This is a sub-entry of
See there for background and context.
This entry considers equivariant cohomology from the perspective of algebraic geometry.
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the following are rough unpolished notes taken more or less verbatim from some seminar talk – needs attention, meaning: somebody should go through this and polish
The slogan is: for $A$ a cohomology theory, finding equivariant cohomology $A$-theory corresponds to finding group scheme over $A({*})$.
The main example we’ll be looking at here is complex K-theory.
Remark if $A_G$ is an equivariant cohomology theory and if $Y \to X$ is a $G$-principal bundle then we want that $A_G(Y) \simeq A(X)$. So whenever we have a $G$-space where the $G$-action is free enough.
Let $X$ be a $G$-space, form the Borel construction $\mathcal{E}G \times_G X$ with $\mathcal{E}G \to \mathcal{B}G$ the universal principal bundle.
Then we can define
notice that $\mathcal{E}G \times_G X$ is the realization of the action groupoid $X//G$. This Borel equivariant cohomology theory is what is discussed currently at the entry equivariant cohomology. The following will actually define a refinement of the discussion currently at equivariant cohomology.
Problem If the cohomology theory is given by a geometric model, such as topological K-theory in terms of vector bundles or elliptic cohomology potentially in a geometric model for elliptic cohomology, then the above notion of equivariant cohomology need not coincide with the cohomology theory given by the equivariant version of these geometric models. In particular, equivariant vector bundles are geometric cocycles of equivariant K-theory $K_G$ and there is a morphism
but it is not an isomorphism. Instead, $K_G$ is a completion of $K_G^{Bor}$.
Here $K_G(X)$ is the Grothendieck group of equivariant vector bundles over the G-space $X$ (say $G$ is a compact Lie group).
Definition An equivariant vector bundle over $X$ is
a G-space $E$ and $G$-equivariant map $p : E \to X$ such that this is a (complex, here) vector bundle of finite rank
for each $g \in G$ the map $g : E_x \to E_{g x}$ is linear.
Morephism are the obvious $G$-equivariant morphisms of vector bundles.
examples*
$X$ a trivial $G$-space,then a $G$-equivariant vector bundle is a family of complex representations;
for $E \to X$ a vector bundle the $k$th tensor power of $E$ is a $\Sigma_k$-equivariant vector bundle;
if $G$ acts smoothly on $X$ then the complexified tangent bundle $T X \otimes \mathbb{C} \to X$ is a $G$-equivariant vector bundle.
remark The category of $G$-equivariant vector bundle, has
direct sum $\oplus$
tensor product $\otimes$
And we can pull back $Vect^G$ \to $Vecg^H$ along any group homomorphism $\phi : H \to G$
So we are entitled to say
definition the Grothendieck group of $Vect^G(X)$ is
With the remaining tensor product $\otimes$ this yields a commutative ring.
proposition
if $X = pt$ then $K_G(X) \simeq Rep(G)$ is the representation ring of $G$.
if $G$ acts freely on $X$, then $K_G(X) \simeq K(X/G)$.
So in particular
so we get a map
theorem (Atiyah-Segal) This $\alpha$ induces an isomorphism
where
where $I_G = ker(Rep(G) \simeq K_G({*}) \to K_G(\mathcal{E}G) \simeq K(\mathcal{B}G) \stackrel{\epsilon}{\to} \mathbb{Z})$
consider $X = {*}$, $G = S^1$, $\mathbb{C}P^\infty \simeq \mathcal{B}S^1$
since $S^1$ is an abelian group, every irreducible representation is 1-dimensional
$\chi : S^1 \hookrightarrow \mathbb{C}^times$
goal now find an algebraic interpretation of $\alpha$ such that
and
adopt the functor of points perspective
a functor.
For $A \in$ CRing get a spectrum
for $X$ a functor, it is an affine scheme if it is a representable functor in that there is $A$ with $X \simeq Spec A$.
examples
$\mathbf{A}^n(R) := R^n$
$\hat \mathbf{A}^n(R) := Nil(R)^n$
$G_m(R) := R^\times \hookrightarrow \mathbf{A}^1(R)$
$\mathbb{P}^n(R) := R^{n+1}/\sim$, where $\sim$ is multiplication by $R^\times$
proposition
$G_m$ is affine.
proof $A = \mathbb{Z}[x,x^{-1}]$, let $u \in Spec(\mathbf{A}(R))$ a map $u A \to R$, then define
by
Conversely, given $v \in G_m(R) = R^\times$, define
by
with
endofproof
similarly, $\mathbf{A}^n \simeq Spec \mathbb{Z}[x_1, \cdots, x_n]$
Given a functor $X : CRing \to Set$ define the ring of funtions $\mathcal{o}_X$ as
in the functor category.
notice notation: this is global sectins of the structure sheaf, not the structure sheaf itself, properly speaking
we have
so that in particular
definition
Let $X$ be an affine scheme and $Y : CRing \to Set$ a functor with a natural transformation $p : Y \to X$. A system of formal coordinates is a sequence of maps
such that
is an isomorphism. A $Y$ that admits a system of formal coordinates is a formal scheme over $X$.
warning very restrictive definition. See formal scheme
A formal group $G$ over a scheme $X$ is a one-dimensional formal scheme with specified abelian group structure on each fiber $p^{-1}\{x\}$.
This means that there is a natural map
and a natural map $\zeta : X \to G$ which maps $x \mapsto 0 \in p^{-1}\{x\}$.
definition (formal multiplicative group)
define $\hat G_m$ on each $R \in CRing$ by
which is a group under multiplication.
there is an isomorphism of underlying formal schemes
We compute $\mathcal{o}_{\hat \mathbf{A}^1} \simeq \mathcal{o}_{\hat G_m}$ in two ways:
Recall that $\hat \mathbf{A}^1$ can be defined as $Spf \; \mathbb{Z} [t]$, so the global sections of the structure sheaf (which is what we have been calling $\mathcal{o}$) is
We can also see this in the functor of points perspective. Consider the functor $\mathrm{Spec} \; \mathbb{Z} [t]/ (t^n)$, then for any ring $R$ $$\hat \mathbf{A}^1 (R) = \lim_\rightarrow \mathrm{Spec} \; \mathbb{Z} [t]/ (t^n) (R).$$ By the universal property of colimits we have
recall we have a morphism $\alpha : Rep(S^1) \to K(\mathbb{C}P^\infty)$
such that
is the canonical inclusion
exercise
is the natural transformation $R^\times \to R$
and
is the natural transformation
so that we get a map
by sending $x$ to $1 + t$, and this corresponds to taking the germ of functions at $1 \in G_m$
given $G$ an algebraic group such that the formal spectrum $Spf A(\mathbb{C}P^\infty)$ is the completion $\hat G$, define $A_{S^1}({*}) := \mathcal{o}_{G}$ then passing to germs gives a completion map