# Endomorphisms of Equivariant Algebraic $K$-theory

K Arun Kumar and Girja S Tripathi

###### Abstract.

In this paper we study equivariant algebraic $K$-theory for an action of a
finite constant group scheme in the equivariant motivic homotopy theory. We
prove that the equivariant algebraic $K$-theory is represented by an
equivariant ind-scheme defined by Grassmannians. Using this result we show
that in the category of equivariant motivic spaces the set of endomorphisms of
the motivic space defined by $K_{0}(G,-)$ coincides with the set of
endomorphisms of infinite Grassmannians in the equivariant motivic homotopy
category. To this end we explicitly recall the folklore computation of
equivariant $K$-theory of Grassmannians.

##  1\. Introduction

The equivariant motivic homotopy theory has been studied by Krishna, Ostvaer,
Heller, Voinegeau and Hoyois in [2], [3] and [1]. Let $G$ be a finite constant
group scheme over a Noetherian separated base scheme $S$. Denote the category
of $G$-equivariant separated schemes of finite type over $S$ by $Sch^{G}_{S}$
and the subcategory of smooth $S$-schemes by $Sm^{G}_{S}$. Let $M^{G}(S)$
(respectively $M^{G}_{\bullet}(S)$) denote the category of motivic
(respectively pointed motivic) spaces over $S$ and $\mathcal{H}^{G}(S)$
(respectively $\mathcal{H}^{G}_{\bullet}(S)$) the equivariant motivic
(respectively pointed motivic) homotopy category of $G$-schemes over $S$. For
the main results of this paper we need the assumption that the order $|G|$ of
the group be a unit in $\Gamma(S,\ \mathcal{O}_{S})$. To avoid repeating this
we assume throughout the paper that all the schemes we consider have this
property and the order $|G|$ will be denoted by g. The $G$-equivariant
algebraic $K$-theory was defined by Thomason by applying Quillen’s
Q-construction to the exact category of $G$-equivariant vector bundles. In
this paper we use an affine version and define the equivariant algebraic
$K$-theory presheaf $\mathcal{K}\in M^{G}_{\bullet}(S)$ by taking the nerve of
a functorial version of $S^{-1}S$-construction applied to the split exact
categories $\mathscr{S}(G,\ \mathrm{Spec\hskip 1.42262pt}R)$ of
$G$-equivariant vector bundles on affine schemes given by the Spec of rings of
global sections of a $G$-equivariant $S$-schemes- under the assumption that
the order $|G|$ of the group is a unit in $\Gamma(S,\ \mathcal{O}_{S})$) the
categories $\mathscr{S}(G,\ \mathrm{Spec\hskip 1.42262pt}R)$ are split exact.

For the ring $R=\Gamma(S,\ \mathcal{O}_{S})$, we have the regular
representation $\rho_{R}$ of $G$. By the regular representation of $G$ over
$S$, denoted by $\rho_{S}$, we will mean $\pi^{*}\rho_{R}$, where
$\pi:S\rightarrow\mathrm{Spec\hskip 1.42262pt}\ \Gamma(S,\ \mathcal{O}_{S})$
is the natural map. This defines Grassmannian schemes $Gr(m,\hskip
1.42262pt\rho_{S}^{n})$ over $\mathrm{Spec\hskip 1.42262pt}\ \Gamma(S,\
\mathcal{O}_{S})$ parametrizing rank $m$ direct summands in
$\rho_{S}^{n}=\rho_{S}\oplus\cdots\rho_{S}\ (n\ \mathrm{copies})$ and the ind-
schemes $Gr(m,\hskip 1.42262pt\infty)$ and $Gr_{G}$ via colimits induced by
the standard inclusions $\rho_{S}^{n}\hookrightarrow\rho_{S}^{n+1},\
v\mapsto(v,0)$. Following the methods from the paper [8] we prove in Theorems
3.18 and 3.19 that in the equivariant motivic homotopy category the infinite
Grassmannian $\mathbb{Z}\times Gr_{G}$ is a geometric model for equivariant
algebraic $K$-theory and hence describes the equivariant algebraic $K$-groups
in $\mathcal{H}^{G}_{\bullet}(S)$.

######  Theorem 1.

Let $R$ be a commutative connected $G$-equivariant regular noetherian ring of
finite Krull dimension and $G\to\mathrm{Spec\hskip 1.42262pt}R$ a constant
group scheme such that $|G|$ is a unit in $R$. Then there are weak
equivalences of simplicial sets

$\mathscr{G}r(\Delta
R)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{S}^{+}(\Delta
R)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\widetilde{\mathcal{K}}(\Delta
R).$

As a consequence for a regular noetherian base scheme $S$ of finite Krull
dimension and a constant group $S$-scheme $G\to S$ such that $|G|$ is a unit
in $\Gamma(S,\ \mathcal{O}_{S})$ there are isomorphisms

$K_{n}(G,\hskip 1.42262ptX)\simeq[X_{+}\wedge S^{n},\ \mathbb{Z}\times
Gr_{G}]_{\mathcal{H}^{G}_{\bullet}(S)}$

in the pointed equivariant motivic homotopy category for $G$-equivariant
smooth $S$-schemes.

Using the above result we describe in Theorem 4.1 the set of endomorphisms of
the presheaf (on $Sm^{G}_{S}$) defined by equivariant algebraic $K$-groups
$K_{0}(G,\ -)$ in the category $M^{G}(S)$ of motivic spaces. More generally we
prove the following main theorem.

######  Theorem 2.

For a regular base scheme $S$ and $G\to S$ a constant group $S$-scheme such
that $|G|$ is a unit in $\Gamma(S,\ \mathcal{O}_{S})$ there is a canonical
bijection

$[(\mathbb{Z}\times Gr_{G})^{n},\mathbf{R}\Omega^{k}\mathbb{Z}\times
Gr_{G}]_{\mathcal{H}^{G}(S)}\xrightarrow{\sim}Hom_{M^{G}(S)}(K_{0}(G,\
-)^{n},K_{k}(G,\ -))$

for each $n,k\in\mathbb{N}$. In particular,

$[\mathbb{Z}\times Gr_{G},\mathbb{Z}\times
Gr_{G}]_{\mathcal{H}^{G}(S)}\xrightarrow{\sim}End_{M^{G}(S)}(K_{0}(G,\ -))$

when $n=1$ and $k=0$.

This bijection for $K_{0}(G,\ -)$ respects algebraic structures, see Corollary
6.3.

######  Theorem 3.

In the category $\mathcal{H}^{G}(S)$ there exists a unique structure of a
special $\lambda$-ring with duality on $\mathbb{Z}\times Gr_{G}$ such that the
corresponding $\lambda$-ring with duality structure on $K_{0}(G,X)$ is the
standard one for all $X\in Sm^{G}_{S}$.

We prove this by computing the ring structure of $K_{*}(G,\
Gr(m,\rho^{n})^{r})$. First we explicit recall the projective bundle formula
as a $K_{*}(G,\hskip 1.42262ptS)$-algebra isomorphism which allows us to
compute $K_{*}(G,\ Gr(m,\rho^{n}))$. The computation of $K_{*}(G,\
Gr(m,\rho^{n})^{r})$ follows by adapting a technique of Joshua from [4] for
describing the Künneth spectral sequence of linear varieties.

##  2\. Brief recollection of Equivariant motivic homotopy category

For a Noetherian separated scheme $S$ and a finite group scheme $G\rightarrow
S$, the category $M^{G}(S)$ of $G$-equivariant motivic spaces over $S$ is the
category $\Delta^{op}PShv(Sm_{S}^{G})$ of simplicial presheaves of sets on
$Sm^{G}_{S}$. There is a global model structure $M^{G}_{\mathrm{gl}}(S)$ on
$M^{G}(S)$ with global weak equivalences (and global fibrations) given by
schemewise weak equivalence (and schemewise Kan fibrations), and cofibrations
defined by the left lifting property.

The $G$-equivariant local model structure
$M_{\mathrm{Nis}}^{G}(S):=L_{\Sigma_{\mathrm{Nis}}^{\mathrm{hp}}}M^{G}_{\mathrm{gl}}(S)$
is the left Bousfield localization of $M^{G}_{\mathrm{gl}}(S)$ with respect to
the class of morphisms

(2.1) $\Sigma_{\mathrm{Nis}}^{\mathrm{hp}}:=\\{Q^{\mathrm{hp}}\rightarrow
X\\}_{Q}\cup\\{\emptyset\rightarrow h_{\emptyset}\\}$

where $Q$ runs through all the cartesian squares

(2.2) ${B}$${Y}$${A}$${X}$$\scriptstyle{p}$$\scriptstyle{i}$

with $p$ etale and $j$ open immersion such that
$(Y-B)_{\mathrm{red}}\cong(X-A)_{\mathrm{red}}$, and $Q^{\mathrm{hp}}$ is the
homotopy pushout of $Q$ in $M_{\mathrm{gl}}^{G}(S)$. Here $\emptyset$ is the
initial object in the category $\Delta^{op}PShv(Sm_{S}^{G})$ and
$h_{\emptyset}$ is the simplicial presheaf represented by the empty scheme.

The collection of all the cartesian squares (2.2) forms a $cd$-structure which
is complete, regular and bounded. This $cd$-structure induces the equivariant
Nisnevich topologies on $Sch_{S}^{G}$ and $Sm_{S}^{G}$. We will denote the
corresponding sites by $[Sch^{G}_{S}]_{Nis}$ and $[Sm^{G}_{S}]_{Nis}$
respectively. It has been shown in [2, Proposition 2.17] or [3, Theorem 3.14]
that these sites have enough points. We particularly note the following [3,
Theorem 3.14].

######  Theorem 2.1.

Let $G$ be a finite group scheme over $S$. A map
$\phi:\mathcal{F}_{1}\rightarrow\mathcal{F}_{2}$ of sheaves of sets on
$[Sch^{G}_{S}]_{Nis}$ (respectively $[Sm^{G}_{S}]_{Nis}$) is an isomorphism if
and only if $\phi_{R}:\mathcal{F}_{1}(\mathrm{Spec\hskip
1.42262pt}R)\rightarrow\mathcal{F}_{2}(\mathrm{Spec\hskip 1.42262pt}R)$ is an
isomorphism for all (respectively all smooth) semilocal affine $G$-schemes
$\mathrm{Spec\hskip 1.42262pt}R$ over $S$ with a single closed orbit.

######  Remark 2.2.

This theorem enables us to recognise Nisnevich equivariant simplicial weak
equivalences in the Nisnevich local model structure. This is a reason for our
choice of the simpler affine version of equivariant $K$-theory presheaf in
3.4. See remark 3.6. This fact has also been used in the proof of the Theorem
3.19, which in turn relies on Theorem 3.18.

The $G$-equivariant motivic model structure
$M^{G}_{\mathrm{mot}}(S):=L_{\Sigma_{\mathbb{A}^{1}}}M^{G}_{\mathrm{Nis}}(S)$
is the left Bousfield localizaiton of $M^{G}_{\mathrm{Nis}}(S)$ with respect
to the class of all the projections

$\Sigma_{\mathbb{A}^{1}}:=\\{X\times\mathbb{A}^{1}\rightarrow X:X\in
Sm_{S}^{G}\\}.$

The $G$-equivariant motivic homotopy category $\mathcal{H}^{G}(S)$ is the
homotopy category associated to the equivalent motivic model structure
$M^{G}_{\mathrm{mot}}(S)$. The pointed $G$-equivariant motivic homotopy
category $\mathcal{H}^{G}_{\bullet}(S)$ is obtained by a similar procedure
from the category of pointed simplicial presheaves
$\Delta^{op}PShv_{\bullet}(Sm_{S}^{G})$.

##  3\. Equivariant algebraic $K$-theory and Grassmannians

For a scheme $X\in Sm^{G}_{S}$ with $G$-action the category $VB(G,\hskip
1.42262ptX)$ of $G$-vector bundles over $X$ is an exact category. The
equivariant algebraic $K$-theory $K_{i}(G,\hskip 1.42262ptX)$ of a $G$-scheme
$X$ is defined as the $i^{\mathrm{th}}$-homotopy group of the nerve of
Quillen’s $Q$-construction applied to the exact category $VB(G,\hskip
1.42262ptX)$. Thus,

$K_{i}(G,\hskip 1.42262ptX)=\pi_{i}(\mathscr{N}Q(VB(G,\hskip 1.42262ptX)),0),\
\ \ \ i\geq 0.$

The category $VB(G,\hskip 1.42262ptX)$ is a symmetric monoidal category with
respect to direct sum of vector bundles, and the category
$\mathscr{S}(G,\hskip 1.42262ptX)=\mathrm{Isom}(VB(G,\hskip 1.42262ptX))$
having the same objects as $VB(G,\hskip 1.42262ptX)$ but morphisms only the
isomorphisms is also a symmetric monoidal category with the induced direct sum
operation. We will now show that for an affine $G$-equivariant scheme this
category is split exact and hence we can use the $S^{-1}S$-construction to
define equivariant algebraic $K$-theory for affine $G$-schemes.

######  Lemma 3.1.

For an affine $G$-scheme $X$, under the assumption that $|G|$ is invertible in
$\Gamma(S,\hskip 1.42262pt\mathcal{O}_{S})$, every short exact sequence of
equivariant vector bundles splits equivariantly.

###### Proof.

Given a short exact sequence
$0\rightarrow\mathcal{E}\xrightarrow{i}\mathcal{L}\xrightarrow{\phi}\mathcal{M}\rightarrow
0$ choose a (non-equivariant) splitting $s:\mathcal{M}\rightarrow\mathcal{L}$
and observe that the morphism

$\bar{s}:\mathcal{M}\rightarrow\mathcal{L},\ \ \
a\mapsto\frac{1}{|G|}\sum_{g\in G}g\cdot s(g^{-1}\cdot a)\ \ \ \ \mathrm{for}\
a\in\mathcal{M}(X),\ \ X\in Sm^{G}_{S}$

of vector bundles is a $G$-equivariant splitting of $\phi$. ∎

Thus, for an affine $G$-scheme $X=\mathrm{Spec}\ R$ such that $|G|$ is a unit
in $R$ the category $VB(G,\hskip 1.42262ptX)$ is split exact. If necessary, to
emphasis the $G$-action $\tau$ on a scheme $\mathrm{Spec\hskip 1.42262pt}R$ or
($X$ in general) we will use notations like $(\mathrm{Spec\hskip
1.42262pt}R,\tau)$ or $R_{\tau}$ (or $(X,\hskip 1.42262pt\tau)$ or
$X_{\tau}$), otherwise no mention of the action $\tau$ will be made.

Let us denote the category $\mathscr{S}(G,\hskip 1.42262pt\mathrm{Spec}\ R)$
by $\mathscr{S}_{R}$. Recall that objects in the category
$\mathscr{S}_{R}^{-1}\mathscr{S}_{R}$ are pairs $(A_{0},\hskip
1.42262ptA_{1})$ of objects in the category $\mathscr{S}_{R}$; and, a moprhism
$(A_{0},\hskip 1.42262ptA_{1})\rightarrow(B_{0},\hskip 1.42262ptB_{1})$
consists of an equivalence class of triples $(C,a_{0},\hskip 1.42262pta_{1})$
where $C\in\mathscr{S}_{R}$ and $a_{i}:A_{i}\oplus C\rightarrow B_{i},\ \
i=0,\hskip 1.42262pt1$, are isomorphisms in $\mathscr{S}_{R}$. In what follows
we will describe a small category $\mathscr{S}_{R}^{+}$ equivalent to the
group completion $\mathscr{S}_{R}^{-1}\mathscr{S}_{R}$ and take equivariant
algebraic $K$-theory presheaf as the resulting presheaf of simplicial sets on
$Sm^{G}_{S}$ by taking the nerve of $\mathscr{S}_{R}^{+}$.

######  Definition 3.2 (Regular representation of $G$ over a scheme $S$).

For an equivariant affine $G$-scheme $(\mathrm{Spec\hskip 1.42262pt}R,\tau)$,
let $\rho_{{}_{R_{\tau}}}$ (or simply $\rho_{{}_{R}}$) be the skew group-ring
$R_{\tau}G$ viewed as the twisted regular representation of $G$ over $R$. The
$G$-action on $\rho_{{}_{R_{\tau}}}$ is given by

$g^{\prime}\cdot(\Sigma_{g\in G}x_{g}\cdot e_{g})=\Sigma_{g\in
G}\tau_{g^{\prime}}(x_{g})\cdot e_{g^{\prime}g},$

where $\\{e_{g}:g\in G\\}$ is the standard basis of $\rho_{{}_{R_{\tau}}}$.
For the base scheme $S$ by the regular representation $\rho_{S}$ of $G$ over
$S$ we will mean $\pi^{*}\rho_{{}_{R}}$ the pullbackof $\rho_{{}_{R}}$ via the
natural map $\pi:S\rightarrow\mathrm{Spec\hskip 1.42262pt}\ \Gamma\hskip
1.42262pt(S,\hskip 1.42262pt\mathcal{O}_{S})$, where $R=\Gamma\hskip
1.42262pt(S,\hskip 1.42262pt\mathcal{O}_{S})$. We have direct sums
$\rho_{S}^{n}$ and embeddings $\rho_{S}^{n}\hookrightarrow\rho_{S}^{n+1},\
v\mapsto(v,0)$ of regular representations. Let $\rho_{S}^{\infty}$ be the
increasing union of these embeddings as a $G$-equivariant quasi-coherent
$\mathcal{O}_{S}$-module.

######  Lemma 3.3.

For an affine $G$-scheme $\mathrm{Spec\hskip 1.42262pt}R$, under the
assumption that $|G|$ is invertible in $R$, every $G$-equivariant finitely
generated projective $R$-module is a summand in a direct sum of regular
representations.

###### Proof.

Let $\tau:G\rightarrow\mathrm{Aut}_{Sch_{S}}(\mathrm{Spec\hskip 1.42262pt}R)$
denote the $G$-action on $\mathrm{Spec\hskip 1.42262pt}R$. Given an
equivariant finitely generated projective $R$-module $P$ choose an epimorphism
$\alpha:R^{n}\rightarrow P$. The free rank 1 module $R$ is a $G$-equivariant
summand in $\rho_{{}_{R_{\tau}}}$ via the inclusion

$R\longrightarrow\rho_{{}_{R_{\tau}}},\ \ \ x\mapsto\sum_{g\in G}\tau(x)e_{g}$

and hence $R^{n}\hookrightarrow
R^{n}\otimes_{R}\rho_{{}_{R_{\tau}}}=\rho_{{}_{R_{\tau}}}^{n}$ is a
$G$-equivariant direct summand. The induced surjective map

$\widetilde{\alpha}:\rho_{{}_{R_{\tau}}}^{n}\rightarrow P\ \ \ \ \ (x,\
\sum_{g\in G}x_{g}\cdot e_{g})\mapsto\sum_{g\in G}x_{g}\cdot\alpha(x)$

is $G$-equivariant and we have an exact sequence of $G$-equivariant finitely
generated projective $R$-modules $\mathrm{ker}\hskip
1.42262pt\widetilde{\alpha}\xrightarrow{\ \ \
}\rho_{{}_{R_{\tau}}}^{n}\xrightarrow{\ \ \widetilde{\alpha}\ \ }P.$ Choosing
an equivariant splitting by Lemma 3.1 we see that $P$ is a direct summand of
$\rho_{{R_{\tau}}}^{n}$. ∎

######  Definition 3.4 (Equivariant algebraic $K$-theory presheaf
$\mathcal{K}$).

Let $\mathscr{S}_{R}^{+}\subset\mathscr{S}_{R}^{-1}\mathscr{S}_{R}$ be the
full subcategory of pairs $(A,B)$ where
$A\subset\rho_{R}^{\infty}\oplus\rho_{R}^{\infty}=(\rho_{R}^{\infty})^{2}$ and
$B\subset\rho_{R}^{\infty}\oplus\rho_{R}^{\infty}\oplus\rho_{R}^{\infty}=(\rho_{R}^{\infty})^{3}$
are $G$-invariant finitely generated projective submodules of
$(\rho_{R}^{\infty})^{2}$ and $(\rho_{R}^{\infty})^{3}$, respectively. The
category $\mathscr{S}_{R}^{+}$ is small, functorial in $R$, and by Lemma 3.3
it is equivalent to $\mathscr{S}_{R}^{-1}\mathscr{S}_{R}$. The equivariant
algebraic $K$-theory $\mathcal{K}\in\Delta^{op}PShv(Sm^{G}_{S})$ is defined by

$\mathcal{K}(X)=\mathscr{N}(\mathscr{S}_{R}^{+}),\ \ \ \ \mathrm{where}\
R=\Gamma(X,\ \mathcal{O}_{X})$

for $X\in Sm^{G}_{S}$. For an affine smooth $G$-scheme $X=\mathrm{Spec}\ R$,
it’s equivariant algebraic $K$-groups are the homotopy groups of the
simplicial set $\mathcal{K}(X)$ at the basepoint $0$. Thus,

$K_{i}(G,\hskip 1.42262ptR)=\pi_{i}(\mathcal{K}(\mathrm{Spec}\ R),0)\ \ \ \
i\geq 0.$

######  Remark 3.5.

It is immediate to see that for an algebraic group $G$ over a field $F$ the
group $K_{0}(G,\hskip 1.42262ptF)$, the Grothendieck group of the monoid of
isomorphism classes of representations of $G$ over $F$, is the representation
ring of $G$ over the field $F$.

######  Remark 3.6.

For a $G$-scheme $X$, not necessarily affine, its equivariant algebraic
$K$-groups $K_{i}(G,\hskip 1.42262ptX)$ have been defined by Thomason as
Quillen’s algebraic $K$-groups of the exact category $VB(G,\hskip 1.42262ptX)$
of $G$-vector bundles over $X$, i.e., as the homotopy groups of
$\mathscr{N}Q(VB(G,\hskip 1.42262ptX))$. To conform with this equivariant
algebraic $K$-theory as a presheaf on $Sm^{G}_{S}$, say $\mathcal{K}^{G}$,
should be defined by

$\mathcal{K}^{G}(X)=\mathscr{N}Q(VB(G,\hskip 1.42262ptX)),\ \ \forall X\in
Sm^{G}_{S}.$

The equivariant algebraic $K$-theory presheaf $\mathcal{K}$ defined here in
(3.4) via the categorical group completion
$\mathscr{S}_{R}^{+}\subset\mathscr{S}_{R}^{-1}\mathscr{S}_{R}$ corresponding
to the ring of global sections for a $G$-equivariant scheme in $Sm^{G}_{S}$
suffices for our purpose of proving the representability result in this
section: There is a natural transformation
$\mathcal{K}\rightarrow\mathcal{K}^{G}$ defined by the map of schemes
$X\rightarrow\mathrm{Spec\hskip 1.42262pt}(\Gamma(X,\hskip
1.42262pt\mathcal{O}_{X}))$ for a given scheme $X\in Sm^{G}_{S}$. This
transformation defines a global weak equivalence for affine schemes and hence
an equivariant Nisnevich and $\mathbb{A}^{1}$-weak equivalence as well by
Theorem 2.1.

######  Definition 3.7 (Reduced equivariant algebraic $K$-theory).

For a ring $R$ consider the full subcategory

$\widetilde{\mathscr{S}_{R}^{+}}\subset\mathscr{S}_{R}^{+}$

of objects $(A,B)\in\mathscr{S}_{R}^{+}$ such that
$A\subset\rho_{R}^{\infty}=0\oplus\rho_{R}^{\infty}\subset(\rho_{R}^{\infty})^{2}$,
$B\subset A\oplus\rho_{R}^{\infty}\subset
0\oplus(\rho_{R}^{\infty})^{2}\subset(\rho_{R}^{\infty})^{3}$ and $A,B$ have
the same constant rank. The reduced equivariant algebraic $K$-theory presheaf
$\widetilde{\mathcal{K}}$ is defined by

$\widetilde{\mathcal{K}}(X)=\mathscr{N}(\widetilde{\mathscr{S}_{R}^{+}}),\ \ \
\ \mathrm{where}\ X\in Sm^{G}_{S}\ \mathrm{and}\ R=\Gamma(X,\
\mathcal{O}_{X}).$

For an affine smooth $G$-scheme $X=\mathrm{Spec\hskip 1.42262pt}\hskip
1.42262ptR$, its reduced equivariant algebraic $K$-groups
$\widetilde{K}_{i}(G,\hskip 1.42262ptR)$ are the homotopy groups of
$\widetilde{\mathcal{K}}(X)$ with respect to the base point $0$. Thus,

$\widetilde{K}_{i}(G,\hskip
1.42262ptR)=\pi_{i}(\widetilde{\mathcal{K}}(\mathrm{Spec}\ R),0)\ \ \ \ i\geq
0.$

###  3.1. Reduced and unreduced equivariant algebraic $K$-theories

Consider the monoid $\mathbb{N}$ of non-negative natural numbers as a discrete
symmetric monoidal category with respect to addition. A ring $R$ is a
$G$-invariant summand of $\rho_{R}$ via

$R\subset\rho_{R},\ \ \ x\mapsto\sum_{g\in G}x\cdot e_{g},\ \ \ x\in R,$

and, it defines a $G$-invariant submodule $R^{n}\subset\rho^{n}_{R}.$ There is
a map of presheaves from the categorical group completion of $\mathbb{N}$ to
$\mathcal{K}$:

$\mathbb{N}^{-1}\mathbb{N}\longrightarrow\mathcal{K}:(n,\hskip
1.42262ptm)\mapsto(R^{n},\hskip 1.42262ptR^{m})$

where the factor $R^{n}$ embeds in $\rho^{n}_{R}\oplus
0\subset(\rho^{\infty}_{R})^{2}$ and the factor $R^{m}$ embeds in
$\rho^{m}_{R}\oplus 0\oplus 0\subset(\rho^{\infty}_{R})^{3}.$ The full
subcategory $\widetilde{\mathscr{S}_{R}^{+}}\subset\mathscr{S}_{R}^{+}$
defines a map of presheaves

(3.1)
$\mathbb{N}^{-1}\mathbb{N}\times\widetilde{\mathcal{K}}\longrightarrow\mathcal{K}:((n,\hskip
1.42262ptm),(A,\hskip 1.42262ptB))\mapsto(R^{n}\oplus A,\hskip
1.42262ptR^{m}\oplus B).$

######  Lemma 3.8.

The above map 3.1 is an isomorphism in the motivic homotopy category

$\mathbb{Z}\times\widetilde{\mathcal{K}}\xrightarrow{\ \sim\ }\mathcal{K}.$

###### Proof.

Similar arguments as in [8, Lemma 4.4] apply here as well. ∎

###  3.2. Grassmannians indexed over representations

In 3.2 we have defined the representations $\rho^{n}_{S}$ and the quasi-
coherent $G$-equivariant $\mathcal{O}_{S}$-module $\rho^{\infty}_{S}$. Now we
will define a presheaf of categories on $Sm^{G}_{S}$ using these
representations useful in relating $K$-theory presheaf with Grassmannians.

######  Definition 3.9 (Presheaf of categories $\varrho$).

For a scheme $X\in Sm^{G}_{S}$ let $R=\Gamma(X,\hskip
1.42262pt\mathcal{O}_{X})$ be the ring of global sections. Define a category
$\varrho_{X}$ as follows. Its objects are finitely generated $G$-equivariant
projective $R$-submodules $V\subset\rho^{\infty}_{R}$ having constant rank;
and, for objects $V,\hskip 1.42262ptV^{\prime}\subset\rho^{\infty}_{R}$ a
morphism from $V$ to $V^{\prime}$ exists only when $V\subset V^{\prime}$ is a
subbundle and in that case a morphism is a choice of $G$-equivariant section
of the quotient map $V^{\prime}\rightarrow V^{\prime}/V$. With respect to
pullbacks of sheaves under maps of schemes, $X\mapsto\varrho_{X}$ defines a
presheaf of filtered categories. For a morphism $V\rightarrow V^{\prime}$ in
$\varrho_{X}$, i.e., an inclusion $V\subset V^{\prime}$ together with a choice
of section $V^{\prime}/V\hookrightarrow V^{\prime}$, let us denote the
$G$-equivariant quotient $V^{\prime}/V$ without any mention of the section:
Here the choice of section is needed to provide a functorial composition in
$\varrho_{X}$.

######  Remark 3.10.

For each $X\in Sm^{G}_{S}$ the subcategory consisting of the set
$\\{\rho^{n}_{R}|\ n\geq 0\\}$ as the set of objects and morphisms generated
by $\rho_{R}^{n}\hookrightarrow\rho_{R}^{n+1}=\rho_{R}\oplus\rho^{n}_{R},\
v\mapsto(0,v)$ is a cofinal subcategory in view of lemma 3.3.

######  Definition 3.11 (Grassmannians of summands in regular
representations).

Given integers $m\geq 0,\ n\geq 1,$ we have the scheme $Gr(m,\hskip
1.42262pt\rho_{S}^{n})$ parametrizing the direct summands of rank $m$ of the
direct sum $\rho_{S}^{n}$ of regular representation $\rho_{S}$ of $G$ over
$S$. Recall that for better readability we have denoted the order of the group
$G$ by g. The scheme $Gr(m,\hskip 1.42262pt\rho_{S}^{n})$ is the usual
Grassmannian $Gr(m,\hskip 1.42262ptn\cdot\textsl{g})$ over $S$ and it is a
$G$-equivariant smooth $S$-scheme. The embeddings
$\rho_{S}^{n}\hookrightarrow\rho_{S}^{n+1},\ v\mapsto(v,0)$ define a map of
Grassmannians $Gr(m,\hskip 1.42262pt\rho_{S}^{n})\hookrightarrow Gr(m,\hskip
1.42262pt\rho_{S}^{n+1})$, and one of the versions of $G$-equivariant infinite
Grassmannians is the ind-scheme

(3.2) $Gr(m,\hskip 1.42262pt\infty)=Gr(m,\hskip
1.42262pt\rho_{S}^{\infty})=\operatorname*{colim}_{n\geq 1}\ [Gr(m,\hskip
1.42262pt\rho_{S}^{n})\hookrightarrow Gr(m,\hskip 1.42262pt\rho_{S}^{n+1})].$

The $G$-invariant subspace

$\mathcal{O}_{S}\hookrightarrow\rho_{S},\ \ \ x\mapsto\sum_{g\in G}x\cdot
e_{g}$

defines an embedding
$\mathcal{O}_{S}\oplus\rho_{S}^{n}\hookrightarrow\rho_{S}\oplus\rho_{S}^{n}$
of $G$-equivariant $\mathcal{O}_{S}$-modules, and in turn, maps of $G$-schemes
$Gr(m,\hskip 1.42262pt\rho_{S}^{n})\hookrightarrow Gr(m+1,\hskip
1.42262pt\rho_{S}\oplus\rho_{S}^{n})=Gr(m+1,\hskip 1.42262pt\rho_{S}^{n+1})$
and ind-schemes $Gr(m,\hskip 1.42262pt\rho_{S}^{\infty})\hookrightarrow
Gr(m+1,\hskip 1.42262pt\rho_{S}\oplus\rho_{S}^{\infty})$. Another version of
infinite Grassmannians $Gr_{G}$ is the ind-scheme

(3.3) $Gr_{G}=Gr(\infty,\hskip 1.42262pt\infty)=\operatorname*{colim}\
[Gr(m,\hskip 1.42262pt\rho_{S}^{\infty})\hookrightarrow Gr(m+1,\hskip
1.42262pt\rho_{S}\oplus\rho_{S}^{\infty})\hookrightarrow\cdots].$

We will use the same notations $Gr(m,\hskip 1.42262pt\rho_{S}^{n}),\
Gr(m,\hskip 1.42262pt\rho_{S}^{\infty})$ and $Gr_{G}$ for the corresponding
representable simplicial presheaves as motivic spaces defined by Grassmannians
via the Yoneda embedding.

######  Remark 3.12 (Better behaved version of infinite Grassmannian
presheaf).

More generally one can define Grassmannians $Gr(m,\hskip
1.42262pt\mathcal{F})$ parametrizing local direct summands of rank $m$ in a
representation of $G$ on a quasi-coherent $\mathcal{O}_{S}$-module
$\mathcal{F}$. In particular, given an object $V\subset\rho^{\infty}_{R}$ in
$\varrho_{S}$ having the constant rank $|V|$, we have the Grassmannian
$Gr(|V|,\hskip 1.42262ptV\oplus\rho^{\infty}_{R})$ as in 3.2, and for a
morphism $V\rightarrow V^{\prime}$ (with a chosen section
$V^{\prime}/V\rightarrow V^{\prime}$) in $\varrho_{S}$ a map of Grassmannians
$Gr(|V|,\hskip 1.42262ptV\oplus\rho_{R}^{\infty})\to Gr(|V^{\prime}|,\hskip
1.42262ptV^{\prime}\oplus\rho_{R}^{\infty})\ :\ E\mapsto(V^{\prime}/V)\oplus
E.$ The colimit $\operatorname*{colim}_{V\in\varrho_{S}}Gr(|V|,\hskip
1.42262ptV\oplus\rho^{\infty}_{R})$ can also be considered as a version of
infinite Grassmannian. In view of remark 3.10 we can identify this colimit
with $Gr_{G}$ defined in 3.3 and we will not make any distinction between
these two. Thus,

$Gr_{G}=\operatorname*{colim}_{V\in\varrho_{S}}Gr(|V|,\hskip
1.42262ptV\oplus\rho^{\infty}_{R})=Gr(\infty,\hskip 1.42262pt\infty).$

Following these considerations we have yet another version of infinite
Grassmannians by considering the inclusions
$\rho_{R}^{n}\hookrightarrow\rho_{R}^{n+1}=\rho_{R}\oplus\rho^{n}_{R},\
v\mapsto(0,v)$ in 3.10 and taking the colimit

$Gr_{G}=\operatorname*{colim}\ [Gr(m,\hskip
1.42262pt\rho_{S}^{\infty})\hookrightarrow Gr(m+\textsl{g},\hskip
1.42262pt\rho_{S}\oplus\rho_{S}^{\infty})\hookrightarrow\cdots$

of induced maps.

###  3.3. Grassmannians as equivariant $K$-theory in motivic homotopy category

We will now describe a presheaf $\mathcal{S}$ of categories on $Sm^{G}_{S}$
which will provide a categorical version of group completion used to define
the equivariant algebraic $K$-theory in 3.4, and enable us to see the
Grassmannian ind-scheme $Gr_{G}$ considered in 3.12 as a geometric model of
equivariant algebraic $K$-theory.

######  Definition 3.13 (Presheaves $\mathcal{S}(\mathcal{F})$ of categories).

Let $X$ be an equivariant scheme in $Sm^{G}_{S}$ with $R=\Gamma(X,\hskip
1.42262pt\mathcal{O}_{X})$, and $\mathcal{F}$ a representation of $G$ on a
quasi-coherent $\mathcal{O}_{S}$-module. Then $p^{*}\mathcal{F}$ is a a quasi-
coherent sheaf on $\mathrm{Spec\hskip 1.42262pt}\Gamma(X,\hskip
1.42262pt\mathcal{O}_{X})$, where $p:\mathrm{Spec\hskip
1.42262pt}\Gamma(X,\hskip 1.42262pt\mathcal{O}_{X})\rightarrow S$ is the
induced morphism of schemes. Denote by $\mathcal{S}(\mathcal{F})(X)$ the
category having as objects all $G$-equivariant locally free submodules
$E\subset p^{*}\mathcal{F}$ and the set of morphism $E\rightarrow E^{\prime}$
being the set of $G$-equivariant isomorphisms (not needed to be compatible
with embeddings into $p^{*}\mathcal{F}$). Via pullback of sheaves of modules
the assignment $X\mapsto\mathcal{S}(\mathcal{F})(X)$ defines a presheaf of
categories $\mathcal{S}(\mathcal{F})$. For a nonnegative integer $d$ denote
the subpresheaf of $\mathcal{S}(\mathcal{F})$ defined by full subcategories
consisting of submodules of constant rank $d$ by $\mathcal{S}(d,\hskip
1.42262pt\mathcal{F})$. In particular, for $G$-equivariant locally free
submodule $V\subset\rho^{\infty}_{S}$, we have presheaf
$\mathcal{S}(V\oplus\rho^{\infty}_{S})$ and subpresheaves of categories
$\mathcal{S}(|V|,\
V\oplus\rho^{\infty}_{S})\subset\mathcal{S}(V\oplus\rho^{\infty}_{S})$.

######  Definition 3.14 (Group completion of presheaf
$\mathcal{S}(\rho^{\infty}_{S})$ of categories).

Set

$\mathcal{S}^{+}=\operatorname*{colim}_{V\subset\rho^{\infty}_{S}}\mathcal{S}(|V|,V\oplus\rho^{\infty}_{S})$

where for $V\subset V^{\prime}$, the transition map $\mathcal{S}(|V|,\hskip
1.42262ptV\oplus\rho^{\infty}_{S})\to\mathcal{S}(|V^{\prime}|,\hskip
1.42262ptV^{\prime}\oplus\rho^{\infty}_{S})$ is defined by
$E\mapsto(V^{\prime}/V)\oplus E$ on objects and by $g\mapsto
1_{V^{\prime}-V}\oplus g$ on morphisms. The set of objects of
$\mathcal{S}(\mathcal{F})(X)$ is $Gr(\mathcal{F}_{X})$ and the set of objects
$\mathcal{S}(d,\hskip 1.42262pt\mathcal{F})(X)$ is $Gr(d,\hskip
1.42262pt\mathcal{F}_{X})$. Inclusion of zero-simplices defines a map of
simplicial presheaves

$Gr(|V|,\
V\oplus\rho_{S}^{\infty})\to\mathcal{S}(|V|,V\oplus\rho^{\infty}_{S}),$

and passing to the colimit over the index category $\varrho$ defines a map
simplicial presheaves

(3.4) $Gr_{G}\to\mathcal{S}^{+}$

from the representable presheaf of ind-Grassmannians $Gr_{G}$ into the group
completion $\mathcal{S}^{+}$.

The presheaf of categories $\varrho$ defined in 3.9 allows us to view infinite
Grassmannian described in 3.12 and the group completion in 3.14 as colimits
over the same filtering category. Now we will introduce the version of
$\varrho$ useful in contructing these homotopy colimits.

######  Definition 3.15 (Category $\varrho^{h}$ for homotopy colimit
description).

Define the category $\varrho_{S}^{h}$ as follows. Its objects are
$G$-equivariant locally free $\mathcal{O}_{S}$-submodules
$V\subset\rho^{\infty}_{S}$ having constant rank; and, for objects $V,\hskip
1.42262ptV^{\prime}\subset\rho^{\infty}_{S}$ the set of morphisms from $V$ to
$V^{\prime}$ is the set of $G$-equivariant inclusions $V\rightarrow
V^{\prime}$ together with a section $V^{\prime}/V\rightarrow V^{\prime}$, and
these inclusions are not required to commute with embeddings into
$\rho^{\infty}_{S}$. Here composition is composition of embeddings. With
respect to pullbacks of sheaves under maps of schemes
$X\mapsto\varrho_{\mathrm{Spec\hskip 1.42262pt}R}^{h}$ where $R=\Gamma(X,\
\mathcal{O}_{X})$ defines the presheaf of categories $\varrho^{h}$.

######  Remark 3.16.

The category $\varrho_{S}^{h}$ is the analog of the category
$\langle\mathcal{S}(\rho^{\infty}_{S})(S),\mathcal{S}(\rho^{\infty}_{S})(S)\rangle$
in [7], and the category $\varrho_{S}$ defined in 3.9 is naturally a
subcategory of $\varrho_{S}^{h}$ both having the same set of objects but
morphisms in $\varrho_{S}$ required to be compatible with embeddings in
$\rho_{S}^{\infty}$.

###  3.4. Equivariant algebraic $K$-theory as a homotopy colimit

Using group completion presheaves defined in 3.13 now we define a functor from
$\varrho^{h}$ into the category of small categories

$\mathcal{S}(-,\
-\oplus\rho^{\infty}_{S}):\varrho^{h}\rightarrow\operatorname{Cat},\ \ \
V\mapsto\mathcal{S}(|V|,\ V\oplus\rho^{\infty}_{S})$

and a $G$-equivariant embedding $\alpha:V\to V^{\prime}$ being sent to the
functor

$\begin{array}[]{ll}\mathcal{S}(|V|,\
V\oplus\rho^{\infty}_{S})\to\mathcal{S}(|V^{\prime}|,\
V^{\prime}\oplus\rho^{\infty}_{S}):&E\mapsto(V^{\prime}/V)\oplus\
\widetilde{\alpha}(E)\\\ &e\mapsto
1_{(V^{\prime}/V)}\oplus\widetilde{\alpha}e\end{array}$

where $\widetilde{\alpha}=\alpha\oplus
1_{\rho_{S}^{\infty}}:V\oplus\rho_{S}^{\infty}\to
V^{\prime}\oplus\rho_{S}^{\infty}$. Following [9] we now have an equality of
categories

$\widetilde{\mathcal{K}}=\operatorname*{hocolim}_{V\in\varrho^{h}}\mathcal{S}(|V|,\
V\oplus\rho^{\infty}_{S})$

and hence a homotopy colimit description of reduced equivariant algebraic
$K$-theory presheaf. Replacing $\mathcal{S}(|V|,\ V\oplus\rho^{\infty}_{S})$
with the category $\mathcal{S}(V\oplus\rho^{\infty}_{S})$ of $G$-equivariant
locally free submodules of constant rank in $V\oplus\rho^{\infty}_{S}$ and
taking the homotopy colimit as above defines a model for the equivariant
algebraic $K$-theory presheaf $\widetilde{K}$.

We similarly define a functor $Gr(-,\
-\oplus\rho_{S}^{\infty}):\varrho^{h}\rightarrow\operatorname{Cat}$.

######  Definition 3.17.

In subsection 3.4 we have described (reduced) equivariant algebraic $K$-theory
a homotopy colimit. To relate it with Grassmannians we will now express the
presheaves $Gr_{G}$ and $\mathcal{S}^{+}$ also as homotopy colimits.

We have defined $Gr_{G}$ and $\mathcal{S}^{+}$ as the colimits of presheaves
of sets $Gr(|V|,V\oplus\rho_{S}^{\infty})$ and of categories
$\mathcal{S}(|V|,\ V\oplus\rho_{S}^{\infty})$ over the category $\varrho$.
Considering presheaves of sets as presheaves of (discrete) categories and
presheaves of categories as simplicial sets (via the nerve functor) and thus
sets as (constant) simplicial sets, we thus have presheaves of categories and
the corresponding homotopy colimits $\mathscr{G}r$ and $\mathscr{S}^{+}$
defined as

$\mathscr{G}r\ =\
\operatorname*{hocolim}_{V\subset\rho_{S}^{\infty}}Gr(|V|,V\oplus\rho_{S}^{\infty})$
$\mathscr{S}^{+}\ =\
\operatorname*{hocolim}_{V\subset\rho_{S}^{\infty}}\mathcal{S}(|V|,V\oplus\rho_{S}^{\infty}).$

Since the categories in $\varrho$ are filtered the natural maps from homotopy
colimit to colimit are weak equivalences

(3.5) $\mathscr{G}r\stackrel{{\scriptstyle\sim}}{{\to}}Gr_{G},\hskip
12.91663pt\mathscr{S}^{+}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{S}^{+}$

of presheaves of simplicial sets.

In definition 3.14 and 3.4 note that the set of objects in presheaves of
categories $\mathcal{S}(V\oplus\rho^{\infty}_{S})$ coincides with the presheaf
of Grassmannians in 3.12. Viewing the presheaf $Gr_{G}$ as a presheaf of
discrete categories we have a natural transformation of functors

$Gr(-,\ -\oplus\rho_{S}^{\infty})\to\mathcal{S}(-,\oplus\rho^{\infty}_{S})$

as inclusion of zero-simplices $Gr(|V|,\
V\oplus\rho_{S}^{\infty})\to\mathcal{S}(|V|,V\oplus\rho^{\infty}_{S})$. This
gives us a map of presheaves of categories on homotopy colimit

(3.6) $\mathscr{G}r\to\mathscr{S}^{+}.$

The inclusion of subcategories $\varrho\subset\varrho^{h}$ defines a functor

$\operatorname*{hocolim}_{V\subset\rho_{S}^{\infty}}\mathscr{S}(|V|,V\oplus\rho_{S}^{\infty})\longrightarrow\operatorname*{hocolim}_{V\in\varrho^{h}}\mathcal{S}(|V|,\
V\oplus\rho^{\infty}_{S})$

and hence by definition a map of presheaves

(3.7) $\mathscr{S}^{+}\to\widetilde{\mathcal{K}}.$

Let $\Delta R$ be the $G$-equivariant simplicial ring
$n\mapsto\Delta^{n}R=R[T_{0},...,T_{n}]/\langle T_{0}+\cdots+T_{n}-1\rangle$
for the trivial $G$-action.

######  Theorem 3.18.

Let $R$ be a commutative connected $G$-equivariant regular noetherian ring in
which $|G|$ is a unit. Then the maps (3.6) and (3.7) induce weak equivalences
of simplicial sets

$\mathscr{G}r(\Delta
R)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\mathscr{S}^{+}(\Delta
R)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\widetilde{\mathcal{K}}(\Delta
R).$

###### Proof.

To prove this theorem the general machinery developed in [8, Sections 6 and 7]
for the proof of representability of Grothendieck-Witt theory by orthogonal
Grassmannian can be applied in this case as well. The details are left out to
the reader. It should be noted that under the assumption that $|G|$ is a unit
is the base scheme, to show the equivalence $\mathscr{S}^{+}(\Delta
R)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\widetilde{\mathcal{K}}(\Delta
R)$ we need the fact that very $G$-equivariant projective $R$-module is a
summand of a direct sum of regular representations. ∎

An important consequence of the theorem 3.18 is the analogous description
well-known for algebraic $K$-theory in motivic homotopy category.

######  Theorem 3.19.

Let $S$ be a regular scheme such that $\frac{1}{|G|}\in\Gamma(S,\
\mathcal{O}_{S})$. Let $X\in Sm^{G}_{S}$ be a $G$-equivariant smooth scheme
over $S$. Then, there are isomorphisms

$K_{n}(G,\hskip 1.42262ptX)\simeq[X_{+}\wedge S^{n},\mathbb{Z}\times
Gr_{G}]_{\mathcal{H}_{\bullet}^{G}(S)}.$

###### Proof.

We have Nisnevich local weak equivalence
$\mathcal{K}\rightarrow\mathcal{K}^{G}$ between our affine version of
equivariant $K$-theory presheaf $\mathcal{K}$ and the version
$\mathcal{K}^{G}$ given by Thomason. The presheaf $\mathcal{K}^{G}$ is
representable in the equivariant motivic homotopy category since it is
homotopy invariant and satisfies equivariant Nisnevich descent. By the above
theorem 3.18 the presheaf $\mathcal{K}^{G}$ is simplicially weakly equivalent
to the representable presheaf $Gr_{G}$ after evaluating at $\Delta R$. Thus in
view of [3, Theorem 3.14] the proof of the thoerem is complete. ∎

##  4\. Operations in equivariant algebraic $K$-theory

The main theorem of this section is the following.

######  Theorem 4.1.

Let $G\to S$ be a constant group $S$-scheme with
$\textsl{g}^{-1}\in\Gamma(S,\mathcal{O}_{S})$. There is a canonical bijection

$[(\mathbb{Z}\times Gr_{G})^{n},\mathbf{R}\Omega^{k}\mathbb{Z}\times
Gr_{G}]_{\mathcal{H}^{G}(S)}\xrightarrow{\sim}Hom_{M^{G}(S)}(K_{0}(G,-)^{n},K_{k}(G,-))$

for each $n,k\in\mathbb{N}$. In particular we get,

$[\mathbb{Z}\times Gr_{G},\mathbb{Z}\times
Gr_{G}]_{\mathcal{H}^{G}(S)}\xrightarrow{\sim}End_{M^{G}(S)}(K_{0}(G,-))$

when $n=1$ and $k=0$.

First note that, as in the non-equivariant case, there is a functor
$\mathcal{H}^{G}(S)\to M^{G}(S)$ given by $E\mapsto[-,E]_{\mathcal{H}^{G}(S)}$
where $[-,E]_{\mathcal{H}^{G}(S)}$ is well defined as an object in $M^{G}(S)$.
Let us denote this functor by $\underline{\pi}_{0}$. We then have a map of
sets

(4.1) $\alpha_{F,E}:[F,E]_{\mathcal{H}^{G}(S)}\to
Hom_{M^{G}(S)}(\underline{\pi}_{0}F,\underline{\pi}_{0}E)$

for any pair of objects $E,F\in\mathcal{H}^{G}(S)$. We have
$\underline{\pi}_{0}\mathbf{R}\Omega^{k}K(G,-)\cong K_{k}(G,-)$ in $M^{G}(S)$.
As $\mathbb{Z}\times Gr_{G}$ represents equivariant algebraic $K$-theory in
$\mathcal{H}^{G}(S)$, the map 4.1 becomes

$[(\mathbb{Z}\times Gr_{G})^{n},\mathbf{R}\Omega^{k}\mathbb{Z}\times
Gr_{G}]_{\mathcal{H}^{G}(S)}\xrightarrow{\alpha}Hom_{M^{G}(S)}(K_{0}(G,-)^{n},K_{k}(G,-))$

when applied to the pair $((\mathbb{Z}\times
Gr_{G})^{n},\mathbf{R}\Omega^{k}\mathbb{Z}\times Gr_{G})$. To prove Theorem
4.1 we just need to show that this map is a bijection. We will do this by
proving some general results below along the lines of [6]. First we have the
equivariant analogue of [6, Lem. 1.2.1].

######  Lemma 4.2.

Let $S$ be a Noetherian scheme and $G\to S$ be a group $S$-scheme. Let $E$ be
a group object in $\mathcal{H}^{G}(S)$. Let $(X_{i})_{i\in I}$ be a directed
system of objects in $Sm^{G}_{S}$ ordered by the directed set $I$ which
contains $\mathbb{N}$ as a cofinal subset. Let $X=\operatorname*{colim}_{i\in
I}X_{i}$ be the colimit as objects in $M^{G}(S)$. Then, there exists a short
exact sequence of groups,

$0\to{\lim_{i\in
I}}^{1}[X_{i},\mathbf{R}\Omega^{1}E]_{\mathcal{H}^{G}(S)}\to[\operatorname*{colim}_{i\in
I}X_{i},E]_{\mathcal{H}^{G}(S)}\to\lim_{i\in
I}[X_{i},E]_{\mathcal{H}^{G}(S)}\to 0.$

In particular, there is an isomorphism of groups

$[\operatorname*{colim}_{i\in
I}X_{i},E]_{\mathcal{H}^{G}(S)}\xrightarrow{\sim}\lim_{i\in
I}[X_{i},E]_{\mathcal{H}^{G}(S)}$

whenever $([X_{i},\mathbf{R}\Omega^{1}E]_{\mathbb{A}^{1},G})_{i\in I}$ is a
Mittag-Leffler system.

###### Proof.

Let $E_{f}\in M^{G}(S)$ be a fibrant model of $E$ with respect to the
equivariant motivic model structure. We then have for any $F\in M^{G}(S)$ a
bijection,

$[F,E]_{\mathcal{H}^{G}(S)}\cong[F,E_{f}]_{\mathcal{H}^{G}(S)}\cong
Hom_{M^{G}(S)}(F,E_{f})/\sim$

where $\sim$ denotes the equivalence relation induced by simplicial homotopy.
For any $Y\in Sm^{G}_{S}$ we then have isomorphisms of groups

$[Y,\mathbf{R}\Omega^{n}E]_{\mathcal{H}^{G}(S)}\cong[S^{n}\wedge
Y_{+},E]_{\mathcal{H}^{G}(S)}\cong[S^{n}\wedge
Y_{+},E_{f}]_{\mathcal{H}^{G}(S)}\cong\pi_{n}E_{f}(Y)$

for all $n\geq 0$. Applying this to the $X_{i}$’s we get
$[X_{i},\mathbf{R}\Omega^{n}E]_{\mathcal{H}^{G}(S)}\cong\pi_{n}E_{f}(X_{i})$
for each $i,n\geq 0$. Also note that
$\mathbf{Hom}_{M^{G}(S)}(\operatorname*{colim}_{i\in
I}X_{i},E_{f})\cong\lim_{i\in I}E_{f}(X_{i})$. Using these two results we have
reduced the problem to finding a short exact sequence

$0\to{\lim_{i\in I}}^{1}\pi_{1}E_{f}(X_{i})\to\pi_{0}(\lim_{i\in
I}E_{f}(X_{i}))\to\lim_{i\in I}\pi_{0}(E_{f}(X_{i})\to 0$

which is a classical result. ∎

We will only be concerned with the cases where the $\lim^{1}$ term vanishes so
we give this case a separate name along the lines of [6].

######  Definition 4.3.

Let $E$ and $(X_{i})_{i\in I}$ be as in Lemma 4.2. We will say that the direct
system $(X_{i})_{i\in I}$ does not unveil phantoms in $E$ if
$([X_{i},\mathbf{R}\Omega^{n}E]_{\mathbb{A}^{1},G})_{i\in I}$ is a Mittag-
Leffler system for all $n$.

For simplicity we fix a functorial factorisation
$E\xrightarrow{\eta_{E}}E_{f}\to*$ which gives us a fibrant replacement
functor $E\mapsto E_{f}$ and a natural transformation
$\eta:1_{M^{g}(S)}\Rightarrow(-)_{f}$ which is a weak equivalence at each
object. For each $E\in M^{G}(S)$, $\eta_{E}$ induces a morphism
$E\to\underline{\pi}_{0}E$ in $M^{G}(S)$ given by

$E(X)\xrightarrow{\eta_{E}}E_{f}(X)\to[X,E]_{\mathbb{A}^{1},G}=\underline{\pi}_{0}E(X)$

for all $X\in Sm^{G}_{S}$. Before we prove our main general theorem (Theorem
4.7), we will need an equivariant version of Jouanolou’s trick. However,
unlike the non-equivariant case, not every $G$-scheme can be covered by
$G$-invariant affine opens. It is however true that in the case of constant
finite groups schemes every $G$-scheme has an equivariant _Nisnevich_ cover
comprised of affine schemes [2, Lem 2.20]. We generalise the notion of affine
bundles accordingly.

######  Definition 4.4.

An $N$-affine bundle over a $G$-equivariant scheme $X$ is a $G$-equivariant
$X$-scheme $Y\to X$ such that $Y\times U_{i}\to U_{i}$ is a $G$-vector bundle
for some Nisnevich cover of $X$, $\\{U_{i}\to X\\}_{i\in I}$.

It follows from this definition that if $\pi:Y\to X$ is an $N$-affine bundle
then $\pi$ is a $G$-motivic weak equivalence.

######  Theorem 4.5.

Let $S$ be a scheme with an ample family of line bundles and $p:G\to S$ a
constant finite group scheme over $S$ with
$\textsl{g}^{-1}\in\Gamma(S,\mathcal{O}_{S})$. There exists an $N$-affine
bundle $W\to S$ where $W$ is affine.

###### Proof.

By our hypothesis, there exist a family of line bundles
$\\{\mathcal{L}_{1},\mathcal{L}_{2},\ldots,\mathcal{L}_{n}\\}$ over $S$ and
global sections $f_{i}\in\Gamma(S,\mathcal{L}_{i})$ such that $S_{f_{i}}$ are
affine and cover $S$. We then have a morphism of vector bundles

$(f_{1},f_{2},\ldots,f_{n}):\mathcal{O}_{S}\rightarrowtail\oplus^{n}_{i=1}\mathcal{L}_{i}$

which induces a morphisms of $G$-vector bundles,

$\hat{f}=p_{*}p^{*}(f_{1},f_{2},\ldots,f_{n})\circ
s_{G}:\mathcal{O}_{S}\rightarrowtail p_{*}\mathcal{O}_{G}\rightarrowtail
p_{*}p^{*}(\oplus^{n}_{i=1}\mathcal{L}_{i})$

where $\mathcal{O}_{S}$ has the trivial $G$-action. Set
$\mathcal{E}=p_{*}p^{*}(\oplus^{n}_{i=1}\mathcal{L}_{i})$. As $\hat{f}$ splits
on each $S_{f_{i}}$ so does $p_{*}p^{*}\hat{f}$ and the map
$\mathcal{O}_{S}\to p_{*}\mathcal{O}_{G}$ is split when $\textsl{g}=|G|$ is
invertible in $\Gamma(S,\mathcal{O}_{S})$. This means the quotient
$\mathcal{E}/\mathcal{O}_{S}$ is a $G$-vector bundle. We can identify
$\mathbb{P}((\mathcal{E}/\mathcal{O}_{S})^{\vee})$ with a $G$-invariant closed
subscheme of $\mathbb{P}(\mathcal{E}^{\vee})$. By the proof of [11, Prop 4.4]
we see that
$W=\mathbb{P}(\mathcal{E}^{\vee})-\mathbb{P}((\mathcal{E}/\mathcal{O}_{S})^{\vee})$
is affine and the pullbacks $W\times S_{f_{i}}\to S_{f_{i}}$ are isomorphic to
the projections $V(\mathcal{E}/\mathcal{O}_{S})\times S_{f_{i}}\to S_{f_{i}}$.
By [2, Lem 2.20] we have an affine $G$-Nisnevich cover $\\{(S_{f_{i}})_{G}\to
S\\}$ where there is a (non-equivariant) factorisation

${(S_{f_{i}})_{G}}$${S}$${S_{f_{i}}}$

for each $i$. Therefore, $W\times(S_{f_{i}})_{G}\to(S_{f_{i}})_{G}$ is
isomorphic to the projections
$V(\mathcal{E}/\mathcal{O}_{S})\times(S_{f_{i}})_{G}\to(S_{f_{i}})_{G}$. ∎

If $S$ is a regular scheme then every separated smooth scheme over $S$ has an
ample family of line bundles and hence every scheme considered in this paper
satisfies the conditions for Theorem 4.5.

######  Corollary 4.6.

Let $S$ be a regular scheme and $G\to S$ be a constant finite group scheme
over $S$ with $\textsl{g}^{-1}\in\Gamma(S,\mathcal{O}_{S})$. For every $X\to
S\in Sm^{G}_{S}$, there exists an $N$-affine bundle $W\to X$ where $W$ is
(globally) affine.

We now have sufficient results to prove the following.

######  Theorem 4.7.

Let $S$ be a Noetherian scheme and $G\to S$ be a group $S$-scheme. Let $E$ be
a group object in $\mathcal{H}^{G}(S)$. Let $(X_{i})_{i\in I}$ be a directed
system of objects in $Sm^{G}_{S}$ which does not unveil phantoms in $E$. Let
$X=\operatorname*{colim}_{i\in I}X_{i}$ be the colimit in $M^{G}(S)$. If
$X(U)\to\underline{\pi}_{0}X(U)$ is onto for all affine $U\in Sm^{G}_{S}$ then
the map

$[X,E]_{\mathcal{H}^{G}(S)}\to
Hom_{M^{G}(S)}(\underline{\pi}_{0}X,\underline{\pi}_{0}E)$

as given in 4.1 is a bijection.

###### Proof.

As $X$ does not unveil phantoms in $E$, we have an isomorphism of groups (and
hence a bijection) $[X,E]_{\mathcal{H}^{G}(S)}\cong\lim_{i\in I}[X_{i},E]$.
This fits into a commutative diagram,

${{[X,E]}_{\mathcal{H}^{G}(S)}}$${Hom_{M^{G}(S)}(\underline{\pi}_{0}X,\underline{\pi}_{0}E)}$${\lim_{i\in
I}[X_{i},E]_{\mathcal{H}^{G}(S)}}$$\scriptstyle{\alpha_{X,E}}$$\scriptstyle{\sim}$$\scriptstyle{\beta}$

where $\beta$ is the composite map

$Hom_{M^{G}(S)}(\underline{\pi}_{0}X,\underline{\pi}_{0}E)\to
Hom_{M^{G}(S)}(X,\underline{\pi}_{0}E)\cong\lim_{i\in
I}[X_{i},E]_{\mathcal{H}^{G}(S)}.$

It follows that $\alpha_{X,E}$ is an injection and $\beta$ is a surjection. To
show $\alpha_{X,E}$ is a bijection it is enough to show that $\beta$ is an
injection. Given any pair $f_{0},f_{1}\in
Hom_{M^{G}(S)}(\underline{\pi}_{0}X,\underline{\pi}_{0}E)$ which are mapped
onto the same element in $Hom_{M^{G}(S)}(X,\underline{\pi}_{0}E)$, we have the
diagram

$X\to\underline{\pi}_{0}X\xrightarrow[f_{1}]{f_{0}}\underline{\pi}_{0}E$

where $X(U)\to\underline{\pi}_{0}X(U)$ is a surjection for all $U$ affine.
Hence $f_{0}$ and $f_{1}$ are equal when evaluated on affine $G$-schemes. As
$\underline{\pi}_{0}X$ and $\underline{\pi}_{0}E$ are invariant under
equivariant motivic equivalences, Corollary 4.6 implies that $f_{0}=f_{1}$ and
we are done. ∎

##  5\. K-theory of Grassmannians

For any $X\in Sm^{G}_{S}$, there is a canonical map of sets

(5.1) $Hom_{PSh(Sm^{G}_{S})}(X,\mathbb{Z}\times Gr^{G})\to K_{0}(G,X)$

induced by the map in Theorem 3.18.

######  Theorem 5.1.

For all affine $X\in Sm^{G}_{S}$, the map 5.1 is onto.

###### Proof.

Let $X\cong Spec(R)$. As before let $\rho$ denote the regular $G$-bundle over
$X$. We denote the kernel of the canonical splitting $s:\rho\to R$ by $I$ as
it is the augmentation ideal. The morphism
$Hom_{M_{\bullet}^{G}(S)}(X_{+},\mathbb{Z}\times Gr^{G})\to K_{0}(G,X)$ is
given by

(5.2) $(i,M\rightarrowtail(\rho)^{\oplus n})\mapsto[M]-n[\rho]+i[R]$

on each $\\{i\\}\times Gr(r\textsl{g},(\rho)^{\oplus n})$. Let $[P]-[Q]\in
K_{0}(G,Spec(R))$. By 3.3, $Q$ is a direct summand of $\rho^{\oplus n}$ for
some $n$. Therefore $[Q]=n[\rho]-[Q^{\prime}]$ for some $G$-vector bundle
$Q^{\prime}$ and we have

$[P]-[Q]=[P]-n[\rho]+[Q^{\prime}]=[P\oplus
Q^{\prime}]-n[\rho]=[P^{\prime}]-n[\rho]$

in $K_{0}(G,Spec(R))$. If rank of $P^{\prime}$ is $p=b\textsl{g}+r$ with
$r<\textsl{g}$ then we can add and subtract $(\textsl{g}-r)$ copies of $[R]$
to get

$[P^{\prime}]-n[\rho]=[P^{\prime}\oplus
R^{\oplus\textsl{g}-r}]-n[\rho]-(\textsl{g}-r)[R]$

where $P^{\prime}\oplus R^{\oplus\textsl{g}-r}$ then has rank a multiple of g.
To finish the proof we need to show that every element in $K_{0}(G,Spec(R))$
of the form $[M]-n[\rho]-i[R]$ where rank of $M$ is a multiple of g can be
written in the form given in 5.2. Let $rank(M)=m\textsl{g}$ we then have two
cases, $m<n$ and $m>n$. If $m<n$ then

$[M]-n[\rho]-i[R]=[M]+(n-m)\textsl{g}[R]-(n-m)\textsl{g}[R]-n[\rho]-i[R]\\\
=[M\oplus R^{\oplus(n-m)\textsl{g}}]-n[\rho]-(n\textsl{g}-m\textsl{g}+i)[R]$

which is in the desired form. When $m>n$ then

$[M]-n[\rho]-i[R]=[M]+(m-n)\textsl{g}[I]-(m-n)\textsl{g}[I]-n[\rho]-i[R]\\\
=[M\oplus
I^{\oplus(m-n)\textsl{g}}]-(m-n)\textsl{g}[\rho]-n[\rho]+((m-n)\textsl{g}-i)[R]\\\
=[M\oplus
I^{\oplus(m-n)\textsl{g}}]-((m-n)\textsl{g}+n)[\rho]+((m-n)\textsl{g}-i)[R]$

which is of the desired form. ∎

To prove Theorem 4.1 we need to calculate the $K$-groups of the Grassmannian
schemes. We have from [10, Thm 3.1] the $G$-equivariant projective bundle
theorem.

######  Theorem 5.2 (Projective bundle theorem).

Let $(X,\sigma)$ be a $G$-equivariant $S$-scheme and let $(\mathcal{E},\tau)$
be a $G$-equivariant vector bundle over $(X,\sigma)$ of constant rank $n$. The
functor

$\prod^{n}_{i=1}VB(G,X,\sigma)\to VB(G,\mathbb{P}(\mathcal{E}),p(\tau))$

given by
$(E_{1},E_{2},\dots,E_{n})\mapsto\oplus^{n}_{i=1}f^{*}E_{i}\otimes\mathcal{O}(-i)$
induces a homotopy equivalence
$K(G,X,\sigma)^{n}\xrightarrow{\sim}K(G,\mathbb{P}(\mathcal{E}),p(\tau))$.

We can go further and describe $K_{*}(G,\mathbb{P}(\mathcal{E}),p(\tau))$ as a
$K_{*}(G,X,\sigma)$-algebra along the same lines as the non-equivariant case.

######  Corollary 5.3.

Let $(X,\sigma)$ and $(\mathcal{E},\tau)$ be as above. The homotopy
equivalence
$K(G,X,\sigma)^{n}\xrightarrow{\sim}K(G,\mathbb{P}(\mathcal{E}),p(\tau))$
given in Theorem 5.2 induces an isomorphism

$K_{*}(G,\mathbb{P}(\mathcal{E},p(\tau))\cong
K_{*}(G,X,\sigma)\otimes_{K_{0}(G,X,\sigma)}K_{0}(G,\mathbb{P}(\mathcal{E},p(\tau)))$

of $K_{*}(G,X,\sigma)$-algebras.

###### Proof.

First by Theorem 5.2 we have a commutative diagram of
$K_{0}(G,X,\sigma)$-modules

${K_{0}(G,X,\sigma)^{n}}$${K_{*}(G,X,\sigma)^{n}}$${K_{0}(G,\mathbb{P}(\mathcal{E},p(\tau))}$${K_{*}(G,\mathbb{P}(\mathcal{E},p(\tau))}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

where the vertical maps are isomorphisms and the bottom map is a ring
homomorphism. Using the extension-restriction adjunction with respect to the
ring homomorphism $K_{0}(G,X,\sigma)\to K_{*}(G,X,\sigma)$ we get the
following commutative diagram

${K_{*}(G,X,\sigma)\otimes_{K_{0}(G,X,\sigma)}K_{0}(G,X,\sigma)^{n}}$${K_{*}(G,X,\sigma)^{n}}$${K_{*}(G,X,\sigma)\otimes_{K_{0}(G,X,\sigma)}K_{0}(G,\mathbb{P}(\mathcal{E},p(\tau)))}$${K_{*}(G,\mathbb{P}(\mathcal{E},p(\tau)))}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$

of $K_{*}(G,X,\sigma)$-modules. Here again the bottom map is a ring
homomorphism. Note that the top map is bijection and hence so is the bottom
map. ∎

Along the lines of the non-equivariant case we can give an explicit
description of the ring $K_{0}(G,\mathbb{P}(\mathcal{E},p(\tau)))$ using
$\lambda$-operations. For any $(X,\sigma)$ let the $n^{th}$ lambda operation
map $\lambda^{n}:K_{0}(G,X,\sigma)\to K_{0}(G,X,\sigma)$ be given by the
exterior powers,
$[(\mathcal{E},\tau)]\mapsto[(\wedge^{n}\mathcal{E},\wedge^{n}\tau)]$. The
collection $(\lambda^{n})_{n\in\mathbb{N}}$ defines a $\lambda$-ring structure
on $K_{0}(G,X,\sigma)$.

######  Corollary 5.4.

Let $(X,\sigma)$ be a $G$-equivariant $S$-scheme and let $(\mathcal{E},\tau)$
be a $G$-equivariant vector bundle over $(X,\sigma)$ of constant rank $n$. Let
$f:\mathbb{P}(\mathcal{E})\to X$ be the corresponding projective bundle. The
$K_{0}(G,X,\sigma)$-algebra homomorphism $c:K_{0}(G,X,\sigma)[t]\to
K_{0}(G,\mathbb{P}(\mathcal{E}),p(\tau))$ given by $t\mapsto[\mathcal{O}(-1)]$
induces an isomorphism of rings,

$\displaystyle K_{0}(G,X,\sigma)[t]/(\theta)\cong
K_{0}(G,\mathbb{P}(\mathcal{E}),p(\tau)),$
$\displaystyle\mathrm{where}\>\>\theta=\sum_{i=0}^{n}(-1)^{i}\lambda^{n-i}[(\mathcal{E},\tau)]t^{i}$

and hence by Corollary 5.3 we get an isomorphism of graded
$K_{*}(G,X,\sigma)$-algebras,

$K_{*}(G,X,\sigma)[t]/(\theta)\cong K_{*}(G,\mathbb{P}(\mathcal{E}),p(\tau)).$

###### Proof.

Let $c(\mathcal{E},\tau)=[(f^{*}\mathcal{E},f^{*}\tau)]$ and
$t=[\mathcal{O}(-1)]$ in $K_{0}(G,\mathbb{P}(\mathcal{E}),p(\tau))$. Consider
the element $(c(\mathcal{E},\tau)-t\in
K_{0}(G,\mathbb{P}(\mathcal{E}),p(\tau))$. We have
$(c(\mathcal{E},\tau)-t)=[\mathcal{F},\hat{\tau}]$ where $\mathcal{F}$ is the
quotient,

$\mathcal{O}(-1)\rightarrowtail f^{*}\mathcal{E}\twoheadrightarrow\mathcal{F}$

over $\mathbb{P}(\mathcal{E})$ and hence
$\lambda^{n}(c(\mathcal{E},\tau)-t)=[(\wedge^{n}\mathcal{F},\wedge^{n}\hat{\tau})]=0$.
Using the properties of $\lambda$-structures we get

$0=\lambda^{n}(c(\mathcal{E},\tau)-t)=\sum_{i=0}^{n}\lambda^{i}(c(\mathcal{E},\tau))\lambda^{n-i}(-t).$

in $K_{0}(G,\mathbb{P}(\mathcal{E}))$. As $t$ is a line bundle we have
$\lambda^{1}(t)=t$ and $\lambda^{i}(t)=0$ for $i\geq 1$. By the properties of
$\lambda$-operations we have $\lambda^{i}(-t)=(-1)^{i}t^{i}$ for all $i\geq
0$. Replacing this into the equation above we get

$t^{n}=\sum_{i=1}^{n}\lambda^{i}(c(\mathcal{E},\tau))(-1)^{n-i}t^{n-i}$

and by Theorem 5.2 we know that $K_{0}(G,\mathbb{P}(\mathcal{E}),p(\tau))$ is
a free $K_{0}(G,X,\sigma)$-module with basis $\\{t,t^{2},\ldots,t^{n-1}\\}$.
Hence we are done. ∎

Just like the non-equivariant case we can use this to calculate
$K_{n}(G,Gr(r,\mathcal{E},\tau))$ for all Grassmannians. Let
$(\mathcal{E},\tau)$ be a rank $n$ equivariant vector bundle over $(X,\mu)$.
For every partition $\pi=(p_{1},p_{2},\ldots,p_{r})$ of $n$, the flag bundle
$f:Flag(\pi,\mathcal{E})\to X$ has an induced $G$-action. We then have the
canonical equivariant flag

$F_{1}\hookrightarrow F_{2}\hookrightarrow\ldots\hookrightarrow F_{r}\cong
f^{*}\mathcal{E}$

where $rk(F_{i})=\sum_{j\leq i}p_{j}$. When the partition is of the form
$\pi=(1,\ldots,1,n-r)$ we denote this bundle by $Flag(1^{r},\mathcal{E})$.
These bundles are in fact iterated projective bundles,

$\mathbb{P}(\mathcal{E})\leftarrow
Flag((1,1,n-1),\mathcal{E})\leftarrow\ldots\leftarrow
Flag((1^{r},n-1),\mathcal{E})$

over $X$. We then get the following corollary.

######  Corollary 5.5.

Let $(X,\sigma)$ be a $G$-equivariant $S$-scheme and let $(\mathcal{E},\tau)$
be a $G$-equivariant vector bundle over $(X,\sigma)$ of constant rank $n$. Let
$Flag(1^{r},\mathcal{E})$ denote the corresponding flag bundle with canonical
flag

$F_{1}\hookrightarrow F_{2}\hookrightarrow\ldots\hookrightarrow F_{r}\cong
f^{*}\mathcal{E}$

where $f:Flag(1^{r},\mathcal{E})\to X$ is the structure map. The K-theory ring
$K^{G}_{*}(Flag(1^{r},\mathcal{E}))$ is generated as a $K^{G}_{*}(X)$-algebra
by the elements $t_{i}=[F_{i}]-[F_{i-1}]$ in
$K_{0}(G,Flag(1^{r},\mathcal{E}))$. Furthermore,
$K_{*}(G,Flag(1^{r},\mathcal{E}))$ is a free $K_{*}(G,X)$-module with basis,

$\\{t_{1}^{a_{1}}t_{2}^{a_{2}}\ldots t_{r}^{a_{r}}|0\leq a_{i}\leq n-i,i\leq
r\\}$

and hence of rank $\frac{n!}{n-r!}$.

We can use this result to compute the K-theory ring of any Grassmannian.

######  Theorem 5.6.

Let $f:Gr(r,\mathcal{E})\to X$ be a Grassmannian over any scheme $X$ with
canonical subbundle $\mathcal{U}_{r}\hookrightarrow f^{*}\mathcal{E}$. The
$K_{*}(G,X)$-algebra $K_{*}(G,Gr(r,\mathcal{E}))$ is faithfuly flat and
generated by $\lambda^{i}=[\Lambda^{i}\mathcal{U}_{r}]$, for $1\leq i\leq r$,
in $K_{0}(G,Gr(r,\mathcal{E}))$.

###### Proof.

Given a Grassmannian $Gr(r,\mathcal{E})$, the projective bundle
$\mathbb{P}(\mathcal{U}_{r})\to Gr(r,\mathcal{E})\to X$ is the flag bundle
$Flag((1,r,n-r-1),\mathcal{E})\to X$. We can than recursively construct a
sequence

$X\leftarrow Gr(r,\mathcal{E})\leftarrow
Flag((1,r,n-r-1),\mathcal{E})\leftarrow
Flag((1,1,r,n-r-2),\mathcal{E})\leftarrow\ldots Flag((1^{r},\mathcal{E}))$

of flag bundles. Let the canonical flag over $Flag((1^{r},\mathcal{E}))$ be
given by

$F_{1}\hookrightarrow F_{2}\hookrightarrow\ldots\hookrightarrow
F_{r}\hookrightarrow f^{*}\mathcal{E}$

with $F_{r}=\mathcal{U}_{r}$. By Corollary 5.5 the morphism
$K_{*}(G,Gr(r,\mathcal{E}))\to K_{*}(G,Flag(1^{r},\mathcal{E}))$ is an
inclusion and $K_{*}(G,Flag(1^{r},\mathcal{E}))$ is generated as a
$K_{*}(G,Gr(r,\mathcal{E}))$-algebra by $t_{i}=[F_{i}]-[F_{i-1}]$ for $0<i\leq
r$. Also, the image of $\lambda^{i}=[\Lambda^{i}\mathcal{U}_{r}]$ in
$K_{*}(G,Flag(1^{r},\mathcal{E}))$ is the $i^{th}$ elementary symmetric
polynomial $e_{i}(t_{1},\ldots,t_{r})$. We therefore get a morphism
$K_{*}(G,X)[e_{1},\ldots,e_{r}]\to K_{*}(G,Gr(r,\mathcal{E}))$ such that the
following diagram commutes

${K_{*}(G,X)[e_{1},\ldots,e_{r}]}$${K_{*}(G,X)[t_{1},\ldots,t_{r}]}$${K_{*}(G,Gr(r,\mathcal{E}))}$${K_{*}(G,Flag(1^{r},\mathcal{E}))}$

where the right vertical map is a quotient map. For any ring $R$ the
polynomial ring $R[t_{1},\ldots,t_{r}]$ is a free module over the ring of
symmetric polynomials $R[e_{1},\ldots,e_{r}]$ with basis

$\\{t_{1}^{a_{1}}t_{2}^{a_{2}}\ldots t_{r}^{a_{r}}|0\leq a_{i}\leq r-i,i\leq
r\\}$

which is precisely the basis of $K_{*}(G,Flag(1^{r},\mathcal{E}))$ as a
$K_{*}(G,Gr(r,\mathcal{E}))$-module. Hence the left vertical map is a
surjection. Faithfully flatness follows from the two out of three property. ∎

##  6\. Proof of theorem 4.1

We now have enough to prove Theorem 4.1. We will need the following lemma.

######  Lemma 6.1.

For each $k$, the directed system $(([-i,i]\times Gr(i,\rho^{2i}))^{k})_{i\in
I}$ does not unveil phantoms in $K(G,\ -)$.

###### Proof.

By Theorem 4.1 we have

$[X,\mathbf{R}\Omega^{n}\mathbb{Z}\times
Gr_{G}]_{\mathcal{H}^{G}(S)}\xrightarrow{\alpha}K_{n}(X)$

for all schemes $X$ and all $n\geq 0$. Hence, it is enough to prove that
$K_{n}(G,(([-i,i]\times Gr(i,\rho^{2i}))^{k})_{i\in I})$ is Mittag-Leffler for
all $n,k\geq 0$. We will achieve this by showing that the morphism of graded
rings

$K_{*}(G,([-i-1,i+1]\times Gr((i+1)g,\rho^{2(i+1)}))^{k})\to
K_{*}(G,([-i,i]\times Gr(ig,\rho^{2i}))^{k})$

is surjective for all $i\geq 0$. We have closed embeddings $Gr(ig,\rho^{n})\to
Gr(ig,\rho^{n+1})$ of $G$-schemes whose open compliment is a $G$-equivariant
vector bundle over $Gr((i-1)g,\rho^{n})$. Therefore along the lines of [4] we
have a spectral sequence

$E^{p,q}_{2}=Tor^{K_{*}(G,X)}_{p,q}(K_{*}(G,Gr(ig,\rho^{n})),K_{*}(G,Gr(jg,\rho^{m})))\Rightarrow
K_{p+q}(G,Gr(ig,\rho^{n})\times Gr(jg,\rho^{m}))$

for all $i,j,n,m\geq 0$. As each $K_{*}(G,Gr(ig,\rho^{n}))$ is a flat
$K_{*}(G,X)$-module, the spectral sequence degenerates and we have

$K_{*}(G,Gr(ig,\rho^{n}))\otimes_{K_{*}(G,X)}K_{*}(G,Gr(jg,\rho^{m}))\cong
K_{*}(G,Gr(ig,\rho^{n})\times Gr(jg,\rho^{m}))$

Hence
$\bigotimes^{k}_{j=0}K_{*}(G,Gr(ig,\rho^{2i}))\xrightarrow{\sim}K_{*}(G,Gr(ig,\rho^{2i})^{k})$
for all $k$ where the tensor product on the left is as $K_{*}(G,X)$-algebras.
Given the morphism of schemes $Gr(ig,\rho^{2i}))^{k}\to
Gr((i+1)g,\rho^{2(i+1)}))^{k}$, the induced ring homomorphism

$f_{i}:K_{*}(G,Gr((i+1)g,\rho^{2(i+1)}))\to K_{*}(G,Gr(ig,\rho^{2i}))$

sends $[\mathcal{U}_{g(i+1)}]-[\rho]$ to $[\mathcal{U}_{gi}]$ in
$K_{0}(G,Gr(ig,\rho^{2i}))$. The homomorphism restricted to $K_{0}(G,-)$
conserves the $\lambda$-ring structure and hence $f_{i}$ is surjective for all
$i$. Together with the result about product and the fact that
$K_{*}(G,X\coprod Y)\xrightarrow{\sim}K_{*}(G,X)\oplus K_{*}(G,Y)$ we get that
the desired ring homomorphism is surjective. ∎

We now have enough to prove the main theorem.

######  Theorem 6.2.

Let $G\to S$ be a constant group $S$-scheme with
$\textsl{g}^{-1}\in\Gamma(S,\mathcal{O}_{S})$. There is a canonical bijection

$[(\mathbb{Z}\times Gr_{G})^{n},\mathbf{R}\Omega^{k}\mathbb{Z}\times
Gr_{G}]_{\mathcal{H}^{G}(S)}\xrightarrow{\sim}Hom_{M^{G}(S)}(K_{0}(G,-)^{n},K_{k}(G,-))$

for each $n,k\in\mathbb{N}$. In particular we get,

$[\mathbb{Z}\times Gr_{G},\mathbb{Z}\times
Gr_{G}]_{\mathcal{H}^{G}(S)}\xrightarrow{\sim}End_{M^{G}(S)}(K_{0}(G,-))$

when $n=1$ and $k=0$.

###### Proof.

By Lemma 6.1 and Lemma 5.1 we can apply Theorem 4.7 to $\mathbb{Z}\times
Gr_{G}=\operatorname*{colim}_{i\in\mathbb{N}}[-i,i]\times Gr(ig,\rho^{2i})$.
Giving us the desired result. ∎

As in the non-equivariant case, the above bijection respects algebraic
structures hence we get the equivariant analogue of [6, Thm 2.3.1].

######  Corollary 6.3.

In the category $\mathcal{H}^{G}(S)$ there exists a unique structure of a
special $\lambda$-ring with duality on $\mathbb{Z}\times Gr_{G}$ such that the
corresponding $\lambda$-ring with duality structure on $K_{0}(G,X)$ is the
standard one for all $X\in Sm^{G}_{S}$.

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