]]_ of $X$. The corresponding [[abelian sheaf cohomology]] is its _[[étale cohomology]]_.

=--

The definition of [[formally étale morphisms of schemes|formally étale]] in components goes like this.

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

A [[morphism]] of [[commutative rings]] $R \longrightarrow A$
is called _[[formally étale morphisms of schemes|formally étale]]_ if for every ring $B$ and for every [[nilpotent ideal]] $I \subset B$ and for every [[commuting diagram]] of the form

$$
  \array{
    B/I &\leftarrow& A
    \\
    \uparrow && \uparrow
    \\
    B &\leftarrow& R
  }
$$

there is a unique diagonal morphism

$$
  \array{
    B/I &\leftarrow& A
    \\
    \uparrow &\swarrow& \uparrow
    \\
    B &\leftarrow& R
  }
$$

that makes both triangles commute.

=--

(e.g. [Stacks Project 57.9, 57.12](#StackProject))

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

So  [[Isbell duality|dually]] this means that $Spec(A) \to Spec(R)$ is formally étale if it has the unique [[right lifting property]] against all [[infinitesimal object|infinitesimal extensions]]

$$
  \array{
    Spec(B_{red}) &\longrightarrow& Spec(A)
    \\
    \downarrow &\nearrow& \downarrow
    \\
    Spec(B) &\longrightarrow & Spec(R)
  }
  \,.
$$

=--

and [[sheafification|locality]] this yields a notion of formally étale morphisms of [[affine varieties]] and of [[schemes]].

It is useful to realize this equivalently but a bit more naturally as follows.

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

Write $CRing_{fin}$ for the [[category]] of [[finitely generated ring|finitely generated]] [[commutative rings]] and write $CRing_{fin}^{ext}$ for the category of [[infinitesimal ring extensions]].
Write

$$
  Red \;\colon\; CRing_{fin}^{ext} \longrightarrow CRing_{fin}
$$

for the [[functor]] which sends an [[infinitesimal ring extension]] to the underlying [[commutative ring]] (in the maximal case this sends a commutative ring to its [[reduced ring]], whence the name of the functor), and write

$$
  i \;\colon\; CRing_{fin} \hookrightarrow CRing_{fin}^{ext}
$$

for the [[full subcategory]] inclusion that regards a ring as the trivial infinitesimal extension over itself.

=--

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

There is an [[adjoint triple]] of [[idempotent monad|idempotent]] ([[comonad|co]]-)[[monads]]

$$
  (Red \dashv \int_{inf} \dashv \flat_{inf}) \;\colon\; PSh((CRing_{fin}^{ext})^{op}) \longrightarrow PSh((CRing_{fin}^{ext})^{op})
$$

where the [[left adjoint]] [[comonad]] $Red$ is given on [[representable functor|representables]] by the [[reduced scheme|reduction]] functor of def. \ref{ReductionOnRings} (followed by the inclusion).

=--

This statement and the following prop. \ref{FormalEtalenessBydeRhamSpace} is a slight paraphrase of an observation due to ([Kontsevich-Rosenberg 04](Q-category#KontsevichRosenbergSpaces)). A closely related adjunction appeared in ([Simpson-Teleman 13](de+Rham+space#SimpsonTeleman)) in the discussion of [[de Rham spaces]]. The general abstract situation of "[[differential cohesion]]" has been discussed in ([Schreiber 13](http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos)).

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

The  functors from def. \ref{ReductionOnRings} form an
[[adjoint pair]] $(Red \dashv i)$ because an extension element
can only map to an extension element; so for $\widehat R \to R$
an [[infinitesimal ring extension]] of $R = Red(\widehat R)$,
and for $S$ a commutative ring with
$i(S) = (S \to S)$ its trivial extension, there is a [[natural isomorphism]]

$$
  Hom_{CRing_{fin}^{ext}}(\widehat R, i(S))
  \simeq
  Hom_{CRing_{fin}}(R,S)
  \,.
$$

This exhibits $CRing_{fin}$ as a [[reflective subcategory]]
of $CRing_{fin}^{ext}$.

$$
  (Red \dashv i)
   \;\colon\;
  CRing_{fin}
   \stackrel{\overset{Red}{\leftarrow}}{\underset{i}{\hookrightarrow}}
  CRing_{fin}^{ext}
  \,.
$$

Via [[Kan extension]] this [[adjoint pair]] induces an [[adjoint quadruple]] of [[functors]] on [[categories of presheaves]]

$$
  PSh(CRing_{fin}^{op})
    \stackrel{\overset{i_!}{\hookrightarrow}}{\stackrel{\overset{i^\ast = Red_!}{\leftarrow}}{\stackrel{\overset{Red^\ast}{\hookrightarrow}}{\underset{Red_\ast}{\leftarrow}}}}
  PSh((CRing_{fin}^{ext})^{op})
  \,.
$$

The [[adjoint triple]] to be shown is obtained from composing these adjoints pairwise.

That $Red$ coincides with the reduction functor on representables is a standard property of [[left Kan extension]] (see [here](Kan+extension#LeftKanOnRepresentables) for details).

=--

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

These considerations make sense in the general abstract context of
"[[differential cohesion]]" where the [[adjoint triple]] of prop. \ref{DifferentialCohesionModality} would be called:

([[reduction modality]] $\dashv$ [[infinitesimal shape modality]] $\dashv$ [[infinitesimal flat modality]]).

=--

Due to the [[full subcategory]] inclusion $i_!$ in the proof of prop. \ref{DifferentialCohesionModality} we may equivalently regard presheaves on $(CRing_{fin})^{op}$ (e.g. [[schemes]]) as presheaves on $(CRing_{fin}^{ext})^{op}$ (e.g. [[formal schemes]]). This is what we do implicitly in the following.

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

A morphism $f \;\colon\; Spec A \to Spec R$ in $CRing_{fin}^{op} \hookrightarrow PSh(CRing_{fin}^{op})$ is
formally étale, def. \ref{ExplicitDefinition}, precisely if it is $\int_{inf}$-[[modal type|modal]] relative $Spec R$, hence if the [[natural transformation|naturality square]] of the [[infinitesimal shape modality]]-[[unit of a monad|unit]]

$$
  \array{
    Spec A &\longrightarrow& \int_{inf} Spec A
    \\
    \downarrow && \downarrow
    \\
    Spec R &\longrightarrow& \int_{inf} Spec R
  }
$$

is a [[pullback]] square.

=--

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

Evaluated on $I \hookrightarrow R  \to R/I \in CRing_{fin}^{ext}$ any object, by the [[Yoneda lemma]] and the $(Red \dashv \int_{inf})$-[[adjunction]] the naturality square becomes

$$
  \array{
    CRing(A,B) &\longrightarrow& CRing(A,B/I)
    \\
    \downarrow && \downarrow
    \\
    CRing(R,B) &\longrightarrow& CRing(R,B/I)
  }
  \,.
$$

in [[Set]]. Chasing elements through this shows that this is a [[pullback]] precisely if the condition in def. \ref{ExplicitDefinition} holds.

=--

The basic stability property of [[étale morphisms]], which we need in the following, immediately follows from this characterization:

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

For $\stackrel{f}{\to} \stackrel{g}{\to}$ two composable morphisms, then

1. if $f$ and $g$ are both (formally) étale, then so is their composite $g \circ f$;

1. if $g$ and $ g\circ f$ are (formally) étale, then so is $f$;

1. the [[pullback]] of a (formally) étale morphism along any morphism is again (formally) étale.

=--

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

With prop. \ref{FormalEtalenessBydeRhamSpace} this is equivalently the
statement of the [[pasting law]] for [[pullback]] diagrams.

=--

Apart from that, for the proofs in the following we need the following basic facts

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

* Every etale morphism is a [[flat morphism]].

* Flat morphism between affines $Spec(B) \to Spec(A)$ is [[faithfully flat]] precisely if it is surjective

=--

We repeatedly use the following example of étale morphisms.

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

Every [[open immersion of schemes]] is an [[étale morphism of schemes]].
In particular a standard open inclusion (a [[cover]] in the [[Zariski topology]]) induced by the [[localization of a commutative ring]]

$$
  Spec(R[S^{-1}]) \longrightarrow Spec(R)
$$

is étale.

=--

(e.g. [[The Stacks Project|Stacks Project, lemma 28.37.9]])

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

By def. \ref{EtaleMoprhism} we need to check that the map $Spec(R[S^{-1}]) \longrightarrow Spec(R)$ is a [[formally étale morphism]] and [[locally of finite presentation]].

The latter is clear, since the very definition of [[localization of a commutative ring]]

$$
  R[S^{-1}] = R[s_1^{-1}, \cdots, s_n^{-1}](s_1 s_1^{-1} - 1, \cdots , s_n s_n^{-1} - 1)
$$

exhibits a [[finitely presented algebra]] over $R$.

To see that it is formally étale we need to check that for every [[commutative ring]] $T$ with [[nilpotent ideal]] $J$ we have a [[pullback]] diagram

$$
  \array{
    Hom(R[S^{-1}], T) &\longrightarrow& Hom(R[S^{-1}],T/J)
    \\
    \downarrow && \downarrow
    \\
    Hom(R, T) &\longrightarrow& Hom(R, T/J)
  }
  \,.
$$

Now by the [[universal property]] of the [[localization of a commutative ring|localization]], a homomorphism $R[S^{-1}] \longrightarrow T$ is a homomorphism $R \longrightarrow T$ which sends all elements in $S \hookrightarrow R$ to invertible elements in $T$. But no element in a [[nilpotent ideal]] can be invertible, Therefore the fiber product of the bottom and right map is the set of maps from $R$ to $T$ such that $S$ is taken to invertibles, which is indeed the top left set.

=--

### Descent theorem and examples of étale sheaves
 {#SheafConditionAndExamples}

Since there are "many more" [[étale morphisms of schemes]] than there are [[open immersions of schemes]], a priori the discussion of [[descent]] over the [[étale site]] is more intricate than that in, say, the [[Zariski topology]]. However, the following proposition drastically reduces the types of étale [[covers]] over which [[descent]] has to be checked in addition to the [[open immersions of schemes|open immersions]]. Then the following [[descent theorem]] effectively solves the descent problem over these remaining covers.

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

For $X$ a [[scheme]], and $A \in PSh(X_{et})$ a [[presheaf]] on its [[étale site]], def. \ref{EtaleSite},
for checking the [[sheaf]] condition it is sufficient to
check [[descent]] on the following two kinds of [[covers]]
in the [[étale site]]

1. jointly surjective collections of [[open immersions of schemes]];

1. single [[faithfully flat morphisms]] between [[affine schemes]]

(all over $X$).

=--

([Tamme, II Lemma (3.1.1)](#Tamme), [Milne, prop. 6.6](#Milne))

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

Suppose given an arbitrary étale [[covering]] $\{X'_i \to X'\}$ over $X$.
We show how to refine it to a more special cover which itslf is the composition of covers of the form as in the statement.

To that end, first choose a cover $\{U'_j \to X'\}$ of $X_i$ by affine [[open immersions of schemes]]. Then pulling back the original cover along that one yields covers

$$
  \{X'_i \times_{X'} U'_j \to U'_j\}
$$

of each of the open affines. By pullback stability, prop. \ref{ClosureForFormallyEtale}, these are still étale maps. Now these patches in turn we cover by open affines

$$
  \{
    \{U'_{i j k} \to X'_i \times_{X'} U'_j \}
  \}
$$

leading to covers

$$
  \{
   U'_{i j k} \to U'_j
  \}
$$

by affines.

(Notice here crucially that while the $U'_{i j k}$ are affine open immersions in $X'_i \times_{X'} U'_j$, after this composition with an [[étale morphism]] they no longer need to be open immersions in $U'_j$, all we know is that the map is étale. This is the source of the second condition in the proposition to be shown, as discussed now. )

Since each $U'_j$, being affine, is a [[quasi-compact scheme]], we may find a finite subcover

$$
  \{
    U'_{j l} \to U'_j
  \}
  \,.
$$

Composed with the original $\{U'_j \to X'\}$ this yields a refinement of the original cover by open affines.

Hence for checking descent it is sufficient to check it for these two kinds of overs. The latter is by open immersions. For the former, we may factor

$\{U'_{j l} \to U'_j\}$  as a collection of open immersions

$$
  \{U'_{j i} \to \coprod U'_{j i}\}
$$

followed by the epimorphism of affines of the form

$$
  \{
    \coprod U'_{j i} \to U'_j
  \}
  \,.
$$

Now this is morphism is etale, hence [[flat morphism|flat]], but also surjective. That makes it a [[faithfully flat morphism]].

=--

Therefore we are led to consider [[descent]] along [[faithfully flat morphisms]] of affines. For these the _[[descent theorem]]_ says that they are [[effective epimorphisms]]:

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

Given a [[commutative ring]] $R$ and an $R$-[[associative algebra]] $A$, hence a [[ring]] [[homomorphism]] $f \colon R \longrightarrow A$, the _[[Amitsur complex]]_ is the [[Moore complex]] of the dual [[Cech nerve]] of $Spec(A) \to Spec(R)$, hence the [[chain complex]]

$$
  0 \to R \stackrel{f}{\to} A \stackrel{1 \otimes id - id \otimes 1}{\longrightarrow} A \otimes_R A \to A \otimes_R A \otimes_R A \to \cdots
  \,.
$$

=--

(See also at _[[Sweedler coring]]_ and at _[[commutative Hopf algebroid]]_ for the same or similar constructions.)

+-- {: .num_prop #DescentTheorem}
###### Proposition
**(descent theorem)**

If $A \to B$ is [[faithfully flat]] then its Amitsur complex is [[exact sequence|exact]].

=--

This is due to ([[Grothendieck]], [[FGA]]1). The following reproduces the proof in low degree following ([Milne, prop. 6.8](#Milne)).

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

We show that

$$
  0 \to A \stackrel{f}{\longrightarrow} B \stackrel{1 \otimes id - id \otimes 1}{\longrightarrow} B \otimes_A B
$$

is an [[exact sequence]] if $f \colon A \longrightarrow B$ is [[faithfully flat]].

First observe that the statement follows if $A \to B$ admits a [[section]] $s \colon B \to A$. Because then we can define a map

$$
  k \colon B \otimes_A B \longrightarrow B
$$

$$
  k \;\colon\; b_1 \otimes b_2 \mapsto b_1 \cdot f(s(b_2))
  \,.
$$

This is such that applied to a [[coboundary]] it yields

$$
  k(1 \otimes b - b \otimes 1) = f(s(b)) - b
$$

and hence it exhibits every [[cocycle]] $b$ as a coboundary $b = f(s(b))$.

So the statement is true for the special morphism

$$
  B \to B \otimes_A B
$$

$$
  b \mapsto b \otimes 1
$$

because that has a section given by the multiplication map.

But now observe that the morphism $B \to B \otimes_A B$ is the [[tensor product]] of the morphism $f$ with $B$ over $A$, hence the [[Amitsur complex]] of this morphism is [[exact sequence|exact]].

Finally, the fact that $A \to B$ is [[faithfully flat]] by assumption, hence that it exhibits $B$ as a [[faithfully flat module]] over $A$, means by definition that the [[Amitsur complex]] for $(A \to B)\otimes_A B$ is exact precisely if that for $A \to B$ is exact.

=--

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

For $Z \to X$ any [[scheme]] over a [[scheme]] $X$, the induced [[presheaf]] on the [[étale site]]

$$
  (U_Y \to X) \mapsto Hom_X(U_Y, Z)
$$

is a [[sheaf]].

=--

This is due to ([[Grothendieck]], [[SGA]]1 exp. XIII 5.3) A review is in ([Tamme, II theorem (3.1.2)](#Tamme), [Milne, 6.2](#Milne)).

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

By prop. \ref{EtaleDescentDetectedOnOpenImmersionCovers} we are reduced to
showing that the represented presheaf satisfies [[descent]] along collections of open immersions and along surjective maps of affines. For the first this is clear (it is [[Zariski topology]]-descent).

For the second case of a [[faithfully flat]] cover of affines $Spec(B) \to Spec(A)$ it follows with the exactness of the corresponding [[Amitsur complex]], by the [[descent theorem]], prop. \ref{DescentTheorem}.

=--

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

This map from $X$-schemes to sheaves on $X_{et}$ is not injective, different $X$-schemes may represent the same sheaf on $X_{et}$.
Unique representatives are given by [[étale morphism of schemes|étale schemess]] over $X$.

=--

(e.g. [Tamme, II theorem 3.1](#Tamme))

We consider some examples of [[sheaves of abelian groups]]
induced by prop. \ref{XSchemesRepresentSheaves} from  [[group schemes]]
over $X$.

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

The [[additive group]] over $X$ is the [[group scheme]]

$$
  \mathbb{G}_a  \coloneqq Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X
  \,.
$$

By the [[universal property]] of the [[pullback]], the corresponding sheaf $(\mathbb{G}_a)_X$ is given by the assignment

$$
  \begin{aligned}
     (\mathbb{G}_a)_X(U_X \to X) & =
     Hom_X(U_X, Spec(\mathbb{Z}[t]) \times_{Spec(\mathbb{Z})} X)
     \\
     & =
     Hom(U_X, Spec(\mathbb{Z}[t]))
     \\
     & =
     Hom(\mathbb{Z}[t], \Gamma(U_X, \mathcal{O}_{U_X}))
     \\
     & = \Gamma(U_X, \mathcal{O}_{U_X})
  \end{aligned}
  \,.
$$

=--

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

In other words, the sheaf represented by the [[additive group]] is the [[abelian sheaf]] underlying the [[structure sheaf]] of $X$, and in particular the structure sheaf is indeed an étale sheaf.

=--

Similarly one finds:

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

The [[multiplicative group]] over $X$

$$
  \mathbb{G}_m \coloneqq Spec(\mathbb{Z}[t,t^{-1}]) \times_{Spec(\mathbb{Z})} X
$$

represents the sheaf $(\mathbb{G}_m)_X$ given by

$$
  (\mathbb{G}_m)_X(U_X)
  \mapsto
  \Gamma(U_X, \mathcal{O}_{U_X})^\times
  \,.
$$

=--

(e.g. [Tamme, II, 3](#Tamme))

### Base change and sheaf cohomology
 {#BaseChange}

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

For $f \colon X \longrightarrow Y$ a [[homomorphism]] of [[schemes]], there is induced a [[functor]] on the [[categories]] underlying the [[étale site]]

$$
  f^{-1} \;\colon\; Y_{et} \longrightarrow X_{et}
$$

given by sending an [[object]] $U_Y \to Y$ to the [[fiber product]]/[[pullback]] along $f$

$$
  f^{-1} \colon (U_Y \to Y) \mapsto (X \times_Y U_Y \to X)
  \,.
$$

=--

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

The morphism in def. \ref{BaseChangeOnSites} is a [[morphism of sites]]
and hence induces a [[geometric morphism]] between the étale toposes

$$
  (f^\ast \dashv f_\ast)
  \;\colon\;
  Sh(X_{et})
  \stackrel{\overset{f^\ast}{\leftarrow}}{\underset{f_\ast}{\longrightarrow}}
  Sh(Y_{et})
  \,.
$$

Here the [[direct image]] is given on a [[sheaf]] $\mathcal{F} \in Sh(X_{et})$ by

$$
  f_\ast \mathcal{F} \;\colon\;  (U_Y \to Y) \mapsto \mathcal{F}(f^{-1}(U_Y)) = \mathcal{F}(X \times_X U_Y)
$$

while the [[inverse image]] is given on a [[sheaf]] $\mathcal{F} \in Sh_(Y_{et})$ by

$$
  f^\ast \mathcal{F} \;\colon\; (U_X \to X) \mapsto \underset{\underset{U_X \to f^{-1}(U_Y)}{\longrightarrow}}{\lim} \mathcal{F}(U_Y)
  \,.
$$

=--

By the discussion at _[morphisms of sites -- Relation to geometric morphisms](morphism+of+sites#RelationToGeometricMorphisms)_.
See also for instance ([Tamme I 1.4](#Tamme)).

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

The $q$th [[derived functor]] $R^q f_\ast$ of the [[direct image]] functor of def. \ref{DirectAndInverseImageAlongMapOfBases} sends $\mathcal{F} \in Ab(Sh(X_{et}))$ to the [[sheafification]] of the [[presheaf]]

$$
  (U_Y \to Y)
  \mapsto
  H^q(X \times_Y U_Y, \mathcal{F})
  \,,
$$

where on the right we have the degree $q$ [[abelian sheaf cohomology]] [[cohomology group|group]] with [[coefficients]] in the given $\mathcal{F}$ ([[étale cohomology]]).

=--

(e.g. [Tamme, I (3.7.1), II (1.3.4)](#Tamme), [Milne, 12.1](#Milne)).

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

We have a [[commuting diagram]]

$$
  \array{
    Ab(PSh(X)) &\stackrel{(-)\circ f^{-1}}{\longrightarrow}& Ab(PSh(Y))
    \\
    \uparrow^{\mathrlap{inc}} && \downarrow^{L}
    \\
    Ab(Sh(X)) &\stackrel{f_\ast}{\longrightarrow}& Ab(Sh(Y))
  }
  \,,
$$

where the right vertical morphism is [[sheafification]]. Because $(-) \circ f^{-1}$ and $L$ are both [[exact functors]] it follows that for $\mathcal{F} \to I^\bullet  $ an [[injective resolution]] that

$$
  \begin{aligned}
     R^p f_\ast(\mathcal{F})
     & :\simeq
     H^p( f_\ast I)
     \\
     & = H^p(L I^\bullet(f^{-1}(-)))
     \\
     & = L (H^p(I^\bullet)(f^{-1}(-)))
  \end{aligned}
$$

=--

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

For $O_X \stackrel{f^{-1}}{\leftarrow} O_Y \stackrel{g^{-1}}{\leftarrow} O_Z$
two composable [[morphisms of sites]],
the [[Grothendieck spectral sequence]] for the corresponding [[direct images]] is of the form

$$
  E^{p,q}_2 = R^p g_\ast(R^q f_\ast(\mathcal{F}))
  \Rightarrow
  E^{p+q} = R^{p+q}(g f)_\ast(\mathcal{F})
  \,.
$$

For the special case that $S_Z = \ast$ and $g^{-1}$ includes an [[étale morphism of schemes|étale morphism]] $U_Y \to Y$ this yields the [[Leray spectral sequence]]

$$
  E^{p,q}_2 = H^p(U_Y, R^q f_\ast \mathcal{F})
  \Rightarrow
  E^{p+q} = H^{p+q}(U_Y \times_Y X , \mathcal{F})
  \,.
$$

=--

## Étale cohomology
 {#EtaleCohomology}

With some basic facts about [[sheaves]] on the [[étale site]] in hand,  we now consider basics of [[abelian sheaf cohomology]] with [[coefficients]] in some such sheaves.

1. [With coefficients in coherent modules](#WithCoefficientsInCoherentModules)

1. [With coefficients in cyclic groups](#WithCoefficientsInACyclicGroup)

1. [With coefficients in the multiplicative group](#WithCoefficientsInTheMultiplicativeGroup)

This may serve to give a first idea of the nature of [[étale cohomology]]. An outlook on the deep structurual theorems about [[étale cohomology]] is in the next section [below](#MainTheorems).

### With coefficients in coherent modules
 {#WithCoefficientsInCoherentModules}

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

For $X$ a [[scheme]] and $N$ a ([[flat module|flat]]) [[quasicoherent module]] over its [[structure sheaf]] $\mathcal{O}_X$, then this induces an [[abelian sheaf]] on the [[étale site]] by

$$
  N_{et} \;\colon\; (U_X \to X)
   \mapsto
  \Gamma(U_Y, N \otimes_{\mathcal{O}_X} \mathcal{O}_{U_Y})
  \,.
$$

=--

(e.g. [Tamme, II 3.2.1](#Tamme))

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

By prop. \ref{EtaleDescentDetectedOnOpenImmersionCovers}
it is sufficient to test the [[sheaf]] condition on open affine covers
and on singleton covers by  faithfully flat morphisms of affines.
For the first case we have a sheaf since this is just the sheaf condition in the [[Zariski topology]]. For the second case the corresponding Cech complexes are the [[Amitsur complexes]] of a faithfully flat $A \to B$ [[tensor product|tensored]] with $N$. By the [[descent theorem]], prop. \ref{DescentTheorem} this is exact, hence verifies the sheaf condition.

=--

We consider now the étale [[abelian sheaf cohomology]] with coefficients in such coherent modules.

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

A [[cover]] in the [[Zariski topology]] on [[schemes]] is an [[open immersion of schemes]] and hence is in particular an [[étale morphism of schemes]]. Hence the [[étale site]] is finer than the [[Zariski site]] and so every étale [[sheaf]] is a Zarsiki sheaf, but not necessarily conversely.

=--

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

For $X$ a [[scheme]], the inclusion

$$
  \epsilon \;\colon\; X_{Zar} \longrightarrow X_{et}
$$

of the [[Zariski site]] into the [[étale site]] is indeed a [[morphism of sites]]. Hence there is a [[Leray spectral sequence]], remark \ref{GrothendieckAndLeraySpectralSequence}, which computes étale cohomology in terms of Zarsiki cohomology

$$
  E^{p,q}_2 = H^p(X_{Zar}, R^q \epsilon^\ast \mathcal{F})
  \Rightarrow
  E^{p+q} = H^{p+q}(X_{et}, \mathcal{F})
  \,.
$$

=--

This is originally due to ([[Grothendieck]], [[SGA]] 4 (Chapter VII, p355)). Reviews include ([Tamme, II 1.3](#Tamme)).

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

For $N$ a [[quasi-coherent sheaf]] of $\mathcal{O}_X$-[[modules]] and $N_{et}$ the induced étale sheaf (by the discussion at [étale topos -- Quasicohetent sheaves](etale+topos#QuasiCoherentModules)),
then the [[edge morphism]]

$$
  H^p_{Zar}(X, N)
  \longrightarrow
  H^p_{et}(X,N_{et})
$$

of the [[Leray spectral sequence]] of remark \ref{LerayForInclusionOfZariskiIntoEtale} is an  [[isomorphism]] for all $p$, identifying the [[abelian sheaf cohomology]] on the [[Zariski site]] with [[coefficients]] in $N$ with the étale cohomology with coefficients in $N_{et}$.

Moreover, for $X$ affine we have

$$
  H^p_{et}(X, N_{et}) \simeq 0
  \,.
$$

=--

This is due to ([[Grothendieck]], [[FGA]] 1). See also for instance ([Tamme, II (4.1.2)](#Tamme)).

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

By the discussion at _[[edge morphism]]_ it suffices to show that

$$
  R^q \epsilon_\ast (N) = 0 \;\,,\;\;\; for \;\; p \gt 0
  \,.
$$

By prop \ref{DerivedDirectImageByCohomology}, $R^q \epsilon_\ast N$ is the [[sheaf]] on the [[Zariski topology]] which is the [[sheafification]] of the [[presheaf]] given by

$$
  U \mapsto H^q(X_{et}|U, N)
  \,,
$$

hence it is sufficient that this vanishes, or rather, by locality ([[sheafification]]) it suffices to show this vanishes for $X = U = Spec(A)$ an affine [[algebraic variety]].

By the existence of [cofinal affine étale covers](etale+site#CofinalAffineCovers) the [[full subcategory]] $X_{et}^{a} \hookrightarrow X_{at}$ on the étale maps with affien domains, equipped with the induced [[coverage]], is a [[dense subsite]]. Therefore it suffices to show the statement there.
Moreover, by the finiteness condition on [[étale morphisms]]  every cover of $X_{et}^{a}$ may be refined by a finite cover, hence by an affine covering map

$$
  Spec(B) \longrightarrow Spec(A)
  \,.
$$

It follows (by a discussion such as e.g. at [[Sweedler coring]]) that the corresponding [[Cech cohomology]] complex

$$
  N_{et}(Spec(A)) \to C^0(\{Spec(B) \to Spec(A)\}, N_{et}) \to C^1(\{Spec(B) \to Spec(A)\}, N_{et}) \to \cdots
$$

is of the form

$$
  0 \to N \to N \otimes_A B \to N \otimes_{A} B \otimes_A B \to \cdots
  \,.
$$

known as the _[[Amitsur complex]]_ of $A \to B$, tensored with $N$.

Since $A \to B$ is a [[faithfully flat morphism]], it follows again by the [[descent theorem]], prop. \ref{DescentTheorem}  that this is [[exact sequence|exact]], hence that the cohomology indeed vanishes.

=--

### With coefficients in cyclic groups
 {#WithCoefficientsInACyclicGroup}

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

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

Write

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

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

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

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

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

=--

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

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

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

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

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

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

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

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

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

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

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

=--

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

If $X = Spec(A)$ is an affine [[reduced scheme]] of [[characteristic]] a [[prime number]] $p$, then its [[étale cohomology]] with [[coefficients]] in $\mathbb{Z}/p\mathbb{Z}$ is

$$
  H^q(X, (\mathbb{Z}/p\mathbb{Z})_X)
  \simeq
  \left\{
    \array{
      A/(F - id)A & if\; q = 1
      \\
      0 & if \; q \gt 0
    }
  \right.
  \,.
$$

=--

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

Under the given assumptions, the [[Artin-Schreier sequence]] (see there) induces a [[long exact sequence in cohomology]] of the form

$$
  \begin{aligned}
    0 & \to
    H^0(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^0(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^0(X_{et}, \mathcal{O}_X)
   \\
    & \to
     H^1(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^1(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^1(X_{et}, \mathcal{O}_X)
   \\
    & \to
     H^2(X_{et}, \mathbb{Z}/p\mathbb{Z}) \to H^2(X_{et}, \mathcal{O}_X) \stackrel{F-id}{\to} H^2(X_{et}, \mathcal{O}_X) \to \cdots
  \end{aligned}
  \,,
$$

where $F(-) = (-)^p$ is the [[Frobenius endomorphism]].
By prop. \ref{CohomologyWithCoeffsInCoherentModules}
the terms of the form $H^{p \geq 1}(X, \mathcal{O}_X)$ vanish,
and so from [[exact sequence|exactness]] we find an [[isomorphism]]

$$
  H^0(X_{et}, \mathcal{O}_X)/(F-id)(H^0(X_{et}, \mathcal{O}_X))
    \stackrel{\simeq}{\to}
  H^1(X_{et}, \mathbb{Z}/p\mathbb{Z})
  \,,
$$

hence the claimed isomorphism

$$
  A/(F-id)(A)
    \stackrel{\simeq}{\to}
  H^1(X_{et}, \mathbb{Z}/p\mathbb{Z})
  \,.
$$

By the same argument all the higher cohomology groups vanish, as claimed.

=--

### With coefficients in the multiplicative group
 {#WithCoefficientsInTheMultiplicativeGroup}

the étale cohomology groups with [[coefficients]] in the [[multiplicative group]] $\mathbb{G}_m$ in the first few degrees go by special names:

* $H^0_{et}(-, \mathbb{G}_m)$: [[group of units]];

* $H^1_{et}(-, \mathbb{G}_m)$: [[Picard group]] ([[Hilbert's theorem 90]], [Tamme, II 4.3.1](#Tamme));

* $H^2_{et}(-, \mathbb{G}_m)$: [[Brauer group]];

## Outlook: The main theorems
 {#MainTheorems}

What makes [[étale cohomology]] interesting in a broader context is that is verifies a collection of good structural theorems, which we just list now. In their totality these properties make [[étale cohomology]] (in its incarnation as [[ℓ-adic cohomology]]) qualify as a [[Weil cohomology theory]]. This in turn means that using [[étale cohomology]] one can give a [[proof]] of the [[Weil conjectures]] -- a number of [[conjectures]] about properties of the numbers of points in [[algebraic varieties]], hence of the numbers of solutions to certain [[polynomial]] [[equations]] over certain [[rings]] -- , and this was historically a central motivation for introducing [[étale cohomology]] in the first place.

These theorems are

1. [[proper base change theorem]] ([Milne, section 17](#Milne))

1. [[comparison theorem (étale cohomology)]] ([Milne, section 21](#Milne))

1. [[Künneth formula]] ([Milne, section 22](#Milne))

1. cycle map theorem ([Milne, section 23](#Milne))

1. [[Poincaré duality]] ([Milne, section 24](#Milne))

Together these imply the central ingredient for a proof of the [[Weil conjectures]], a Lefschetz fixed-point formula

+ [Künneth formula](#KünnethFormula) + [cycle map](#CycleMap) + [Poincaré duality](#PoincareDuality) $\Rightarrow$ [[Lefschetz fixed-point formula]] ([Milne, section 25](#Milne))

For more on this see... elsewhere.

## References
 {#References}

* [[Günter Tamme]], _[[Introduction to Étale Cohomology]]_
 {#Tamme}

* [[James Milne]], _[[Lectures on Étale Cohomology]]_
 {#Milne}

* [[The Stacks Project]], _Étale cohomology_ ([pdf](http://stacks.math.columbia.edu/download/etale-cohomology.pdf))
  {#StackProject}

[[!redirects basics of étale cohomology]]


basis

There are different notions in mathematics called _basis_.  Generally speaking, these fall into two classes:

*  situations where the term _[[base]]_ is also used:
   *  [[topological base]]
   *  [[basis for a Grothendieck topology]],
   *  etc ...;

*  the generating [[set]] of a [[free object]]:
   *  [[basis of a free module]] (see also [[dual basis]] of a [[projective module]])
   *  [[basis of a vector space]]
   *  [[basis in functional analysis]] (which is applied more generally than to free objects, but analogously)
   *  etc ....

Note that the plurals of 'basis' and 'base' are both spelt 'bases' (although their pronunciations differ).

[[!redirects basis]]
[[!redirects bases]]
[[!redirects basises]]
[[!redirects basis element]]
[[!redirects basis elements]]


basis for a topology > history

< [[basis for a topology]]


basis in functional analysis

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

# Concepts of basis in functional analysis
* table of contents
{: toc}

## Idea

A _basis in functional analysis_ is a [[linear basis]] that is compatible with the [[topological space|topology]] of the underlying [[topological vector space]]. Therefore this is sometimes also referred to as a "topological basis", but beware that this term is also used for referring to the unrelated concept of a  "[[basis for the topology]]".

[[basis|Bases]] in [[linear algebra]] are extremely useful tools for analysing problems.  Using a basis, one can often rephrase a complicated abstract problem in concrete terms, perhaps even suitable for a computer to work with.  A basis provides a way of describing a [[vector space]] in a way that:

1. Is complete: every point in the space can be described in this fashion.
2. Has no redundancies: the description of a point is unique.

When translated into the language of linear algebra, we recover the key properties of a basis: that it be a spanning set and linearly independent.

In infinite dimensions, having a basis becomes more valuable as the spaces get more complicated.  However, the notion of a basis also becomes complex because the question of what makes a description admits different answers depending on whether we want only finite sums, we allow sequences, or we want infinite sums.

## Definitions

+-- {: .num_defn #basis}
Let $V$ be a [[topological vector space]] and $B \subseteq V$ a subset.

1. We say that $B$ is a **[[Hamel basis]]** if:
   1. Every element of $v$ is a finite linear combination of elements of $B$,
   2. If $v = \sum_{b \in B} \alpha_b b$ then the $\alpha_b$ are unique.

   Alternatively, $B$ is [[linearly independent subset|linearly independent]] and $\Span(B) = V$; in other words, the [[span]] of $B$ is $V$ but no [[proper subset]] of $B$ has this property.

2. We say that $B$ is a **topological basis** if:
   1. Every element $v \in V$ is a limit of a [[sequence]] or (more generally) a [[net]] of finite linear combinations of elements of $B$,
   2. No element of $B$ is a limit of a sequence or net of finite linear combinations of the _other_ elements of $B$.

   Alternatively, $B$ is [[total subset|total]] (meaning that its span is [[dense subspace|dense]]) but no proper subset of $B$ is total.

3. We say that $B$ is a **Schauder basis** if:
   1. Every element of $v$ is a (possibly infinite) sum of scales of elements of $B$,
   2. If $v = \sum_{b \in B} \alpha_b b$ then the $\alpha_b$ are unique.
=--

## Properties

1. In the presence of the [[axiom of choice]], Hamel bases always exist.

2. If $B$ is a topological basis, then $B$ has a dual basis.  Since $B \setminus \{b\}$ is not total but $B$ is total, the closure of the span of $B \setminus \{b\}$ must be a codimension $1$ subspace, whence the kernel of a non-trivial continuous linear functional on $V$, say $f_b$.  By scaling, this functional can be assumed to satisfy $f_b(b) = 1$.  Since $B \setminus \{b\} \subseteq \ker f$, $f(b') = 0$ for all $b' \in B$, $b' \ne b$.

3. If $B$ is a Schauder basis then it is a topological basis and so, as mentioned, has a dual basis.  Then the coefficients in the sum $v = \sum \alpha_b b$ must be given by evaluating the dual basis on $v$: $v = \sum f_b(v) b$.

## Examples

1. In $C([0,1],\mathbb{C})$ with the norm ${\|f\|} = \max\{{|f(t)|}\}$:

   1. The monomials are linearly independent and have dense span, but do not form a topological basis as there is a sequence of polynomials with no linear term converging to $t$.

   2. The trigonometric polynomials do form a topological basis.  The dual basis is given by taking the Fourier coefficients of a function.  However, it is not a Schauder basis as there are continuous functions which are not the uniform limit of their Fourier series.

   3. The following is a Schauder basis.  Let $(d_n)$ be the sequence $\{0, 1, \frac{1}{2}, \frac{1}{4}, \frac{3}{4}, \dots\}$.  Define $f_n$ to be the piecewise-linear function with the property that: $f_n(d_n) = 1$ and $f_n(d_k) = 0$ for $k \lt n$, and $f_n$ has the least "breaks".  Then $f_n$ forms a Schauder basis for $C([0,1],\mathbb{C})$.  This is the classical _Faber-Schader_ basis.

## References

* Enflo, P. (1973). A counterexample to the approximation problem in Banach spaces. _Acta Math._, _130_, 309--317.

* Semadeni, Z. (1982). _Schauder bases in Banach spaces of continuous functions_ (Vol. 918). Lecture Notes in Mathematics. Berlin: Springer-Verlag.

category: functional analysis

[[!redirects basis in functional analysis]]
[[!redirects bases in functional analysis]]
[[!redirects basises in functional analysis]]

[[!redirects Schauder basis]]
[[!redirects Schauder bases]]
[[!redirects Schauder basises]]


basis of a free module

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

# Bases of a free module
* table of contents
{: toc}

## Definition

A **basis** of a [[free module|free]] $R$-[[module]] $M$ (possibly a [[vector space]], see [[basis of a vector space]]) is a linear [[isomorphism]] $B\colon M \to \oplus_{i\in I}R$ to a [[direct sum]] of copies of the ring $R$, regarded as a module over itself.

We see how this is equivalent to the classical definition of a basis as a linearly independent spanning set:

+-- {: .num_lemma}
###### Lemma

A basis for a free $R$-module $M$ determines a unique generating set for $M$ of [[linear independence|linearly independent]] elements of $M$.

=--

+-- {: .proof}
###### Proof
Fix a basis $B$ for $M$ over $R$.  Then let $a_i\coloneqq (\delta_{ij})_{j\in I}$ for each $i\in I$.  Since $B$ is an isomorphism, each $a_i$ determines a unique element $b_i \coloneqq B^{-1}(a_i)$.  Since every element of $M$ is of the form $B^{-1}(x)$ for $x\in \oplus_{i\in I}R$, and since every element of $\oplus_{i\in I}R$ can be written as a finite $R$-linear combination of the $a_i$, this proves that $\{b_i\}_{i\in I}$ generates $M$.  To show linear independence, we again apply $B$ and its linearity.  The result is immediate.

=--

## Related concepts

* [[free module]], [[free group]], [[free abelian group]]

* [[linear independence]]

[[!redirects bases of a free module]]


basis of a vector space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Linear algebra
+-- {: .hide}
[[!include higher linear algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

\begin{definition}
\label{BasisOfAVectorSpace}
For $\mathbb{K}$ a [[field]] and $V$ a $\mathbb{K}$-[[vector space]], hence a $\mathbb{K}$-[[module]], a *[[linear basis]]* for $V$ is

1. a [[set]] $B$ (of *basis elements*)

1. a $\mathbb{K}$-[[linear map|linear]] [[isomorphism]]

   $$
     \underset{B}{\oplus}
     \mathbb{K}
     \xrightarrow{\;\; \simeq \;\;}
     V
   $$

to $V$ from the $B$-indexed [[direct sum]] (the [[coproduct]] in $\mathbb{K}$[[Vect]]) of copies of $\mathbb{K}$ (canonically regarded as a vector space over itself), also known as the free $\mathbb{K}$-[[linear span]] of $B$, hence the vector space of free $\mathbb{K}$-[[linear combinations]] of elements of $B$.
\end{definition}

Hence if a basis for $V$ exists it means in particular that $V$ is a [[free module]] over $\mathbb{K}$.

\begin{remark}
**(finiteness of linear combinations)**
\linebreak
Beware that a certain finiteness-condition is hidden in Def. \ref{BasisOfAVectorSpace}: Since a [[linear combination]] is defined to be a [[sum]] of [[finite set|finitely]] many vectors, a basis of a vector space must be such that every vector in the space is the (unique) combination of *finitely* many basis elements -- even if there are infinitely many elements in the basis.

More abstractly this is to do with the appearance of the [[coproduct]] $\amalg_b = \oplus_B$ ([[direct sum]]) in Def. \ref{BasisOfAVectorSpace} instead of the [[product]] $\prod_B$. Many vector spaces in practice arise as (subspaces) of products $\prod_W \mathbb{K}$ ([[function spaces]] $W \to \mathbb{K}$), but if $W$ here is not a [[finite set]] then it is not going to be a basis set.

(On the other hand, if $W$ *is* a [[finite set]], then we have a *[[biproduct]]* $\prod_W \,\simeq\, \coprod_W \,\simeq\, \oplus_W$ in the [[additive category]] $\mathbb{K}$[[Vect]], see [there](additive+category#ProductsAreBiproducts).)
\end{remark}

Related to this are the following phenomena:

\begin{remark}
**(basis and dimension of finitely-generated spaces)**
\linebreak
For every *finitely generated* vector space $V$ (Def. \ref{GeneratedVectorSpace}) it is straightforward to *[[constructive mathematics|construct]]* a linear basis, and to see that the [[cardinality]] of all bases is the same finite [[natural number]] $dim(V) \in \mathbb{N}$, called the *[[dimension of a vector space|dimension]]* of the vector space (whence a *[[finite-dimensional vector space]]*).
\end{remark}

On the other hand:

\begin{remark}
**(Hamel-bases of infinite-dimensional vector spaces)**
\linebreak
While the definition \ref{BasisOfAVectorSpace} applies also to not-necessarily finitely generated vector spaces -- such as for instance the space of ([[continuous function|continuous]]) [[functions]] from a non-finite ([[topological space|topological]]) [[space]] to the ([[topological field|topological]]) [[ground field]] -- it turns out to be subtle and somewhat ill-behaved in this generality.

In fact, in practice infinite-dimensional vector spaces tend to appear and to be understood with [[extra structure]] (typically that of [[topological vector spaces]] such as [[Banach spaces]] or [[Hilbert spaces]]) in which cases there are more appropriate notions of linear bases for them (such as that of [[Schauder bases]], which allow infinite-[[linear combinations]] subject to a condition of [[convergence of a sequence]]).

In order to distinguish the plain notion of basis (Def. \ref{BasisOfAVectorSpace}) from these more refined notions, one also speaks of *Hamel bases* here.

(This is in honor of [Hamel 1905 pp. 460](#Hamel1905) who considered this notion for the special case of $V = \mathbb{R}$ the [[real numbers]] regarded as a [[rational vector space]], hence  over the [[ground field]] $\mathbb{K} = \mathbb{Q}$ of [[rational numbers]]).

Hence, in principle, also a linear basis of a finitely generated vector space is thus a *Hamel basis*, but rarely called this way unless in the context of infinite-dimensional vector spaces.
\end{remark}

\begin{remark}
**(basis theorem and dimension)**
\linebreak
For an infinitely-generated vector space it is *not* in general possible to *[[constructive mathematics|construct]]* a (Hamel-)basis, but the *existence* of such a basis is nevertheless implied, in [[classical mathematics]], by [[Zorn's lemma]] (essentially a form of the [[axiom of choice]]):  This is the content of the *[[basis theorem]]*.

With this classical context understood, it follows that every vector vector space admits a linear basis (even if non-constructible in general) and that each basis is of the same [[cardinality]], then called the [[dimension of a vector space|dimension]] of the vector space.
\end{remark}

## Background definitions

\begin{definition}\label{GeneratedVectorSpace}
**(generated vector space)**
\linebreak
  For $V$ a vector space, a [[subset]] $G \subset V$ of its [[underlying]] set is called a *generating set* or *spanning set* if every element of $V$ can be expressed as a [[linear combination]] of elements of $G$ in $V$, hence if the [[linear span]] of $G$ inside $V$ is all of $V$.

The vector space $V$ is called *finitely generated* if it admits a generating set (spanning set) which is a [[finite set]].
\end{definition}

## Related concepts

* [[orthogonal basis]]

* [[basis in functional analysis]]

* [[Schauder basis]]

* [[dual basis]]

* [[basis]]

* [[mutually unbiased bases]]

* [[Gelfand-Tsetlin basis]] (in [[representation theory]])

* [[seminormal basis]] (in [[representation theory of the symmetric group]])

## References

Lecture notes with much conceptual exposition:

* Karen E Smith, *Bases for infinite-dimensional vector spaces* [[pdf](https://dept.math.lsa.umich.edu/~kesmith/infinite.pdf), [[Smith-InfiniteDimBases.pdf:file]]]

Lecture notes with the proofs concisely spelled out:

* [[Christoph Schweigert]], *Basis und Dimension*, §2.4 in:  *Lineare Algebra*, lecture notes, Hamburg (2022) [[pdf](https://www.math.uni-hamburg.de/home/schweigert/skripten/laskript.pdf)]

See also:

* Wikipedia, *Basis (linear algebra)*

The original discussion (for [[real numbers|$\mathbb{R}$]] regarded as a [[rational vector space]]) after which *Hamel bases* are named:

* {#Hamel1905} [[Georg Hamel]], pp. 460 of: *Eine Basis aller Zahlen und die unstetigen Lösungen der Funktionalgleichung: $f(x + y) = f(x) + f(y)$*, Mathematische Annalen **60** (1905) 459–462 [[doi:10.1007/BF01457624](https://doi.org/10.1007/BF01457624)]

[[!redirects Hamel basis]]
[[!redirects Hamel bases]]
[[!redirects Hamel basises]]

[[!redirects linear basis]]
[[!redirects linear bases]]