Schanuel topos

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[[!redirects Myhill-Schanuel topos]]
[[!redirects Schanuel Topos]]

# Contents
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##Idea
The **Schanuel topos** (also called the _Myhill-Schanuel topos_) is the [[Grothendieck topos]] of "combinatorial functors" whose "typical objects are natural combinatorial constructions such as [[falling factorial|binomial coefficients]]"[^ref]. It plays an important role in [[computer science]] in the theory of [[name binding]] calculi and in [[William  Lawvere]]'s approach to [[petit toposes]]. It can be viewed as a [[category theory|category-theoretic]] variant of the [[Fraenkel-Mostowski model]] of [[set theory]].

[^ref]: Lawvere [1989, p.299](#Law89).

##Definition

The _Schanuel topos_ is the [[sheaf topos]] $Sh(FinSet^{op}_{mono},J)$ where $FinSet^{op}_{mono}$ is the opposite of the [[FinInj|category of finite sets and monomorphisms]] and the coverage is the collection of all sieves generated by singletons $\{f\}\quad.$

##Properties

* The objects of the Schanuel topos are called [[nominal set|nominal sets]] and correspond precisely to the pullback preserving functors $FinSet_{mono}\to Set$.

* Since monos are precisely the morphisms that pull back along themselves to isomorphisms, any pullback preserving functor preserves monos. This affords another description as the category of intersection preserving functors from $FinSet_{mono}\to Set_{mono}$, a special case of [[combinatorial functor|combinatorial functors]].

* The Schanuel topos $Sh(FinSet^{op}_{mono},J)$ is [[atomic topos|atomic]] over $Set$ ([[Stephen Schanuel|S. Schanuel]], cf. (Wraith 1978), p.335) hence Boolean. This fact can be viewed as a reflex of the _urelements_ in Fraenkel-Mostowski set theory.

* $Sh(FinSet^{op}_{mono},J)$ is the [[classifying topos]] $Set[D_\infty]$ for the [[theory of decidable objects#Infinite decidable objects|theory of infinite decidable objects]] $D_\infty$ i.e. for a Grothendieck topos $\mathcal{E}$ geometric morphisms $\mathcal{E}\to Sh(FinSet^{op}_{mono},J)$ correspond to infinite [[decidable objects]] in $\mathcal{E}$. $Sh(FinSet^{op}_{mono},J)$ is equivalent to $Sh_{\neg\neg}([FinSet_mono,Set])$ where $[FinSet_mono,Set]$ classifies [[decidable object|decidable objects]].

* $Sh(FinSet^{op}_{mono},J)$ is the [[category of G-sets|category of continuous actions for the group]] of bijections of $N$ equipped with the topology derived from the product topology for $\prod_{N}N\quad .$

* $Sh(FinSet^{op}_{mono},J)$ is the [[Kleisli category]] of the monad on the topos of [[species]] $Set^{FinSet_{iso}}$ induced by the inclusion of finite sets and bijections $FinSet_{iso}\hookrightarrow FinSet_{mono}$ (cf. Fiore-Menni 2004).

* $Sh(FinSet^{op}_{mono},J)$ is contractible in the sense of [[natural homotopy]] (cf. Joyal-Wraith [1984](#JW84)).

### Remark

For some information on the history of the Schanuel topos see section 10 of Menni ([2009, pp.529f](#Menni09)).

##Related Concepts

* [[nominal set]]

* [[locally decidable topos]]

* [[theory of decidable objects]]

* [[Fraenkel-Mostowski model]]

* [[combinatorial functor]]

* [[permutation model]]

* [[atomic topos]]

* [[species]]

##References

* [[Marcelo Fiore|M. Fiore]], [[Matias Menni|M. Menni]], _Reflective Kleisli Subcategories of the Category of Eilenberg-Moore Algebras for Factorization Monads_ , TAC **15** no. 2 (2004) pp.40-65. ([pdf](http://www.tac.mta.ca/tac/volumes/15/2/15-02.pdf))

* M. J. Gabbay, [[Andrew Pitts|A. M. Pitts]], _A new approach to abstract syntax with variable binding_, Formal Aspects of Computing **13** (2002) pp.341-363. ([draft](http://www.cl.cam.ac.uk/~amp12/papers/newaas/newaas-jv.pdf))

* [[Peter Johnstone]], _Sketches of an [[Elephant]] vols. I,II_, Oxford UP 2002. (pp.79f, 691, 925)

* {#JW84}[[André Joyal]], [[Gavin Wraith]], _Eilenberg-Mac Lane Toposes and Cohomology_ , pp.117-131 in Cont. Math. **92** AMS 1984.

* {#Law89}[[F. William Lawvere]], _Qualitative Distinctions between some Toposes of Generalized Graphs_, Cont. Math. **92** (1989) (p. 298 ff).

* [[Saunders Mac Lane|S. Mac Lane]], [[Ieke Moerdijk|I. Moerdijk]], _Sheaves in Geometry and Logic_, Springer Heidelberg 1994. (pp.155, 158)

* {#Menni03}[[Matías Menni]], _About N-quantifiers_ , Appl. Cat. Struc. **11** (2003) pp.421-445. ([preprint](https://sites.google.com/site/matiasmenni/newQuantifiers.pdf?attredirects=0))

* {#Menni09}[[Matías Menni]], _Algebraic categories whose projectives are explicitly free_ , TAC **22** no.20 (2009) pp.509-541. ([abstract](http://www.tac.mta.ca/tac/volumes/22/20/22-20abs.html))

* John Myhill, _Recursive equivalence types and combinatorial functions_ , Bull. Amer. Math. Soc. **64** (1958) pp.373-376.

* [[Sam Staton]], _Name-passing process calculi: operational models and structural operational semantics_, Technical Report **688** CL University Cambridge 2007. ([pdf](https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-688.pdf))

* [[Gavin Wraith|G. Wraith]], _Intuitionistic Algebra: Some Recent Developments of Topos Theory_ , Proc. ICM Helsinki (1978) pp.331-337. ([pdf](http://www.mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0331.0338.ocr.pdf))


Schanuel's conjecture

[[Stephen H. Schanuel|Schanuel's]] conjecture is a conjecture in transcendental number theory:

> If $z_1, \ldots, z_n$ are [[complex numbers]] that are [[linear independence|linearly independent]] over the [[rational numbers]] $\mathbb{Q}$, then the field $\mathbb{Q}(z_1, \ldots, z_n, \exp(z_1), \ldots, \exp(z_n))$ has [[transcendence degree]] at least $n$. (Here $\exp$ denotes the [[exponential function]].)

Compare the statement of the [Lindemann-Weierstrass theorem](/nlab/show/pi#LindWeier): if $z_1, \ldots, z_n$ are $\mathbb{Q}$-linearly independent **[[algebraic number|algebraic]]** numbers, then $\mathbb{Q}(\exp(z_1), \ldots, \exp(z_n))$ has transcendence degree $n$. Indeed, the Lindemann-Weierstrass theorem is a straightforward consequence of Schanuel's conjecture.

Most known results in transcendental number theory follow from Schanuel's conjecture, and it would imply many more results not yet known. As one illustration: by setting $z_1 = 1$ and $z_2 = \pi i$, it would imply that $\mathbb{Q}(1, \pi i, \exp(1), \exp(\pi i)) = \mathbb{Q}(e, \pi i)$ has transcendence degree $2$, implying that $e$ and $\pi$ are algebraically independent. Meanwhile, much weaker claims such as the irrationality of $e + \pi$ are unknown! This supports the general opinion that we are still very far from deciding Schanuel's conjecture, even if it is widely believed to be true.

Schanuel's conjecture, if proven, would have important consequences for the [[model theory]] of exponential rings. For example, it would imply that the theory of [[real numbers]] as an exponential field is decidable (Tarski's exponential function problem). For more information, see [Wikipedia](http://en.wikipedia.org/wiki/Schanuel%27s_conjecture).


Schanuel's lemma

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## Overview

*Schanuel's lemma* is basic [[lemma]] in [[homological algebra]], useful in the study of [[projective resolutions]].

## Statement

Let $R$ be a [[commutative ring]].

Given two [[short exact sequences]] of $R$-[[modules]],

$$
  0
  \rightarrow
  P_0
  \rightarrow
  P_1
  \rightarrow
  M
  \rightarrow
  0
$$
$$
  0
  \rightarrow
  P_0'
  \rightarrow
  P_1'
  \rightarrow
  M
  \rightarrow
  0
$$

with $P_0,P_1,P_0',P_1'$ [[projective module|projective]], there is
an [[isomorphism]] of the [[direct sums]]

$$
  P_0 \oplus P_1' \;\cong\; P_0'\oplus P_1
  \,.
$$

## Proof

Observe the following commutative diagram

$$
  \array{
     &&  && 0 && 0 &&
     \\
     &&  && \downarrow && \downarrow &&
     \\
      & &  & & P_0' &\to& P_0'&\to& 0
     \\
     &&  && \downarrow && \downarrow &&
     \\
     0 & \to & P_0 &\to& Q &\to& P_1' &\to& 0
     \\
     && \downarrow && \downarrow && \downarrow &&
     \\
     0 & \to & P_0 &\to& P_1 &\to& M &\to& 0
     \\
     &&  && \downarrow && \downarrow &&
     \\
     && && 0 && 0 &&
  }
$$
where the square $Q-P_1'-M-P_1$ is cartesian and the morphisms
$P_0\to Q$ and $P_0'\to Q$ are obtained by the universal property of cartesian square. These morphisms are then necessarily monic and the rows and columns are also exact at $Q$. Thus rows and columns of the diagram are exact. Now, by the projectivity of $P_1'$ and $P_1$ the upper and the left  short exact sequences split, hence
$P_0\oplus P_1' = Q = P_0'\oplus P_1$.

## Literature

Textbook accounts:

* [[Carl Faith]], _Algebra: Rings, Modules, Categories_, vol. 1,  Grundlehren der mathematischen Wissenschaften **190**, Springer (1973) [[doi:10.1007/978-3-642-80634-6](https://doi.org/10.1007/978-3-642-80634-6)]

It is Proposition 2.8.26 in

* Louis Rowen, _Ring theory_, student edition.

See also

* Wikipedia, *[Schanuel's lemma](https://en.wikipedia.org/wiki/Schanuel%27s_lemma)*

[[!redirects Schanuel lemma]]

category: algebra


Schauenburg bialgebroid

## Motivation

Schauenburg bialgebroid, also called Ehresmann-Schauenburg bialgebroid, is a noncommutative generalization of the algebra of functions on an [[Atiyah groupoid]] (also called gauge or Ehresmann groupoid) of a [[principal bundle]] where the principal bundle is replaced by a [[Hopf-Galois extension]] (on the level of algebra of functions).

## Definition

It is an associative [[bialgebroid]] whose structure, given a Hopf-Galois extension, is described in Brzeziński-Wisbauer2003, 34.14. The description there is by first constructing an associated [[coring]], the [[Ehresmann coring]] (which is a more general construction, defined for any coalgebra-Galois extension which is faithfully flat as a left module over the base of the extension), and then making the bialgebroid from it.

## Literature

* [[T. Brzeziński]], R. Wisbauer, _Corings and comodules_, London Math. Soc. Lec. Note Series __309__, Cambridge 2003.

* [[Peter Schauenburg]], _Bialgebras over noncommutative rings and a structure theorem for Hopf bimodules_, Appl. Cat. Str. **6** (1998) 193–222 [[doi:10.1023/A:1008608028634](https://doi.org/10.1023/A:1008608028634)]

* Piotr M. Hajac, Tomasz Maszczyk, _Pullbacks and nontriviality of associated noncommutative vector bundles_, [arxiv/1601.00021](https://arxiv.org/abs/1601.00021)
* Tomasz Brzeziński, Joost Vercruysse, _Bimodule herds_, J. Algebra __321__:9, (2009) 2670-2704 [arXiv:0805.2510](http://arXiv.org/abs/0805.2510) [doi](https://doi.org/10.1016/j.jalgebra.2009.01.020)
* Xiao Han, Giovanni Landi, _On the gauge group of Galois objects_, [arxiv/2002.06097](https://arxiv.org/abs/2002.06097)
* Xiao Han, _Twisted Ehresmann Schauenburg bialgebroids_, [arxiv/2009.02764](https://arxiv.org/abs/2009.02764); _Quantum principal bundles, gauge groupoids and coherent Hopf 2-algebras_, PhD thesis, SISSA 2019/20 [pdf](https://iris.sissa.it/retrieve/handle/20.500.11767/114051/125719/PhD%20Thesis%20Xiao.pdf)
* Ludwik Dabrowski, [[Giovanni Landi]], Jacopo Zanchettin, _Hopf algebroids and twists for quantum projective spaces_, [arXiv:2302.12073](https://arxiv.org/abs/2302.12073)

category: algebra


schedule

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# Schedules
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## Idea
{#idea}

Let $X$ be a [[topological space]] and $\mathcal{U}$ an [[open cover]] thereof.  A ([[continuous function|continuous]]) path $\gamma \colon I \to X$ can pass through many of the elements of $\mathcal{U}$ as it winds its way around $X$.  We can decompose that path into segments such that each segment lies wholly inside one of the [[open sets]] in $\mathcal{U}$.  The **Schedule Theorem** says that this can be done continuously over all paths in $X$.

This was proved by [Dyer and Eilenberg](#DyerEilenberg) and applied to the question of fibrations over numerable spaces.

## Schedules
{#schedules}

The idea of a schedule is that it is a way of decomposing a length into pieces and then assigning a label to each piece.  This clearly fits with the stated purpose of these things since we wish to decompose a path into pieces and assign an open set to each piece.

To make this precise, we start with a set of labels.  Following Dyer and Eilenberg, let us write this as $A$.  Lengths are positive [[real numbers]] and so we also need the set of such, Dyer and Eilenberg denote this by $T$; thus $T \coloneqq \mathbb{R}_{\ge 0}$.

+-- {: mynumdef #schmon}
###### Definition
The **schedule monoid** of $A$ is the [[free monoid]] on the [[set]] $A \times T$.  It is written $S A$.  Its elements are **schedules** in $A$.
=--

A schedule in $A$ is thus a finite [[order|ordered]] list of pairs $(a,t)$ where $a \in A$ and $t \in T$.

There are two notions of *length* for a schedule.  There is the *word length* which simply counts the number of pairs.  Then there is the function $l \colon S A \to T$ defined by $l((a_1,t_1) \cdots (a_k,t_k)) = t_1 + \cdots + t_k$.  There is also a right action of $T$ on $S A$ which simply multiplies all of the lengths: $((a_1,t_1) \cdots (a_k,t_k)) \cdot t = ((a_1,t_1 t) \cdots (a_k,t_k t))$.  Then $l(s t) = l(s) t$.

+-- {: mynumdef #reduced}
###### Definition
A schedule is said to be **reduced** if all of its terms, $(a,t)$, have non-zero length, i.e. $t \gt 0$.  The set of reduced schedules forms a submonoid of $S A$ which is written $R S A$.
=--

The empty schedule is reduced.

There is a [[retraction]] map $\rho \colon S A \to R S A$ defined by removing all terms with zero length part.

The schedule monoid is given a [[topology]] so that the labels are discrete and the lengths topologised as usual.  More concretely, given a word $a_1 a_2 \dots a_k$ of elements in $A$, the set of schedules of the form $(a_1,t_1) (a_2,t_2) \cdots (a_k,t_k)$ is in bijection with $T^k$ and we make that bijection a homeomorphism.  Then $S A$ is topologised by taking the coproduct over the set of words in $A$.  The reduced schedule monoid is topologised as the quotient of this.

## Paths
{#paths}

Let $X$ be a [[topological space]].  Let $P X$ denotes its [[Moore path space]].  Suppose that we have a family $\mathcal{U}$ of subsets of $X$ indexed by some set $A$.  Then we consider a schedule in $A$ as giving an ordered list of these subsets together with the times to be spent in each.  For a path in $X$, and a schedule of the appropriate length, then we can ask whether or not the path *fits* (or obeys) the schedule.  We make that precise as follows.

+-- {: mynumdef #fits}
###### Definition
Suppose that we have $\alpha \in P X$ and $s \in S A$, and suppose that $s = (a_1, t_1) \cdots (a_k,t_k)$.  Then we say that $\alpha$ **fits the schedule** s, written $\alpha \Vert s$, if the following conditions hold:

1. $l(\alpha) = l(s)$
2. We can split $\alpha$ into subpaths according to the times $\{t_i\}$.  Let $\alpha_i$ be the $i$th segment.  Then $\alpha_i \in P U_{a_i}$.
=--

Here, $l \colon P X \to T$ is the function that assigns to a Moore path its length.  The schedule designates a decomposition of $[0,l]$ into subintervals with $t_i$ being the length of the $i$th subinterval.  Then saying that $\alpha$ fits the schedule $s$ means that $\alpha$ spends the $i$th subinterval in the open set $U_{a_i}$.

## Schedule Theorem
{#secschthm}

We can now state the main theorem.

+-- {: .num_theorem #schthm}
###### Theorem
Let $X$ be a [[topological space]].  Let $\mathcal{U}$ be a [[locally finite cover|locally finite]] [[open covering]] of $X$ by [[numerable]] open sets with indexing set $A$.  Then there is a covering $\mathcal{F}$ of $P X$ by [[closed sets]] and a family of [[continuous functions]] $f \colon F \to S A$, indexed by $F \in \mathcal{F}$ such that:

1. for each $\alpha \in P X$, there some finite subfamily $\{F_1, \dots, F_k\} \subseteq \mathcal{F}$ such that $\alpha$ is in the interior of $\bigcup F_j$,
1. for each $\alpha \in F$, $\alpha \Vert f_F(\alpha)$, and
2. for each $\alpha \in F \cap F'$, $\rho(f_F(\alpha)) = \rho(f_{F'}(\alpha))$
=--

The first condition is purely about the covering.  Dyer and Eilenberg use the term *local covering* for a covering by closed sets with this property.

+-- {: .num_corollary #schcor}
###### Corollary
There exists a [[continuous function]] $h \colon P X \to R S A$ such that $\alpha \Vert h(\alpha)$ if $l(\alpha) \gt 0$ and $h(\alpha) = \Lambda$ if $l(\alpha) = 0$.
=--

Here, $\Lambda \in R S A$ is the empty word.

## Globalisation Theorem
{#secglobal}

The original motivation for the notion of *schedules* was to prove the *globalisation theorem* for (Hurewicz) [[Hurewicz fibration|fibrations]].

+-- {: .num_theorem #global}
###### Theorem
Let $p \colon Y \to B$ be a continuous function.  Suppose that $\mathcal{U}$ is a locally finite covering of $B$ by numerable open sets with the property that for each $U \in \mathcal{U}$ then the restriction $p_U \colon Y_U \to U$ is a fibration.  Then $p$ is a fibration.
=--

The link between the globalisation theorem and the schedule theorem is the characterisation of Hurewicz fibrations in terms of [[Hurewicz connections]].

## Proof of the Schedule Theorem
{#secproof}

Let $X$ be a topological space.  Let $\mathcal{U}$ be a locally finite open covering of $X$ by numerable open sets and indexing set $A$.

Let us write $A^*$ for the free monoid on $A$.  Then there is a function $A^* \times T \to S A$ which takes $(a_1 a_2 \cdots a_k, t)$ to the schedule $(a_1,t/k)(a_2,t/k)\cdots (a_k,t/k)$.  We say that a path $\alpha \in P X$ *evenly fits* $s \in A^*$, and write this as $\alpha \Vert_e s$, if it fits the schedule corresponding to $(s,l(\alpha))$.

We need an initial technical result.

+-- {: .num_lemma #even}
###### Lemma
There is a locally finite covering $\mathcal{W} = \{W_s \mid s \in A^*\}$ of $P X$ by numerable open sets such that for $\alpha \in W_s$ then $\alpha$ evenly fits the word $s$.
=--

+-- {: myrmk}
###### Remark
Let us explain why this is a reasonable result.  Consider a path, $\alpha$, of length $l$.  We pull back the cover $\mathcal{U}$ to a cover of $[0,l]$.  Using compactness of $[0,l]$ we can replace the pull-back cover by a finite family of open subintervals of $[0,l]$ which cover $[0,l]$.  Each subinterval is labelled by an element of $\mathcal{U}$ (though a label might be reused).  As the family is finite, the intersections are finite and therefore have a minimum length.  Choose $n$ big enough so that $l/n$ is less than this minimum length.  Then consider the subdivision of $[0,l]$ given by $\{0,1/n,\dots,l/n\}$.  Our conditions on $n$ guarantee that every intersection of subintervals contains at least one of these division points.  We can therefore assign to each subinterval of the form $[k/n, (k+1)/n]$ one of the original family of subintervals that contains it.  Then we can assign the corresponding element of $\mathcal{U}$.  Thus $\alpha$ fits evenly the corresponding word.

Thus the sets $Y_s \coloneqq \{\alpha : \alpha \|_e s\}$ cover $P X$.  That they are open follows from the fact that the condition for membership depends on certain compact sets lying in certain open sets and we use the [[compact-open topology]] on $P X$.

What is more complicated is reducing the family to a locally finite one.
=--

As $\mathcal{W}$ is locally finite and its elements are numerable, we can choose a numeration that is also a partition of unity.  That is, we can choose continuous functions $q_s \colon X \to [0,1]$ with the property that $q_s^{-1}((0,1]) = W_s$ and $\sum_s q_s = 1$.

Let $\mathcal{B}$ be the set of finite subsets of $A^* \setminus \Lambda$ (where $\Lambda$ is the empty word).  For $b \in \mathcal{B}$ we define

$$
\begin{aligned}
D_b &\coloneqq \{\alpha \in P X \mid \sum_{s \in b} q_s(\alpha) = 1 \} \\
&=\{ \alpha \in P X \mid q_s(\alpha) = 0 \; \text{for all}\; s \notin b\}
\end{aligned}
$$

This is a covering of $P X$ by closed sets.  As $\mathcal{W}$ is locally finite, for $\alpha \in P X$ there is some neighbourhood $V$ which meets only a finite number of the $\mathcal{W}$.  These are indexed by elements of $A^*$, indeed of $A^* \setminus \Lambda$, and so the set of indices is an element, say $b$, of$ \mathcal{B}$.  Then for $s \notin b$, $q_s \mid V = 0$ and so for $\beta \in V$, $\sum_{s \in b} q_s(\beta) = 1$, whence $V \subseteq D_b$.  Thus each $\alpha$ is contained in the interior of some $D_b$.

Now let us put a [[total ordering]] on $A^*$.  This induces a total ordering on each $b \in \mathcal{B}$ and thus allows us to define the partial sums of the summation $\sum_{s \in b} q_s$.  Write these as $Q_i$, with $Q_0$ as the zero function.

Fix $b \in \mathcal{B}$ and write it as $b = \{s_1,s_2,\dots,s_k\}$ in the inherited ordering.  Let $e = (l_1,r_1,\dots,l_k,r_k)$ be a list of integers with the property that $1 \le l_i \le r_i \le \#s_i$ where $\#s_i$ is the word length of $s_i$.  Define:

$$
D_{(b,e)} = \left\{ \alpha \in D_b \mid \frac{l_i -1}{\# s_i} \le Q_{i - 1}(\alpha) \le \frac{l_i}{\# s_i} \; \text{and} \; \frac{r_i - 1}{\# s_i} \le Q_i(\alpha) \le \frac{r_i}{\# s_i} \right\}.
$$

This is closed in $D_b$ and the collection $\{D_{(b,e)}\}$ is a finite cover of $D_b$.
The family $\{D_{(b,e)}\}$ ranging over all $b \in \mathcal{B}$ and suitable $e$ is the family $\mathcal{F}$ that we are looking for.  It has the required covering property since the interiors of the $D_b$ cover $P X$.

Define $f_{(b,e)} \colon D_{(b,e)} \to S A$ as follows:

$$
f_{(b,e)}(\alpha) = \sigma_1 \cdots \sigma_k l(\alpha)
$$

where $\sigma_i$ is the schedule with $\# \sigma_i = r_i - l_i + 1$ and $l(\sigma_i) = q_{s_i}(\alpha)$, and if $s_i = a_1 \cdots a_n$ then if $l_i \lt r_i$ we have

$$
\sigma_i = \left( a_{l_1}, \frac{l_i}{n} - Q_{i - 1}(\alpha)\right) \left(a_{l_i+1}, \frac{1}{n} \right) \cdots \left(a_{r_i - 1}, \frac{1}{n} \right) \left( a_{r_i}, Q_i(\alpha) - \frac{r_i - 1}{n} \right)
$$

otherwise, $\sigma_i = (a_{l_i}, Q_i(\alpha) - Q_{i-1}(\alpha))$.

This is continuous and for $\alpha \in D_{(b,e)}$ then $\alpha$ fits $f_{(b,e)}(\alpha)$.  Moreover, for $\alpha \in D_{(b,e)} \cap D_{(b',e')}$ then $\rho f_{(b,e)}(\alpha) = \rho f_{(b',e')}(\alpha)$.

## References
{#refs}

* {#MR0963792} [[Eldon Dyer]], and [[Samuel Eilenberg]]. (1988). Globalizing fibrations by schedules. _Fund. Math._, _130_, 125–136. [MR0963792](http://www.ams.org/mathscinet-getitem?mr=963792){#DyerEilenberg}

[[!redirects schedules]]

[[!redirects schedule theorem]]


Schelling

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

## related $n$Lab entries

* [[C.S. Peirce]]

category: people


schematic algebra

##Motivation##

The functor $Proj$ of a [[graded object|graded]] [[commutative algebra]] is a [[projective variety]]. [[projective scheme|Projective schemes]] are particular cases of [[schemes]], so they have nontrivial covers by open affine subschemes. In the noncommutative case, [[Ore localization]]s (even on one-sided Ore sets) on Ore sets which are multiplicatively generated by finite subsets behave very much like [[Zariski topology|Zariski-open]] subsets. F. van Oystaeyen produced an appropriate notion of a [[cover]] in terms of Ore localizations at two-sided Ore subsets in a graded algebra to define a notion of _schematic algebra_.

##Definition##

...to be written

##Literature##

* F. van Oystaeyen, Algebraic geometry for associative algebras, 287 p. Dekker, 2000.

* F. van Oystaeyen, L. Willaert, Cohomology of schematic algebras, J. algebra 185:1 (1996), p. 74--84

* F. van Oystaeyen, L. Willaert, Grothendieck topology, coherent sheaves and Serre's theorem for schematic algebras, J. Pure and Appl. Algebra, 104 (1995), p. 109--122

* F. van Oystaeyen, L. Willaert, The quantum site of a schematic algebra, Comm. Alg. 24:1(1996), p. 209--222

category: noncommutative geometry
[[!redirects schematic algebras]]


schematic homotopy type

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### $(\infty,1)$-Topos Theory
+--{: .hide}
[[!include (infinity,1)-topos - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _schematic homotopy type_ is a [[geometric ∞-stack]] over a site of [[formal duality|formal duals]] of $k$-[[associative algebras|algebras]] that models a [[homotopy type]] in generalization to how a [[dg-algebra]] models a [[rational space]] in [[rational homotopy theory]] (via the [[fundamental theorem of dg-algebraic rational homotopy theory]]): schematic homotopy types in particular model more general [[fundamental groups]]

(...)

## Definition

Let $k$ be a [[commutative ring]], $T$ the [[Lawvere theory]] of commutative $k$-[[associative algebras]]. Let $\mathbb{U} \subset \mathbb{V}$ be an inclusion of [[universes]]. Let

$$
  T \hookrightarrow C = T Alg_{\mathbb{U}} \hookrightarrow T Alg_{\mathbb{V}}
$$

be the [[site]] on [[formal duality|formal duals]] of small $k$-algebras equipped with the [[fpqc-topology]].

By the general discussion at [[function algebras on ∞-stacks]] we have then  the [[Isbell duality]] [[pair]] of [[adjoint (∞,1)-functors]]

$$
  (\mathcal{O} \dashv Spec) : (T Alg_{\mathbb{V}}^{\Delta})^{op}
    \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}}
  Sh_\infty(C) =: \mathbf{H}
$$

(due to [Toën 2006](#Toen06)) where the [[(∞,1)-topos]] $\mathbf{H}$ is the [[(∞,1)-category of (∞,1)-sheaves]] on $C$.

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

(...) Let $Perf \in \mathbf{H}$ be the stack of [[perfect complex]]es of modules on $C$. (...)

Write $P \subset Mor(\mathbf{H})$ for the [[class]] of [[morphism]]s such that for all $p \in P$ we have that $\mathbf{H}(p,Perf)$ is an [[equivalence in an (∞,1)-category|equivalence]].

=--

This is discussed in ([HirschowitzSimpson, paragraph 21](#HirschowizSimpson)).

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

A [[pointed object|pointed]] **schematic homtopy type** is the [[delooping]] $\mathbf{B}G \in \mathbf{H}$ of an [[∞-group]] $G \in \mathbf{H}$ such that

* $G$ is in the image of $Spec$, in that there is $A \in T Alg^\Delta$ such that $G \simeq Spec A$;

* $\mathbf{B}G$ is a $P$-[[local object]].

=--

This appears as ([Toën 2006, def 3.1.2](#Toen06))

## Properties

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

A schematic homotopy type is in particular a [[geometric ∞-stack]] over $C$.

=--

## Examples

### de Rham schematic homotopy type

For a [[connected]] [[scheme]] $X$ let $X_{dR}$ be its [[de Rham space]]. According to [Toën 2006, sect. 3.5.1](#Toen06) one finds that the functor

$$
 Ho(SchHoType/\mathbb{C}) \to Set
$$

$$
  F \mapsto Ho_{Sh_{(\infty,1)}(Alg_\mathbb{C}^{op})}(X_{dR}, F)
$$

is co-[[representable functor|representable]] by a schematic homotopy type $X^{dR}$. This is the **de Rham schematic homotopy type**. The [[cohomology]] of $X^{dR} \in Sh_{(\infty,1)}$ is the algebraic [[de Rham cohomology]] of $X$.

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

A similar construction exists in every [[cohesive (∞,1)-topos]]. See the discussion in the section cohesive (∞,1)-topos -- de Rham cohomology.

=--

## Related concepts

* [[real homotopy theory]]

## References

An introduction to the general theory

* [[Ludmil Katzarkov]], [[Tony Pantev]], [[Bertrand Toën]], _Schematic homotopy types and non-abelian Hodge theory_, [math.AG/0107129](http://arxiv.org/abs/math/0107129)

* {#Toen06} [[Bertrand Toën]], _Champs affines_, Selecta Math. new series **12** (2006), no. 1, 39-135 ([arXiv:math/0012219](https://arxiv.org/abs/math/0012219), [doi:10.1007/s00029-006-0019-z](https://doi.org/10.1007/s00029-006-0019-z))

The stack $Perf$ of [[perfect complexes]] is discussed for instance in section 21 of

* {#HirschowizSimpson} [[Andre Hirschowicz]], [[Carlos Simpson]], _Descente pour les n-champs_ ([arXiv:math/9807049](http://arxiv.org/abs/math/9807049))

[[!redirects schematic homotopy types]]

[[!redirects de Rham schematic homotopy type]]


scheme

#Contents#
* table of contents
{: toc}

## Idea

A _scheme_ is a [[space]] that _locally_ looks like a particularly simple [[ringed space]]: an [[affine scheme]].
This can be formalised either within the category of [[locally ringed spaces]] or within the category of presheaves of sets on the category of affine schemes $Aff$.

The notion of scheme originated in [[algebraic geometry]] where it is, since [[Grothendieck]]'s revolution of that subject, a central notion.

However, the idea that

+-- {: .standout}

A scheme is a ringed space that is locally isomorphic to an affine space.

=--

is much more general and need not be restricted to the locality in Zariski topology and to a notion of affine spaces that are [[duality|formal duals]] of rings. If one takes another subcanonical Grothendieck topology $\tau$ on $Aff$ then one talks about __$\tau$-locally affine spaces__. More generally one can take another "category of local models" $Loc$ replacing $Aff$ and suitable topology and consider sheaves on it, as locally affine space in this generalized sense. The category $Loc$ can sometimes be represented by ringed spaces of special type and the gluing can be sometimes made in a genuine (classical, not Grothendieck) topology, within the category of ringed spaces.

For instance a smooth [[manifold]] is a ringed space locally isomorphic to a "smooth affine space" $\mathbb{R}^n$, with its standard smooth structure.

The standard concept of scheme in [[algebraic geometry]] is therefore usefully understood as a special case of [[generalized scheme]]s that naturally appear for instance also in [[differential geometry]], in [[synthetic differential geometry]] and many other topics.

## Definition

Throughout this article, "ring" will mean "commutative ring with unit".

### As locally ringed spaces ##

A __scheme__ is a [[locally ringed space]] $(X, \mathcal{O}_X)$ such that, for every point $x$ of $X$, there is an open subset $U$ of $X$ with $x \in U$ such that the locally ringed space $(U, \mathcal{O}_{X} | U)$ is isomorphic to an [[affine scheme]], that is to say, a commutative [[prime spectrum|ring spectrum]] $Spec A = (|Spec A|, \mathcal{O}_{Spec A})$.

Here $\mathcal{O}_{X} | U$ denotes the restriction of $\mathcal{O}_{X}$ to $U$, that is to say, the sheaf $i^{*}(\mathcal{O}_{X})$, where $i: U \hookrightarrow X$ is the inclusion map, and $i^{*}$ is the corresponding [[inverse image]] functor from the category of sheaves on $X$ to the category of sheaves on $U$.

A [[morphism of schemes]] $f : (X,\mathcal{O}_X) \to (Y,\mathcal{O}_Y)$ is a morphism of the underlying [[locally ringed space]]s. This means it is a morphism of [[ringed space|ringed spaces]] such that for each point $x \in X$ the induced map of [[local ring]]s
$$
  (\mathcal{O}_Y)_{f(x)} \to  (\mathcal{O}_X)_x
$$
is _local_ (in that it carries the maximal ideal to the maximal ideal). See [[functor of points]].

### As sheaves on $CRing^{op}$

+-- {: .num_defn}
###### Definition
(k-ring, k-functor,affine k-scheme)

For a ring $k$ the _category of $k$-rings_, denoted by $M_k,$ is defined to be the category of commutative associative $k$-algebras with unit. This is equivalently the [[under category]] $k\downarrow CRing$ of pairs $(R,f:k\to R)$ where $R$ is a commutative ring and $f$ is a ring homomorphism.

The _category of $k$-functors_, denoted by $co Psh (M_k)$, is defined to be the category of covariant functors $M_k\to Set$.

The forgetful functor $O_k:R\mapsto R$ sending a $k$-ring to its underlying set is called the _affine line_.

For the full and faithful contravariant functor
$$Sp_k:\begin{cases}
M_k&\to& co Psh(M_k)
\\
A&\mapsto& M_k(A,-)
\end{cases}$$
$Sp_k A$ (and every isomorphic functor) is called an _affine $k$-scheme_. $Sp_k$ restricts to an equivalence between the categories of $k$-rings and the category $Aff Sch_k$ of affine $k$-schemes. We think of this category as of $M_k^{op}$. The functor $Sp_k$ commutes with limits and scalar extension (see below). Consequently $Aff Sch_k$ is closed under limits and base change.

Note that the affine line $O_k$ defined above is naturally isomorphic to $M_k(k[t],-)$, and so $O_k$ is an affine $k$-scheme.

A _function on a_ $k$-functor $X$ is defined to be an object $f\in O(X):=co Psh (M_k)(X,O_k)$. $O(X)$ is a $k$-ring, naturally in $X$, by componentwise addition and multiplication.

=--

+-- {: .num_remark}
###### Remark
The category of $k$-functors has limits.

The terminal object is $e:R\mapsto\{\varnothing\}$. Products and pullbacks are computed component-wise.
=--

+-- {: .num_remark}
###### Remark
For $\phi:k\to k'$ the ''base change'' functor $(-)\otimes_k k':co Psh(M_k)\to co Psh(M_{k'})$ induced by $(-)\circ \phi:M_k\to M_{k'}$ given by postcompositions with $\phi$ is called _scalar extension_.
=--

Now we come to the definition of not necessarily affine k-schemes

For a $k$-functor $X\in coPsh(M_k)$ and $E\subseteq O(X)$ a set of functions on $X$, we define
$$V(E)(R):=\{x\in X(R) |\forall f\in E, f(x)=0\}$$
and
$$D(E)(R):=\{x\in X(R)|f\in E, \;\text{ the } \; f(x) \; \text{ generate the unit ideal of } \; R\}$$

For a transformation $u:Y\to X$ of $k$-functors and $Z\subseteq X$ a subfunctor we define
$$u^{-1}(Z)(R):=\{y\in Y(R)|u(y)\in Z(R)\}$$

A subfunctor $Y\subseteq X$ is called _open subfunctor_ resp. _closed subfunctor_ if for every transformation $u:T\to X$ we have $u^{-1}(Y)$ is of the form $V(E)$ resp. $D(E)$.

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

A $k$-functor $X$ is called a $k$_-scheme_ if the following two conditions hold:

1. ($X$ is a sheaf for the [[Zariski site|Zariski Grothendieck topology]] on $M_k^{op}$) For all $k$-rings and all families $(f_i)_i$ such that $R=\sum_i R f_i$ we have: if for all $x_i\in R[f_i^{-1}]$ such that the images of $x_i$ and $x_j$ coincide in $X(R[f_i^{-1} f_j^{-1}])$ there is a unique $x\in X(R)$ mapping to the $x_i$.

1. ($X$ has a cover of Zariski open immersions of affine $k$-schemes) The exists a small family $(U_i)_i$ of open affine subfunctors of $X$ such that for all fields $K\in M_k$ we have that $X(K)=\bigcup_i U_i(K)$.

=--

+-- {: .num_remark}
###### Remark
The category of $k$-schemes is closed under finite limits, forming open- and closed subfunctors, and scalar extension. As a subcategory of the category of Zariski sheaves, it is also closed under taking small coproducts.

=--

### Translation between the two approaches

The [[fundamental theorem on morphisms of schemes]]  asserts that there is a [[fully faithful functor]] from the category $Sch$ of schemes to $Psh(Aff) \equiv Psh(CRing^{op}) $,  the category of [[presheaf|presheaves]] on the category of affine schemes, or equivalently on the opposite of the category of commutative rings,
given by
$$(X,\mathcal{O}_X)\mapsto Sch((|Spec (-)|,\mathcal{O}_{Spec(-)}),(X,\mathcal{O}_X))$$

This identifies schemes with those presheaves on [[CRing]]${}^{op}$ that

1. are [[sheaf|sheaves]] with respect to the Zariski [[Grothendieck topology]] on $CRing^{op}$;
2. have a [[cover]] by Zariski-open immersions of [[affine scheme]]s in the category of presheaves over $Aff$.

The standard reference for the functor-of-points approach to schemes is [Demazure-Gabriel](#DG).

+-- {: .num_remark}
###### Remark
Different authors take different approaches to the underlying set-theoretic issues. The astute reader will have noticed that we consider the category of _all_ functors $CRing \to Set$ -- which is not a locally small category. Nor is the category of sheaves on the site $CRing^{op}$ with its Zariski topology actually a Grothendieck topos. With due regard to such set-theoretic issues, this approach seems to be conceptually the simplest. Or, one could keep one's options open: for some suitable small category of rings left to one's discretion, for example the category of [[finitely presented object|finitely presented]] rings, one can consider schemes locally modelled on that category.

Demazure-Gabriel steer a middle course involving [[universes]]: assuming two universes $U$ and $V$ with $\mathbb{N} \in U \in V$, one has a category of "small rings" (belonging to $U$) and a category of sets (belonging to $V$) and one considers functors $M \to Set$ from small rings (called "models") to (not necessarily small) sets. They remark that the device of using universes is really just a convenience that could mostly be dispensed with: one could work within the standard Bernays-Gödel framework by assuming that the models are inclusive enough to hold various standard commutative algebra constructions (e.g., quotients, localizations, completions) while still remaining a small category. However, since they wish to avail themselves of direct limits in the category of models, they choose to work with universes instead.
=--

## Properties

* [[schemes are sober]]

The category of schemes admits [[small coproducts]].

It does not admit [[coequalizers]]:
<https://mathoverflow.net/questions/9961/colimits-of-schemes/23966#23966>

The category of schemes admits [[finite limits]].

It does not admit [[infinite products]]:
<https://mathoverflow.net/questions/9134/arbitrary-products-of-schemes-dont-exist-do-they/65534#65534>

(...)

## Generalizations

In [[algebraic geometry]] this is a basic object of study, since the revolution of [[Grothendieck]]. There are generalizations like [[relative schemes]] (which are just objects in a [[slice category]] $Sch/S$), relative [[noncommutative scheme]]s in [[noncommutative algebraic geometry]] introduced by A. Rosenberg in terms of categories and covers defined using pairs of [[adjoint functors]], the generalized schemes of [[Nikolai Durov]], the [[algebraic stack]]s of [[Deligne-Mumford stack|Deligne-Mumford]] and Artin, the dg-schemes of Kapranov, the [[derived scheme]]s of [[Jacob Lurie]], the higher [[algebraic stack]]s of [[Bertrand Toen|Toën]]--Vezzosi, almost schemes (Ofer Gabber and Lorenzo Ramero), formal schemes (Cartier--Grothendieck), [[locally affine spaces]] in the fpqc, fppf or étale topology (Grothendieck), [[algebraic spaces]], etc. See also [[generalized scheme]], [[simplicial scheme]], [[super-scheme]], [[semiring scheme]].

### Underlying topological space vs. underlying locale

[[Jacob Lurie]] argues that underlying locale point of view is better than underlying topological space point of view, see [[schemes as locally affine structured (∞,1)-toposes]].

## References

Terminology: [[EGA]] says prescheme, for what we call scheme, and says scheme for what we call [[separated scheme]].

Generalizations: [[simplicial scheme]], [[super-scheme]], [[semiring scheme]], [[noncommutative scheme]], [[derived scheme]]

Other related entries include

* [[morphism of schemes]]
* [[Serre's theorem on Proj]], [[Serre's theorem on quasicoherent sheaves on affine schemes]]
* [[locally ringed space]]

### Standard monographs

* Robin Hartshorne, _Algebraic geometry_, Springer

* Qing Liu, _Algebraic geometry and arithmetic curves_, 592 pp. Oxford Univ. Press 2002

* D. Eisenbud, J. Harris, _The geometry of schemes_, Springer Grad. Texts in Math.

* [[David Mumford]], _Red book of varieties and schemes_

* Amnon Neeman, _Algebraic and analytic geometry_, London Math. Soc. Lec. Note Series __345__

* William Fulton, _Intersection theory_, Springer 1984

* Ulrich Görtz, Torsten Wedhorn, _Algebraic geometry I. Schemes with examples and exercises_, Advanced Lectures in Mathematics. Vieweg + Teubner, Wiesbaden, 2010. viii+615 pp. [Springerlink book](http://www.springerlink.com/content/kt5u74/#section=748613&page=1)

* M. Demazure, P. Gabriel, _Groupes algebriques_, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970 (functor of points approach, mainly)
{#DG}

* Michel Demazure, lectures on p-divisible groups [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

* [[EGA]], [[FGA explained]]

### Other references

* Ravi Vakil's Berkeley [course notes](http://math.stanford.edu/~vakil/0708-216)
* [[Paul Goerss]], [[Topological Algebraic Geometry - A Workshop]]  -- at the beginning one finds a quick introduction in the light of its higher categorical generalizations

* Wikipedia: [scheme (mathematics)](http://en.wikipedia.org/wiki/Scheme_%28mathematics%29).

MathOverflow: [arbitrary-products-of-schemes-dont-exist](http://mathoverflow.net/questions/9134/arbitrary-products-of-schemes-dont-exist-do-they), [model-of-a-scheme-regular-over-the-generic-point](http://mathoverflow.net/questions/32196/model-of-a-scheme-regular-over-the-generic-point), [categorical-construction-of-the-category-of-schemes](http://mathoverflow.net/questions/26506/categorical-construction-of-the-category-of-schemes), [when-is-an-algebraic-space-a-scheme](http://mathoverflow.net/questions/4573/when-is-an-algebraic-space-a-scheme), [is-an-algebraic-space-group-always-a-scheme](http://mathoverflow.net/questions/8918/is-an-algebraic-space-group-always-a-scheme), [connections-between-various-generalized-algebraic-geometries-toen-vaquie-durov](http://mathoverflow.net/questions/89475/connections-between-various-generalized-algebraic-geometries-toen-vaquie-durov)

category: algebraic geometry
[[!redirects schemes]]
[[!redirects algebraic scheme]]
[[!redirects algebraic schemes]]


+-- {: .standout}

This is a sub-entry of [[scheme]] and [[derived scheme]].

This entry discusses a higher-categorical perspective on that standard notion.

=--

On p. 37 of

* [[Jacob Lurie]], [[Structured Spaces]] (StSp),

it is argued that the standard definition of [[scheme]] with its insistance on an underlying topological space is slightly misleading:

> it is in some sense coincidental that $Spec A$ is described by a [[topological space]]. What arises more canonically is the [[lattice]] of open subsets of $Spec A$, which is generated by basic open sets of the form $U_f$. This lattice naturally forms a [[locale]], or [[0-topos]]. It happens that this [[locale]] has [[point of a topos|enough points]], and can therefore be described as the lattice of open subsets of a [[topological space]]. However, there are various reasons we might want to disregard this fact:

> a) The existence of [[point of a topos|enough points]] for $Spec A$ is equivalent to the assertion that every nonzero commutative ring contains a prime ideal, and the proof of this assertion requires the [[axiom of choice]].

> b) In _[[relative scheme|relative]]_ situations, the relevant construction may well fail to admit enough points, even if the [[axiom of choice]] is assumed. However, the underlying [[locale]] (and its associated [[sheaf theory]]) are still well-behaved.

> c) If we wish to replace the Zariski topology by some other topology (such as the étale topology), then we are forced to work with $Spec A$ as a [[topos]] rather than simply as a [[topological space]]: the category of étale [[sheaf|sheaves]] on $Spec A$ is [[localic topos|not generated by]] [[subobject]]s of the [[terminal object|final object]].

> When we study [[derived algebraic geometry]], we will want to study [[sheaf|sheaves]] on $Spec A$ of a [[higher category theory|higher categorical]] nature, such as sheaves of [[space]]s or sheaves of [[spectrum|spectra]]. For these purposes it will be most convenient to regard $Spec A$ as an [[(∞,1)-topos|∞-topos]], rather than as a [[topological space]].

From that perspective then a scheme is realized as a [[structured (∞,1)-topos]] with respect to the Zariski [[geometry (for structured (∞,1)-toposes)]] $\mathcal{G}_{Zar}$. See the examples at [[derived scheme]] for more details on this.

+-- {: .un_prop }
###### Proposition ([[Structured Spaces|StSp]], 4.2.9)

Let $Sch_{\leq 0}^{\leq 0}(\mathcal{G}_{Zar}) \subset Sch(\mathcal{G}_{Zar})$ be the full subcategory of all [[derived scheme]]s on the [[n-truncated structured (infinity,1)-topos|0-truncated]] and [[n-localic (infinity,1)-topos|0-localic]] ones. This is canonically equivalent to the ordinary category $Sch(k)$ of [[relative schemes]] over $k$:

$$
  Sch_{\leq 0}^{\leq 0}(\mathcal{G}_{Zar}(k))
  \simeq
  Sch(k)
  \,.
$$
=--

category: algebraic geometry
[[!redirects scheme as structured locally affine infinity topos]]
[[!redirects scheme as locally affine structured infinity topos]]
[[!redirects scheme as a structured locally affine infinity topos]]
[[!redirects scheme as a locally affine structured infinity topos]]
[[!redirects scheme as structured locally affine infinity-topos]]
[[!redirects scheme as locally affine structured infinity-topos]]
[[!redirects scheme as a structured locally affine infinity-topos]]
[[!redirects scheme as a locally affine structured infinity-topos]]
[[!redirects scheme as structured locally affine (infinity,1)-topos]]
[[!redirects scheme as locally affine structured (infinity,1)-topos]]
[[!redirects scheme as a structured locally affine (infinity,1)-topos]]
[[!redirects scheme as a locally affine structured (infinity,1)-topos]]
[[!redirects scheme as structured locally affine ∞-topos]]
[[!redirects scheme as locally affine structured ∞-topos]]
[[!redirects scheme as a structured locally affine ∞-topos]]
[[!redirects scheme as a locally affine structured ∞-topos]]
[[!redirects scheme as structured locally affine (∞,1)-topos]]
[[!redirects scheme as locally affine structured (∞,1)-topos]]
[[!redirects scheme as a structured locally affine (∞,1)-topos]]
[[!redirects scheme as a locally affine structured (∞,1)-topos]]
[[!redirects schemes as structured locally affine infinity toposes]]
[[!redirects schemes as locally affine structured infinity toposes]]
[[!redirects schemes as structured locally affine infinity-toposes]]
[[!redirects schemes as locally affine structured infinity-toposes]]
[[!redirects schemes as structured locally affine (infinity,1)-toposes]]
[[!redirects schemes as locally affine structured (infinity,1)-toposes]]
[[!redirects schemes as structured locally affine ∞-toposes]]
[[!redirects schemes as locally affine structured ∞-toposes]]
[[!redirects schemes as structured locally affine (∞,1)-toposes]]
[[!redirects schemes as locally affine structured (∞,1)-toposes]]
[[!redirects schemes as structured locally affine infinity topoi]]
[[!redirects schemes as locally affine structured infinity topoi]]
[[!redirects schemes as structured locally affine infinity-topoi]]
[[!redirects schemes as locally affine structured infinity-topoi]]
[[!redirects schemes as structured locally affine (infinity,1)-topoi]]
[[!redirects schemes as locally affine structured (infinity,1)-topoi]]
[[!redirects schemes as structured locally affine ∞-topoi]]
[[!redirects schemes as locally affine structured ∞-topoi]]
[[!redirects schemes as structured locally affine (∞,1)-topoi]]
[[!redirects schemes as locally affine structured (∞,1)-topoi]]


schemes are sober

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

#Contents#
* table of contents
{:toc}

## Statement

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

The [[topological space]] underlying a [[scheme]] is a [[sober topological space]]. In particular for [[affine schemes]]: The [[Zariski topology]] on the [[prime spectrum of a commutative ring]] is sober.

=--

For **proof** see [this prop](Zariski+topology#ZariskiTopologyIsSober) at _[[Zariski topology]]_.

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

If instead of the [[prime spectrum of a commutative ring]] one considers only the [[topological subspace]] of [[maximal ideals]] in prop. \ref{SchemesAreSober}, as in [[algebraic geometry]] before the introduction of [[schemes]] by [[Alexander Grothendieck]], then one does not get a sober space. But if the ring is a [[Jacobson ring]] (in that every [[prime ideal]] in the ring is an [[intersection]] of [[maximal ideals]]), then the [soberification](sober+topological+space#Reflection) of the [[topological subspace]] of maximal ideals inside the prime ideals is the full prime spectrum.

=--

## Related statements

* [[Hausdorff implies sober]]

* [[a CW-complex is a Hausdorff space]]

* [[compact subspaces of Hausdorff spaces are closed]]

* [[map from compact space to Hausdorff space]]

## References
 {#References}

* [[The Stacks Project]], _[Tag 01IS](http://stacks.math.columbia.edu/tag/01IS)_


schemes as sheaves on affine schemes

## Idea

Grothendieck introduced spaces as locally representable sheaves over a fixed [[site]] whose objects are interpreted local models. If local models are affine schemes, then [[algebraic scheme]]s (and, more generally, [[algebraic space]]s) can be considered as an example.

## Schemes as presheaves

Given a ring $R$, the correspondence $Y\mapsto Spec(\Gamma_Y \mathcal{O}_Y)$ extends to a [[functor]] $Scheme\to Aff$. A basic result, sometimes referred to as __fundamental theorem on morphisms of schemes__ says that there is a bijection

$$
  CRing(R, \Gamma_Y\mathcal{O}_Y) \cong Scheme(Y, Spec R).
$$

More precisely, for fixed $Y$, and for varying $R$ there is a restricted functor

$$ Scheme(-,Y)|_{Aff^{op}} = h_Y|_{Aff^{op}} = h_Y|_{CRing} : CRing\to Set,$$

and the functor $Y\mapsto h_Y|_{CRing}$ from schemes to presheaves on $Aff = CRing^{op}$ is [[fully faithful functor|fully faithful]]. Thus the general schemes if defined as ringed spaces, indeed form a full subcategory of the category of presheaves on $Aff$.

In fact, the theorem is more general: instead of $Y$ being a scheme, we can take any locally ringed space. Thus the entire category of locally ringed spaces embeds into the category of presheaves on $Aff$.

## Schemes as locally representable sheaves

In fact, one can characterize the essential image of the category of schemes in the category of presheaves over the category of affine schemes, as the full subcategory spanned by those presheaves which are $\tau$-locally representable and which are $\tau$-sheaves, where $\tau$ is the Zariski Grothendieck topology on the category of affine schemes.

See Demazure, Gabriel for functorial approach to schemes, and relations to standard spectra.

## Noncommutative analogues

...(Rosenberg 1998 for fundamental theorem on morphisms, and 1999 for the equivalence of two approaches)

## Derived analogues

...(Toen, Vezzosi etc.)

## References

### Related entries in $n$Lab

* [[affine scheme]], [[algebraic scheme]]
* [[morphism of schemes]]
* [[Yoneda embedding]], [[Isbell duality]]
* [[functorial geometry]]

### Web sources

For the fundamental theorem of morphisms of schemes see for example Theorem 2 in MIT 18-726 AG course notes

* [here](https://ocw.mit.edu/courses/18-726-algebraic-geometry-spring-2009/resources/mit18_726s09_lec06_morphisms)

### Literature

A standard reference to the functorial approach to schemes

* [[Michel Demazure]], [[Pierre Gabriel]], _Groupes algebriques_, tome 1 (later volumes never appeared), Mason and Cie, Paris 1970;  English edition is Introduction to algebraic geometry and algebraic groups, North-Holland, Amsterdam 1980 (North-Holland)

The fundamental theorem of morphisms of schemes has been adapted to noncommutative schemes in

* [[Alexander Rosenberg]], _Noncommutative schemes_, Compos. Math. __112__ (1998) 93--125 [doi](https://doi.org/10.1023/A:1000479824211)

[[!redirects fundamental theorem on morphism of schemes]]
[[!redirects fundamental theorem of morphism of schemes]]


Scherk-Schwarz mechanism

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#### Spin geometry
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#Contents#
* table of contents
{:toc}

## Idea

The _Scherk-Schwarz mechanism_ ([Scherk-Schwarz 79](#ScherkSchwarz79)) is the [[spontaneously broken symmetry|spontaneous]] [[supersymmetry breaking]] by [[KK-compactification]] on a [[circle]] whose [[spin structure]] imposes anti-periodic [[boundary conditions]] for [[fermion fields]].

## Examples and applications

### Witten-Sakai-Sugimoto brane model for quantum chromodynamics

In the [[Witten-Sakai-Sugimoto model]] non-supersymmetric [[QCD]] is [[geometric engineering of QFT|geometrically engineered]] from the [[D=5 super Yang-Mills theory]] on [[D4-branes]] by [[wrapped brane|wrapping]] these on a Scherk-Schwarz circle ([Witten 98, p. 19](#Witten98))

## Related concepts

* [[generalized Scherk-Schwarz mechanism]]

## References

### General

The original articles:

* {#ScherkSchwarz79} [[Joël Scherk]], [[John Schwarz]], _Spontaneous breaking of supersymmetry through dimensional reduction_, Physics Letters B Volume 82, Issue 1, 12 March 1979, Pages 60-64 (doi:10.1016/0370-2693(79)90425-8)

* {#ScherkSchwarz89} [[Joël Scherk]], [[John Schwarz]], _How to get masses from extra dimensions_, Nuclear Physics B Volume 153, 1979, Pages 61-88  ([doi:10.1016/0550-3213(79)90592-3](https://doi.org/10.1016/0550-3213(79)90592-3)), reprinted in: Supergravities in Diverse Dimensions, pp. 1282-1309 (1989) ([doi:10.1142/9789814542340_0083](https://doi.org/10.1142/9789814542340_0083))

### Examples and applications

In [[geometric engineering of QFT|geometrically engineering]] of non-supersymmetric [[QCD]] from the [[D=5 super Yang-Mills theory]] on [[D4-branes]] by [[wrapped brane|wrapping]] these on a Scherk-Schwarz circle (the [[Witten-Sakai-Sugimoto model]]):

* {#Witten98} [[Edward Witten]], p. 19 of: _Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories_, Adv. Theor. Math. Phys.2:505-532, 1998 ([arXiv:hep-th/9803131](https://arxiv.org/abs/hep-th/9803131))

[[!redirects Scherk-Schwarz mechanism]]


Schläfli symbol

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[regular polytope]]

* [[tesselation]]

## References

See also

* Wikipedia, *[Schläfli symbol](https://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol)*

[[!redirects Schlaefli symbol]]


Schneider's theorem

The __Schneider's descent theorem__ is a noncommutative affine analogue of the [[descent along a torsor]] where the structure group is also replaced by a Hopf algebra.
In a left-right version, it says that given a Hopf algebra $H$ and a faithfully flat [[Hopf-Galois extension]] $U\hookrightarrow E$ (where $E$ is a right $H$-comodule algebra), the category of left-right relative Hopf modules ${}_E\mathcal{M}^H$ is equivalent to the category of left $U$-modules ${}_U \mathcal{M}$. The adjoint equivalence is given canonically...

* Hans-Jürgen Schneider, _Principal homogeneous spaces for arbitrary Hopf algebras_,
Israel J. Math. __72__ (1990), no. 1-2, 167–195 [MR92a:16047](http://www.ams.org/mathscinet-getitem?mr=1098988) [doi](http://dx.doi.org/10.1007/BF02764619)
* [[Z. Škoda]], _Some equivariant constructions in noncommutative algebraic geometry_ Georgian Mathematical Journal __16__ (2009), No. 1, 183–202, [arXiv:0811.4770](http://arxiv.org/abs/0811.4770) [MR2011b:14004](http://www.ams.org/mathscinet-getitem?mr=2527623)
* A. Ardizzoni, [[G. Böhm]], C. Menini, _A Schneider type theorem for Hopf algebroids_,
J. Algebra __318__ (2007), no. 1, 225–269 [MR2008j:16103](http://www.ams.org/mathscinet-getitem?mr=2363132) [doi](http://dx.doi.org/10.1016/j.jalgebra.2007.05.017) [math.QA/0612633](http://arxiv.org/abs/math/0612633) (arXiv version is unified, corrected); _Corrigendum_, J. Algebra __321__:6 (2009) 1786-1796 [MR2010b:16060](http://www.ams.org/mathscinet-getitem?mr=2498268) [doi](http://dx.doi.org/10.1016/j.jalgebra.2008.11.040)

[[!redirects Schneider's descent theorem]]


Schouten bracket

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* table of contents
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## Definition

The *Schouten bracket* on [[multivector fields]] is [[Michor (1987)](#Michor87)] the unique (up to a multiplication by a constant) [[natural operation]] on [[multivector fields]]
$$
  \Gamma(\Lambda^k T M)
    \otimes
  \Gamma(\Lambda^l T M)
  \longrightarrow
  \Gamma(\Lambda^{k+l-1} T M)
  \,.
$$

Concretely, it is given in terms of the [[Lie bracket of vector fields]] by:
$$
  [X_1\wedge\cdots\wedge X_k,Y_1\wedge\cdots\wedge Y_l]
  \;=\;
  \sum_{i,j}(-1)^{i+j}
  [X_i, Y_j] \wedge X_1\wedge\cdots\hat X_i\cdots\wedge X_k\wedge Y_1\wedge\cdots \hat Y_j\cdots \wedge Y_l
  .
$$

For [[multivector fields]] regarded as “[[antifields]]” in [[BV-BRST formalism]], the Schouten bracket is called the _[[antibracket]]_.

## Applications

Suppose $P\in\Gamma(\Lambda^2 TM)$.
The the bracket $\{f,g\}=\langle d f\wedge d g,P\rangle$ satisfies the [[Jacobi identity]] (and hence is a [[Poisson bracket]]) if and only if $[P,P]=0$.

## Related concepts

* [[Lie bracket of vector fields]]

* [[Frölicher–Nijenhuis bracket]]

* [[Nijenhuis–Richardson bracket]]

## References

The notion is due to:

* [[Jan Schouten]], _Über Differentialkonkomitanten zweier kontravarianten Grössen_, Indagationes Mathematicae **2** (1940) 449–452

* [[Jan Schouten]], _On the differential operators of the first order in tensor calculus_, In: Convegno Int. Geom. Diff. Italia. (1953) 1–7

* [[Albert Nijenhuis]], _Jacobi-type identities for bilinear differential concomitants of certain tensor fields I_, Indagationes Mathematicae **17** (1955) 390–403 [[doi:10.1016/S1385-7258(55)50054-0](https://doi.org/10.1016/S1385-7258(55)50054-0)]

A coordinate-free treatment is given in

* W. M. Tulczyjew, _The Graded Lie Algebra of Multivector Fields and the Generalized Lie Derivative of Forms_. Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 22:9 (1974), 937–942.  [PDF](https://dmitripavlov.org/scans/tulczyjew-the-graded-lie-algebra-of-multivector-fields-and-the-generalized-lie-derivative-of-forms.pdf).

Characterization as a [[natural operation]] is due to:

* {#Michor87} [[Peter W. Michor]], *Remarks on the Schouten-Nijenhuis bracket*,  In: J. Bureš, [[Vladimir Souček|V. Souček]] (eds.): *Proceedings of the Winter School "Geometry and Physics"* Circolo Matematico di Palermo, Palermo (1987) 207-215 [[dml:701423](https://dml.cz/handle/10338.dmlcz/701423), [pdf](https://dml.cz/bitstream/handle/10338.dmlcz/701423/WSGP_07-1987-1_21.pdf), [[Michor-SchoutenNijenhuisBracket.pdf:file]]]

Textbook account: Chapter 33.2 of

* [[Peter W. Michor]], _Topics in Differential Geometry_, Graduate Studies in Mathematics 93 (2008).  [PDF](https://www.mat.univie.ac.at/~michor/dgbook.pdf).

[[!redirects Schouten-Nijenhuis bracket]]
[[!redirects Schouten–Nijenhuis bracket]]
[[!redirects Schouten-Nijenhuis brackets]]
[[!redirects Schouten–Nijenhuis brackets]]


Schrödinger equation

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#Contents#
* table of contents
{:toc}

## Idea ##
The Schrödinger equation (named after [[Erwin Schrödinger]]) is the [[evolution equation]] of [[quantum mechanics]] in the [[Schrödinger picture]]. Its simplest version results from replacing the classical expressions in the nonrelativistic, mechanical equation for the energy of a point particle, by [[operators]] on a [[Hilbert space]]:

We start with a point particle with mass $m$, impulse $p$ moving in the space $\mathbb{R}^3$ with a given potential function $V$, the energy of it is the sum of kinetic and potential energy:
$$
   E = \frac{p^2}{2 m} + V
$$

Quantizing this equation means replacing the coordinate $x \in \mathbb{R}^3$ with the [[Hilbert space]] $L^2(\mathbb{R}^3)$ and
$$
E \to i \hbar  \frac{\partial}{\partial t}
$$

$$
p \to -i  \hbar \nabla
$$

with the [[Planck constant]] $h$ and
$$
\hbar = \frac{h}{2 \pi}
$$
the reduced Planck constant.

This results in the Schrödinger equation for a single particle in a potential:
$$
i \hbar  \frac{\partial}{\partial t} \psi(t, x) = - \frac{\hbar^2}{2 m} \nabla^2 \psi(t, x) + V(t, x) \psi(t, x)
$$
The last term is the multiplication of the functions $V$ and $\psi$.

The right hand side is called the [[Hamilton operator]] $H$, the Schrödinger equation is therefore mostly stated in this form:
$$
i \hbar \psi_t = H \psi
$$

## Abstract ##
...

## Definition ##
...

## Properties ##

### Decomposition into phase and amplitude

Consider for simplicity, the [[mechanical system]] of a [[particle]] of [[mass]] $m$ propagating on the [[real line]] $\mathbb{R}$ and subject to a [[potential]] $V \in C^\infty(\mathbb{R})$, so that the Schrödinger equation is the [[differential equation]] on [[complex number|complex]]-valued [[functions]] $\Psi \colon \mathbb{R}\times \mathbb{R} \to \mathbb{C}$ given by

$$
  i \hbar
  \frac{\partial}{\partial t}
  \Psi
  =
  \frac{\hbar^2}{2m}
  \frac{\partial^2}{\partial^2 x} \Psi
  +
  V \Psi
  \,,
$$

where $\hbar$ denotes [[Planck's constant]].

By the nature of [[complex numbers]] and by the discussion at _[[phase and phase space in physics]]_, it is natural to parameterize $\Psi$ -- away from its [[zero locus]] -- by a [[complex phase]] function

$$
  S \;\colon\; \mathbb{R}\times \mathbb{R} \longrightarrow \mathbb{R}
$$

and an [[absolute value]] function $\sqrt{\rho}$

$$
  \sqrt{\rho} \;\colon\; \mathbb{R}\times \mathbb{R} \longrightarrow \mathbb{R}
$$

which is positive, $\sqrt{\rho} \gt 0$, as

$$
  \Psi
    \coloneqq
  \exp\left(\frac{i}{\hbar} S / \hbar\right) \sqrt{\rho}
  \,.
$$

Entering this Ansatz into the above Schrödinger equation, that [[complex number|complex]] equation becomes equivalent to the following two [[real number|real]] equations:

$$
  \frac{\partial}{\partial t} S
  =
  -
  \frac{1}{2m} \left(\frac{\partial}{\partial x}S\right)^2
  -
  V
  +
  \frac{\hbar^2}{2m} \frac{1}{\sqrt{\rho}}\frac{\partial^2}{\partial^2 x} \sqrt{\rho}
$$

and

$$
  \frac{\partial}{\partial t} \rho
  =
  -
  \frac{\partial}{\partial x}
  \left(
    \frac{1}{m} \left(\frac{\partial}{\partial x}S\right) \rho
  \right)
  \,.
$$

Now in this form one may notice a similarity of the form of these two equations with other equations from [[classical mechanics]] and [[statistical mechanics]]:

1. The first equation is similar to the [[Hamilton-Jacobi equation]] that expresses the classical [[action functional]] $S$ and the [[canonical momentum]]

   $$
     p \coloneqq \frac{\partial}{\partial x} S
   $$

   except that in addition to the ordinary [[potential energy]] $V$ there is an additional term

   $$
    Q
      \coloneq
    \frac{\hbar^2}{2m} \frac{1}{\sqrt{\rho}}\frac{\partial^2}{\partial^2 x} \sqrt{\rho}
   $$

   which is unlike what may appar in an ordinary [[Hamilton-Jacobi equation]]. The perspective of Bohmian mechanics is to regard this as a correction of [[quantum physics]] to classical [[Hamilton-Jacobi theory]], it is then called the _quantum potential_. Notice that unlike ordinary potentials, this "quantum potential" is a function of the density that is subject to the potential. (Notice that this works only away from the [[zero locus]] of $\rho$.)

1. The second equation has the form of the [[continuity equation]] of the [[flow]] expressed by $\frac{1}{m}p$.

(In the context of _[[Bohmian mechanics]]_ one regard this equivalent rewriting of the [[Schrödinger equation]] as providing a [[hidden variable theory]] formulation of [[quantum mechanics]].)

## Examples ##
...

## Related concepts

* [[Tomonaga-Schwinger equation]]

* [[Schrödinger picture]]

* [[heat equation]]

* [[wave equation]], [[Klein-Gordon equation]]

* [[Dirac equation]]

## References ##

Any introductory textbook about [[quantum mechanics]] will explain the Schrödinger equation (from the viewpoint of physicists mostly).

* Wikipedia: [Schrödinger equation] (http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation)

[[!redirects Schrödinger's equation]]

[[!redirects Schroedinger equation]]
[[!redirects Schroedinger's equation]]


Schrödinger picture > history

< [[Schrödinger picture]]


Schubert calculus

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#Schubert calculus#
* table of contents
{:toc}

## Idea

Schubert calculus is a formal calculus in [[enumerative geometry]], which geometrically reduces to the combinatorics and [[intersection theory]] of so-called __Schubert cells__ in [[Grassmannians]].

Schubert calculus is concerned with the [[ring]] structure on the [[cohomology]] of [[flag varieties]] and [[Schubert varieties]]. Traditionally this is considered for [[ordinary cohomology]] (see [References -- traditional](#ReferencesTraditional)) later also for [[generalized cohomology theories]] (see [References -- In generalized cohomology](#ReferencesInGeneralizedCohomology)), notably in [[complex oriented cohomology theory]] such as [[K-theory]], [[elliptic cohomology]] and [[algebraic cobordism]].

The rigorous foundations of Schubert calculus is the content of the 15th of [[Hilbert's problems]].

## Details

### Flag varieties and Schubert varieties

The basic data to be fixed is a sequence of inclusions

$$
  T \subset B \subset G
$$

where

* $G$ is a [[connected topological space|connected]] [[complex geometry|complex]] [[reductive group|reductive]] [[algebraic group]]

* $B$ is a [[Borel subgroup]]

* $T$ is a [[maximal torus]].

This induces

* the [[Weyl group]] $W_0 = N(T)/T$;

* the [[character lattice]] $\mathfrak{h}_{\mathbb{Z}}^\ast = Hom(T, \mathbb{C}^\times)$;

* the [[cocharacter lattice]] $\mathfrak{h}_{\mathbb{Z}} = Hom(\mathbb{C}^\times, T)$.

* a _[[standard parabolic subgroup]]_ of $G$ is a subgroup $P_J$ including $B$ such that $G/P$ is a [[projective variety]];

   [[parabolic subgroup]] is one [[conjugation|conjugate]] to the standard parabolic subgroup.

* the [[flag variety]] $G/B$;

* the _[[partial flag varieties]]_ $G/P_J$

* the [[Bruhat decomposition]] is the [[coproduct]] decomposition

  $$
    G = \underset{w \in W_0}{\coprod} B w B
  $$

  $$
    G =  \underset{u \in W^J}{\coprod} B u P_j
  $$

  with

  * $W_J \coloneqq \{v \in W_0 | v T  \subset P_J\}$

  * $W^J \coloneqq \{coset\; representatives\; u \; of \; cosets \; in W_0/W_J\}$

* into the [[Schubert varieties]]

  $$
    X_w \coloneqq \overline{B w B} \subset G/B
  $$

  $$
    X_u^J \coloneqq \overline{B u P_J} \subset G/P_J
    \,.
  $$

### Correspondences, pull-push and Schubert classes
  {#Correspondences}

From the above data one obtains [[homomorphisms]] of spaces with
$G$-[[action]] forming [[correspondences]] ("generalized [[twistor correspondence]]")

$$
  \array{
    && G/B
    \\
    & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}}
    \\
    G/P_1
    &&
    &&
    G/P_2
  }
  \,.
$$

e.g. ([Ganter-Ram 12, p.4](#GanterRam12))

For [[fiber integration]] $(p_i)_!$ in [[generalized cohomology theories]] along these maps see ([Ganter-Ram 12, 4.1](#GanterRam12))

Similarly, let

$$
  \sigma_w \;\colon\; X_w \hookrightarrow G/B
$$

be the inclusion of the [[Schubert varieties]], then push-forward of the unit classes allong these inclusions defined _Schubert classes_

$$
  [X_w] \coloneqq (\sigma_w)_!(w)
$$

([Ganter-Ram 12, 5](#GanterRam12))

For [[equivariant K-theory]] this is discussed in ([Ganter 12, 8.2](#Ganter12)). For [[equivariant elliptic cohomology]] in ([Ganter 12, 8.3](#Ganter12))

### Schubert products
 {#SchubertProducts}

With Schubert classes $[X_w]$ defines as above in a [[multiplicative cohomology theory]], the _Schubert product formula_ is

$$
  [X_u][X_v]
  =
  \underset{w \in W_0}{\sum} c^w_{u v} [X_w]
$$

for some [[coefficients]] $\{c^w_{u v}\}$, to be determined.

([Ganter-Ram 12, 6](#GanterRam12))

## Related concepts

* [[Borel-Weil-Bott theorem]]

  * [[orbit method]]

  * [[Dirac induction]]

* [[quantum Schubert calculus]]

* [[twistor space]],

* [[twistor correspondence]], [[horocycle correspondence]], [[Grothendieck-Springer correspondence]]

## References

### Traditional
 {#ReferencesTraditional}

* [eom]: Frank Sotile, [Schubert calculus](http://www.encyclopediaofmath.org/index.php/Schubert_calculus)
* wikipedia [Schubert calculus](http://en.wikipedia.org/wiki/Schubert_calculus)
* H. Schubert, _Kalkül der abzählenden Geometrie_, Springer (1879) (Reprinted (with an introduction by S. Kleiman) 1979), [MR0555576](http://www.ams.org/mathscinet-getitem?mr=0555576)

* S.L. Kleiman, D. Laksov, _Schubert calculus_, Amer. Math. Monthly __79__ (1972) pp. 1061–1082, [MR0323796](http://www.ams.org/mathscinet-getitem?
mr=0323796), [jstor](http://www.jstor.org/stable/2317421)

### In generalized cohomology theory
 {#ReferencesInGeneralizedCohomology}

Discussion of Schubert calculus in [[generalized cohomology theories]] is in

* [[Paul Bressler]], Sam Evens. _The Schubert calculus, braid relations,
and generalized cohomology_. Trans. Amer. Math. Soc.,
317(2):799–811, 1990
 {#BresslerEvens90}

* [[Paul Bressler]], Sam Evens, _Schubert calculus in complex cobordism_ Trans. Amer. Math. Soc., 331(2):799–813, 1992

* Baptiste Calmès, Victor Petrov,  Kirill Zainoulline, _Invariants,
torsion indices and oriented cohomology of complete flags_
May 200 ([web](http://front.math.ucdavis.edu/0905.1341))

* Jens Hornbostel, Valentina Kiritchenko, _Schubert calculus for
algebraic cobordism_. J. Reine Angew. Math., 656:59–85, 2011

* [[Nora Ganter]], [[Arun Ram]], _Generalized Schubert calculus_, J. Ramanujan Math. Soc. 28A (Spec. Issue-2013) 1-42 [arxiv/1212.5742](http://arxiv.org/abs/1212.5742)
  {#GanterRam12}

* [[Nora Ganter]], _The elliptic Weyl character formula_ ([arXiv:1206.0528](http://arxiv.org/abs/1206.0528))
 {#Ganter12}

[[!redirects Hilbert's 15th problem]]

[[!redirects Schubert class]]
[[!redirects Schubert classes]]

[[!redirects generalized Schubert calculus]]

category: combinatorics, algebraic geometry


Schubert variety

#Contents#
* table of contents
{:toc}

## Idea

A certain sub-variety of a [[Grassmannian]].

## Related concepts

* [[Schubert calculus]]
* [[quantum Schubert cell]]

## References

* Wikipedia, _[Schubert variety](http://en.wikipedia.org/wiki/Schubert_variety)_

For an [[ind-object|ind]]-version see

* Lucas Fresse, Ivan Penkov, _Schubert decompositions for ind-varieties of generalized flags_, [arxiv/1506.08263](http://arxiv.org/abs/1506.08263)

[[!redirects Schubert varieties]]
[[!redirects Schubert cell]]


Schur function

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#Contents#
* table of contents
{:toc}

## Idea

The *Schur polynomials* $s_\lambda$, are [[polynomials]] in $n$ [[variables]] indexed by [[partitions]] $\lambda = \lambda_1 \geq \lambda_2 \geq \ldots \geq \lambda_n$ of $n$, which constitute a [[linear basis]] of the [[ring]] of [[symmetric polynomials]] in $n$ variables

## Definition

Given the [[partition]] $\lambda_1\geq \lambda_2\geq \ldots \geq \lambda_n$, the corresponding __Schur polynomial__ is defined as follows. First define the $n\times n$-[[determinant]] (for any partition $\alpha$ in $n$ parts)

$$
  a_\alpha \;\coloneqq\; det\big(x_i^{\alpha_j}\big).
$$

Let $\delta=(n-1, n-2, \dots, 1, 0)$. Then the Schur polynomial attached to $\lambda$ is the quotient

$$
  s_{\lambda}(x_1,x_2,\dots,x_n)
  =
  a_{\lambda+\delta}/a_{\delta}.
$$

As is usual in the theory of [[symmetric functions]] one can also deal with [[formal power series]] in infinitely many variables. To make this precise one uses an [[inverse limit]] (see [Macdonald 95](#Macdonald95)) and obtains a __Schur function__ $s_{\lambda}$ for each partition, depending on countably many variables $x_1, x_2,\dots,x_n,x_{n+1}, \dots$.

## Properties

### In terms of semistandard Young tableaux
 {#InTermsOfSemistandardYoungTableaux}

A *[[semistandard Young tableau]]* is a [[Young tableau]] such that its values are:

1. weakly increasing to the right (along rows),

1. strictly increasing downwards (along columns).

For example:

$$
\array{
  1 & 1 & 1 & 3
  \\
  2 & 3
  \\
  4
}
$$

Given a [[semistandard Young tableau]] $T$, we write $X^T$ for  the [[monomial]] which contains one factor of the [[variable]] $x_k$ for each occurrence of $k$ in the Young tableau:

\[
  \label{MonomialAssociatedWithSemistandardYoungTableau}
  x^T
  \;\coloneqq\;
  x_1^{\#1s}
  x_1^{\#2s}
  \cdots
  \,.
\]

We write

\[
  \label{SetOfSemistandardYoungTableaux}
  ssYT_\lambda \;\supset\; ssYY_\lambda(N)
\]

for the set of semistandard Young tableaux whose underlying [[partition]] (i.e. forgetting its labels) is $\lambda$, and, respectively, for its [[subset]] on those whose labels are bounded as $T_{i,j} \leq N$.

\begin{prop}
  For $n \in \mathbb{N}$ and $\lambda$ a [[partition]] of $n$, the corresponding Schur polynomial $s_\lambda$ is equal to the [[sum]] over the [[monomials]] (eq:MonomialAssociatedWithSemistandardYoungTableau) associated with all [[semistandard Young tableaux]] (eq:SetOfSemistandardYoungTableaux) of shape $\lambda$:

\[
  s_\lambda
  \;\;=\;\;
  \underset{
    {T \in ssYT(\lambda)},
  }{\sum}
  x^T
  \,.
\]
\end{prop}

([Sagan 01, Def. 4.4.1](#Sagan01), review in [Sagan Enc., p. 1](#SaganEnc))

This means that the Schur polynomial $s_\lambda$ in $N$ variables is the sum over semistandard Young tableaux (eq:SetOfSemistandardYoungTableaux) of shape $\lambda$ and with labels bounded as $T_{i,j} \leq N$:

\[
  \label{SchurPolynomialWithFiniteNumberOfvariablesAsSumOverssYT}
  s_\lambda
  \big(
    x_1, \cdots, x_N
  \big)
  \;\;=\;\;
  \underset{
    {T \in ssYT_\lambda(N)},
  }{\sum}
  x^T
  \,.
\]

In particular, the evaluation of a Schur polynomial at $N$ unit values is the number of [[semistandard Young tableaux]] with labels $\leq N$:

\[
  \label{SchurPolynomialCountsNumberOfSemiStandardYoungTableaux}
  s_\lambda
  \big(
    x_1\!=\!1, \cdots, x_N \!=\! 1
  \big)
  \;\;=\;\;
  \underset{
    {T \in ssYT_\lambda(N)},
  }{\sum}
  1
  \;\;=\;\;
  \left\vert
    ssYT_\lambda(N)
  \right\vert
  \,.
\]

An immediate consequence is:

\begin{prop}\label{CoefficientsOfMonomialsAreNaturalNumbers}
  The [[coefficient]] of any [[monomial]] $x_1^{n_1} x_2^{n_2} \cdots$ appearing in a Schur polynomial is non-negative.
\end{prop}

### In terms of $Sym(n)$-Characters
 {#InTermsOfSymnCharacters}

We discuss an expression of the Schur polynomials as [[symmetric polynomials]] with [[coefficients]] in the [[character of a linear representation|character]]-values of the symmetric group; this is Frobenius' formula in Prop. \ref{FrobeniusFormula} below.

First to set up some notation:

For the present purpose, a *[[partition]]* of a [[natural number]] $n \in \mathbb{N}$ is a weakly decreasing sequence

$$
  \lambda = (\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k)
$$

of [[natural numbers]] whose [[sum]] equals $n$:

$$
  \lambda_1 + \lambda_2 + \cdots + \lambda_k \;=\; n
  \,.
$$

By the [[representation theory of the symmetric group]], such partitions label its [[irreducible representations]] in the form of [[Specht modules]] $Specht^{(\lambda)}$. We write

\[
  \label{SpechtCharacter}
  \chi^{(\lambda)}(-)
  \;\coloneqq\;
  Tr_{Specht^{(\lambda)}}(-)
\]

for the [[irreducible character|irreducible]] [[character of a linear representation|character]] corresponding to this [[Specht module]].

We write

$$
  \array{
    Sym(n)
    &
      \overset{CycLengths}{\longrightarrow}
    &
    Part(n)
    \\
    \big\downarrow & \nearrow_{\mathrlap{\simeq}}
    \\
    Sym(n)/_{ad}Sym(n)
  }
$$

for the function that sends a [[permutation]] $\sigma$ of $n$ elements to the [[partition]]

\[
  \label{CycleLengths}
  CycLengths(\sigma)
  \;=\;
  (l_1 \geq l_2 \geq \cdots \geq l_{\left\vert Cycles(\sigma)\right\vert})
\]

of $n$ given by the lengths of its [[permutation cycles]] ([this Prop.](symmetric+group#ConjugacycClassesOfSymmetricGroupCorrespondToCycleSet)).

Given any such [[partition]] $(l_1 \geq l_2 \geq \cdots \geq l_k)$, we consider the following [[symmetric polynomial]] in $n$ [[variables]]:

\[
  \label{SymmetricFunctionIndecedByPartition}
  p_{
    (l_1 \geq l_2 \geq \cdots \geq l_k)
  }
  (x_1, x_2, \cdots, x_n)
  \;\;\;\coloneqq\;\;\;
  \big(
    x_1^{l_1} + \cdots + x_n^{l_1}
  \big)
  \cdot
  \big(
    x_1^{l_2} + \cdots + x_n^{l_2}
  \big)
  \cdots
  \big(
    x_1^{l_k} + \cdots + x_n^{l_k}
  \big)
  \,.
\]

\begin{prop}\label{FrobeniusFormula}
**(Frobenius formula)**

For $n \in \mathbb{N}$ and $\lambda$ a [[partition]] of $n$, the Schur polynomial $s_\lambda$ is equal to the [[sum]] over [[permutations]] $\sigma$ of the character values $\chi^{(\lambda)}$ (eq:SpechtCharacter) at $\sigma$ times the [[symmetric polynomial]] (eq:SymmetricFunctionIndecedByPartition) which is indexed by the cycle lengths (eq:CycleLengths) of $\sigma$:

$$
  s_\lambda
   \;=\;
  \frac{1}{n!}
  \underset
    {\sigma \in Sym(n)}
    {\sum}
  \chi^{(\lambda)}(\sigma)
  \cdot
  p_{CycLengths(\sigma)}
  \,.
$$
\end{prop}

([Sagan 01, Thm. 4.6.4](#Sagan01), review in [Sagan Enc., Thm. 3](#SaganEnc))

\linebreak

### Schur-Weyl duality

The Schur polynomials  are precisely the [[irreducible characters]] of [[finite dimensional vector space|finite dimensional]] polynomial [[linear representation|representations]] of [[general linear group|GL(n)]].

Also, the character $\chi_{\lambda}$ of $V_\lambda$, the irreducible representation of $S_k$ attached to $\lambda$ (for $k$ the size $|\lambda|$ of the partition) maps to the Schur polynomial under the character map
$ch$ from virtual characters to symmetric polynomials.

This correspondence between [[linear representations]] of the [[symmetric groups]] and the [[general linear groups]] is called *[[Schur-Weyl duality]]*.

## Generalizations via Schur functors

[[Schur functors]] may be viewed as a [[categorification]] of Schur functions. In fact, the Schur functors make sense in more general [[symmetric monoidal categories]] than [[VectorSpaces]]. It is a theorem in the case of vector space that the [[trace]] of
a Schur functor $\mathbf{S}_\lambda(V)\stackrel{\mathbf{S}_\lambda(g)}\to \mathbf{S}_\lambda(V)$ on an endomorphism $g\in GL(V)$ is the Schur function of the eigenvalues of $g$. Considering the trace of a Schur functor makes sense in a general situation allowing for Schur functors and for the trace ([[rigid monoidal category]]); of course choosing appropriate variables to express the trace may depend on a context.

## Related concepts

* [[Schur positivity]]

* [[representation theory of the symmetric group]]

Generalizations:

* [[Jack polynomial]]

* [[Macdonald polynomial]]

* [[noncommutative Schur function]]

* [[quasisymmetric Schur function]]

\linebreak

## References

The concept first appears in work by [[Carl Jacobi]] on [[determinants]].

It is named after:

* [[Issai Schur]], *Über eine Klasse von Matrizen die sich einer gegeben Matrix zuordnen lassen*, Inaugural-Dissertation, Berlin (1901) JMF 32.0165.04

See also

* [[Richard Stanley]], Section II of: *Theory and application of plane partitions 1*, Studies in Applied Math. **50** 2 (1971), 167-188 ([pdf](http://www-math.mit.edu/~rstan/pubs/pubfiles/12-1.pdf), [[StanleyPlanePartitions1.pdf:file]])

Textbook accounts:

* {#Macdonald95} [[Ian G. Macdonald]], Section I.3 of: _Symmetric functions and Hall polynomials_, Oxford Math. Monographs, 2nd enlarged ed. 1995 ([ISBN:9780198739128](https://global.oup.com/academic/product/symmetric-functions-and-hall-polynomials-9780198739128?cc=ae&lang=en&))

* {#Stanley99} [[Richard Stanley]], Sections 7.10, 7.15 in: *Enumerative combinatorics 2*, Cambridge University Press (1999, 2010) ([doi:10.1017/CBO9780511609589](https://doi.org/10.1017/CBO9780511609589), [webpage](http://www-math.mit.edu/~rstan/ec/))

* {#Sagan01} [[Bruce Sagan]], Section 4.4 of: _The symmetric group_, Springer 2001 ([doi:10.1007/978-1-4757-6804-6](https://link.springer.com/book/10.1007/978-1-4757-6804-6), [pdf](http://math.sfsu.edu/federico/Clase/RepTh/sagan.pdf))

* Stuart Martin, _Schur algebras and representation theory_, Cambridge Univ. Press 1994

Survey:

* {#SaganEnc} [[Bruce E. Sagan]], _Schur functions_, in: [[Michiel Hazewinkel]] (ed.), *[[eom|Encyclopaedia of Mathematics]]*, Springer 2002 ([pdf](http://www.mth.msu.edu/~sagan/Papers/Old/schur.pdf), [[SaganSchurFunctions.pdf:file]])

* Wikipedia, *[Schur polynomial](https://en.wikipedia.org/wiki/Schur_polynomial)*

See also:

* Olivier Blondeau-Fournier, Pierre Mathieu, _Schur superpolynomials: combinatorial definition and Pieri rule_, [arxiv/1408.2807](http://arxiv.org/abs/1408.2807)

* Miles Jones, Luc Lapointe, _Pieri rules for Schur functions in superspace_, [arxiv/1608.08577](https://arxiv.org/abs/1608.08577)

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