Bundles

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and

Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu

Last updates: 20 November 2009

Bundles

Grothendieck groups

Let $𝒜$ be an abelian category. The Grothendieck group of $𝒜$ is the group $K\left(𝒜\right)$ generated by

$\left[M\right]\text{,}\text{for}M\in 𝒜\text{,}\text{with relations}\left[M_1\right]=\left[M_2\right]\text{,}\text{if}{M}_1\cong M_2$

and

$\left[M\right]=\left[M_1\right]+\left[M_2\right]\text{,}\text{if there exists an exact sequence}0⟶M_1⟶M⟶M_2⟶0.$

Let $X$ be a space and let $\mathrm{pt}$ be the space with a single point.

• The representation ring, or character ring, of $G$ is the Grothendieck group $K_G\left(\mathrm{pt}\right)$ of the category of $G$-modules.
• The (topological) $K$-theory of $X$ is the Grothendieck group $K\left(X\right)$ of the category of vector bundles on $X$.
• The (algebraic) $K$-theory of $X$ is the Grothendieck group $K\left(X\right)$ of the category of coherent sheaves on $X$.
• The (topological) $G$-equivariant $K$-theory of $X$ is the Grothendieck group $K_G\left(X\right)$ of the category of $G$-equivariant vector bundles on $X$.
• The (algebraic) $G$-equivariant $K$-theory of $X$ is the Grothendieck group $K_G\left(X\right)$ of the category of $G$-equivariant coherent sheaves on $X$.

Bundles

Let $X$ and $F$ be spaces. A fibre bundle on $X$ with fiber $F$ is a space $E$ with a surjective map $E\stackrel{p}{\to }X$ such that $p^{–1}\left(x\right)\simeq F$, for $x\in X$, and if $x\in X$ there is a neighborhood $U$ of $x$ and an isomorphism

$$\begin{array}{ccc}U\times F& \stackrel{\phi }{⟶}& p^{–1}\left(U\right)\\ {\mathrm{pr}}_1\downarrow & & \downarrow p\\ U& =& U\end{array}$$

Let $G$ be a group which acts on $F$. A $G$-bundle with fibre $F$ is a bundle $E\stackrel{p}{\to }X$ such that the transition functions $g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to \mathrm{Aut}\left(F\right)$ given by

$$g_{\alpha \beta }\left(x\right)={\phi }_{\alpha }{\phi }_{\beta }^{–1}{\mid }_{\left\{x\right\}\times F}$$

are morphisms $g_{\alpha \beta }:U_{\alpha }\cap U_{\beta }\to G$. A principal $G$-bundle is a fibre bundle with fibre $G$ and a $G$-action $E\times G⟶E$.

Let $X$ be a space and let $V$ be a vector space. A vector bundle on $X$ with fiber $V$ is a space $E$ with a surjective map $E\stackrel{p}{\to }X$ such that $p^{–1}\left(x\right)\simeq V$, for $x\in X$, and there is a open cover of $X$ and isomorphisms

$$\begin{array}{ccc}{U}_{\alpha }\times V& \stackrel{{\phi }_{\alpha }}{⟶}& p^{–1}\left(U_{\alpha }\right)\\ {\mathrm{pr}}_1\downarrow & & \downarrow p\\ U_{\alpha }& =& U_{\alpha }\end{array}$$

with $\phi :x\times V\to p^{–1}\left(x\right)$ a linear isomorphism. A vector bundle is a $\mathrm{GL}\left(V\right)$-bundle.

A section of $E$ is a morphism $s:X\to E$

$\begin{array}{c}E\\ p\downarrow \uparrow s\\ X\end{array}\text{such that}p\circ s={\mathrm{id}}_X$

The sheaf of sections of $E$ is given by

$ℰ\left(U\right)=\left\{\mathrm{ \; sections\; of\; }E\stackrel{p}{\to }U\right\}$

with ${𝒪}_X\left(U\right)$-action given by $\left(fs\right)\left(x\right)=f\left(x\right)s\left(x\right),$ for $f\in {𝒪}_X\left(U\right),s\in ℰ\left(U\right)$. Then

$\begin{array}{ccc}\left\{\text{vector bundles on }X\right\}& ⟷& \left\{\text{locally free sheaves on }X\right\}\\ E& \mapsto & ℰ\end{array}$

is an equivalence of categories.

Let $G$ be a topological group. A $G$-space is a topological space $X$ with a $G$-action. More precisely, a $G$-space is a topological space $X$ with a continuous map

$G\times X⟶X\text{such that}1x=x\text{and}{g}_1\left(g_{2}x\right)=\left(g_1{g}_2\right)x,$

for $x\in X$ and $g_1,g_2\in G$.

Let $X$ be a $G$-space. A $G$-equivariant vector bundle on $X$ is a $G$-space $E$ with a surjective map $E\stackrel{p}{\to }X$ such that

• If $x\in X$ the fiber $E_x=p^{–1}\left(x\right)$ is isomorphic to ${ℂ}^m$,
• $E$ is locally trivial, i.e. If $x\in X$ then there exists an open set $U$ containing $x$ and a homeomorphism

  $\begin{array}{ccc}U\times F& \stackrel{\phi }{⟶}& p^{–1}\left(U\right)\\ {\mathrm{pr}}_1\downarrow & & \downarrow p\\ U& =& U\end{array}$

  such that if $u\in U$ then $\phi :{ℂ}^m\stackrel{\simeq }{\to }{p}^{–1}\left(u\right)$,
• If $e\in E$ and $g\in G$ then $p\left(ge\right)=gp\left(e\right)$,
• If $g\in G$ and $x\in X$ then $g:E_x\to E_{gx}$ is a morphism of vector spaces.

Sheaves

Let $X$ be a topological space. A sheaf on $X$ is a contravariant functor

$$\begin{array}{rrcl}{𝒪}_X:& \left\{\text{\hspace{0.17em}open sets of }X\right\}& ⟶& \left\{\text{ rings }\right\}\\ & U& ⟼& {𝒪}_X\left(U\right)\end{array}$$

such that if $\left\{U_{\alpha }\right\}$ is an open cover of $U$ and $f_{\alpha }\in {𝒪}_X\left(U\right)$ are such that

$$\text{if }{f}_{\alpha }{\mid }_{U_{\alpha }\cap U_{\beta }}=f_{\beta }{\mid }_{U_{\alpha }\cap U_{\beta }}\text{for all }\alpha ,\beta ,\text{then there is a unique }f\in {𝒪}_X\left(U\right)$$

such that $f_{\alpha }=f{\mid }_{U_{\alpha }}$, for all $\alpha$.

Let $A$ be a ring. An $A$-module is free if there is an isomorphism

$A^{\oplus n}\simeq M,\text{for some}n\in {\mathbb{Z}}_{\ge 0}.$

An $A$-module $M$ is finitely generated, or of finite type, if there is a surjective morphism

$A^{\oplus n}\to M,\text{for some}n\in {\mathbb{Z}}_{\ge 0}.$

An $A$-module $M$ is finitely presented if there is an exact sequence

$R\to G\to M\to 0,\text{with }R\text{ and }G\text{ free }A\text{-modules.}$

A locally free sheaf on $X$ is a sheaf $ℱ$ of ${𝒪}_X$-modules such that if $x\in X$ then there is a neighborhood $U$ of $x$ with

$ℱ\left(U\right)\simeq {𝒪}_X^{\oplus n}$.

A coherent sheaf on $X$ is a locally finitely presented sheaf of ${𝒪}_X$-modules, i.e. a sheaf $ℱ$ of ${𝒪}_X$-modules such that

• $ℱ$ is of finite type, i.e. $ℱ\left(U\right)$ is generated by a finite set of sections (there exists a surjection ${𝒪}_X(U{)}^{\oplus n}\to ℱ(U)$),
• For each open set $U$ of $X$ and each homomorphism ${𝒪}_X\left(U{)}^{\oplus n}\stackrel{\phi }{\to }ℱ\right(U)$, $\ker \phi$ is of finite type.

Let $G$ be a group that is also a space and let $X$ be a space with a $G$-action $G\times X\stackrel{a}{⟶}X.$ A $G$-equivariant sheaf is a sheaf $ℱ$ of ${𝒪}_X$-modules with an isomorphism

$a^{*}ℱ\stackrel{\phi }{⟶}{\mathrm{pr}}_2^{*}ℱ\text{such that}{p}_{23}^{*}\phi \circ ({\mathrm{id}}_G\times a{)}^{*}\phi =(m\times {\mathrm{id}}_X{)}^{*}\phi \text{and}\phi {\mid }_{1\times X}={\mathrm{id}}_{ℱ},$

where $p_{23}:G\times G\times X⟶G\times X$ is given by $p_{23}(g_1,g_2,x)=(g_2,x)$ and we identify $a^{*}ℱ{\mid }_{1\times X}=ℱ$ and ${\mathrm{pr}}_2^{*}ℱ{\mid }_{1\times ℱ}=ℱ$.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)