Algebraic K-functors for $\Gamma$-rings

TL;DR

This paper extends algebraic K-theory to $ ext{Gamma}$-rings, introducing new K-functors and proving extended versions of key conjectures related to Witt and Chow groups.

Contribution

It introduces $ ext{Gamma}$-algebraic K-theory, defines Milnor K-theory and Swan's K-functors for $ ext{Gamma}$-rings, and proves extended conjectures.

Findings

Extended Matsumoto conjecture related to symbol groups

Proved Milnor conjectures for Witt and Chow groups

Developed foundational theory for $ ext{Gamma}$-rings in algebraic K-theory

Abstract

This is an attempt to extend to algebraic K-theory our approach to group actions in homological algebra that could be called an introduction to $\Gamma$-algebraic K-theory. For $\Gamma$-rings the Milnor algebraic K-theory and Swan's algebraic K-functors are introduced and investigated. Particularly the Matsumoto conjecture related to the symbol group, and the Milnor conjectures related to Witt and Chow groups are extended and proven.

Equations

$$E:1\to A\to B\stackrel{\tau }{\to }G\to 1$$

$${}^{\sigma }{(}^{g}a){=}^{\sigma }g{(}^{\sigma }a)$$

$$H_n^{\Gamma }(G,A)=To{r}_n^{\mathcal{P}}(\mathbb{Z},A),H_{\Gamma }^n(G,A)=Ex{t}_{\mathcal{P}}^n(\mathbb{Z},A),n\ge 0,$$

$$H_n^{\Gamma }(G,A)\cong L_{n-1}^{\mathbb{F}}(I(G){\otimes }_{G⋉\Gamma }A),H_{\Gamma }^n(G,A)\cong R_{\mathbb{F}}^{n-1}De{r}_{\Gamma }(G,A)$$

$$1\to A\to B\stackrel{\tau }{\to }G\to 1$$

$$0\to V\to H_2^{\Gamma }(B)\to H_2^{\Gamma }(G)\to A/[B,A{]}_{\Gamma }\to H_1^{\Gamma }(B)\to H_1^{\Gamma }(G)\to 0,$$

$$0\to Ker\Gamma (\tau )/\Gamma Ker\tau \to (Ker\tau {)}_{\Gamma }\to G_{\Gamma }\to G_{\Gamma }^{\prime }\to 0.$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$(G/H{)}_{\Gamma }=G/(H\cdot \Gamma G).$$

$$e\to SL(A)\to GL(A)\stackrel{det}{\to }{A}^{\ast }\to 0,$$

$$0\to SL(A)/E(A)\to K_1(A)\to A^{\ast }\to 0$$

$$K_1^{\Gamma }(A)\approx (SL(A)/E(A){)}_{\Gamma }\oplus (A^{\ast }{)}_{\Gamma }\cong (SL(A)/(E(A)\cdot \Gamma SL(A)))\oplus (A^{\ast }{)}_{\Gamma },$$

$$0\to Ker\Gamma (\varphi )/\Gamma Ker(\varphi )\to Sy{m}^{\Gamma }(F)\to K_2^{\Gamma }(F)\to 0,$$

$$0\to K_2(A)\to U(A)\stackrel{\tau }{\to }[A^{\ast },A^{\ast }]\to e,$$

$$0\to Ker\Gamma (\tau )/\Gamma (K_2(A))\to (K_2(A){)}_{A^{\ast }}\to Sy{m}^{A^{\ast }}(A)\to$$

$$[A^{\ast },A^{\ast }]/\Gamma ([A^{\ast },A^{\ast }])\to 0,$$

$$F_{\star }^{+}(A)=F_{\star }(A)\stackrel{\tau }{\to }A,F_0(A)=F(A),F_n=F(F_{n-1}(A))for\ge 1,$$

$$...\to {\pi }_{n+1}(L_{\ast }^{\Gamma })\to {\pi }_n(\Gamma L_{\ast })\to {\pi }_n(L_{\ast })\to {\pi }_n(L_{\ast }^{\Gamma })\to {\pi }_{n-1}(\Gamma L_{\ast })\to ...,$$

$${\pi }_n(\Gamma L_{\ast })\to ({\pi }_n(L_{\ast }){)}_{\Gamma }\to {\pi }_n(L_{\ast }^{\Gamma })\to {\pi }_{n-1}(\Gamma L_{\ast }).$$

$$L^{\Gamma }{T}_n(A)={\pi }_n((T{F}_{\ast }(A){)}^{\Gamma }),n\ge 0.$$

$$K_1^{\Gamma }(A)\otimes K_1^{\Gamma }(A)\to K_2^{\Gamma }(A)$$

$$diag(x\otimes I^{\prime };x^{-1}\otimes I^{\prime };I\otimes I^{\prime })\ast diag(I\otimes y,I\otimes I^{\prime };I\otimes y^{-1})\in K_2(A).$$

$$K_n^M(A)=(A^{\star }{)}^{\otimes n}/\{a_1\otimes a_2...\otimes a_n:a_i+a_{i+1}=1\},$$

$$((A^{\star }{)}^{\otimes n}/\{a_1\otimes a_2...\otimes a_n:a_i+a_{i+1}=1\}{)}_{\Gamma },$$

$$I(F)\supset I^2(F)\supset ...\supset I^n(F)\supset ...$$

$$P_F^n=\{a_1,...,a_{n+1}\}/\sim ,a_i\in F,$$

$$(K_n^M(F)/2K_n^M(F){)}_{\Gamma }\cong (I^n/I^{n+1}{)}_{\Gamma }$$

$$(K_n^M(F){)}_{\Gamma }\cong (C{H}^n(F,n){)}_{\Gamma }$$

$$0\to (K_2(F)\cap \Gamma St(F))/\Gamma K_2(F)\to C{H}^{n,\Gamma }(F,n)\to K_2^{\Gamma }(F)\to 0.$$

Full text

Hvedri Inassaridze

1. title

Algebraic K-functors for $\Gamma$-rings

2. abstract

This is an attempt to extend to algebraic K-theory our approach to group actions in homological algebra that could be called introduction to $\Gamma$-algebraic K-theory. For $\Gamma$-rings the Milnor algebraic K-theory and Swan’s algebraic K-functors are introduced and investigated. Particularly, the Matsumoto conjecture related to symbol group, and the Milnor conjectures related to Witt algebras and Chow groups are extended and proved.

3. introduction

We continue the study of our approach to group actions in homological algebra for the case of algebraic K-theory, that was started in [8] and continued in [9,10]. The originality of our approach to the study of homological properties of groups and rings consists of the definition of a new and natural, unexpected action of a group $\Gamma$ on the classical chain complexes defining classical homology of a group $G$ and of a ring $R$ respectively, induced by its given action on the group $G$ and on the ring $R$ respectively. By taking the homology groups of the tensor product of the classical chain complex under this action of $\Gamma$ with the coefficient group, we have defined new homology and cohomology groups of $\Gamma$-groups and $\Gamma$-rings, called $\Gamma$-equivariant (co)homology. These new tools allowed us to extend important homological properties of groups and rings to the case of $\Gamma$-groups and $\Gamma$-rings, opening a new direction in homological algebra that we called $\Gamma$-homological algebra. The recently developed investigation of equivariant cohomological dimensions is closely related to $\Gamma$-homological algebra [3,4].

We are trying to investigate the influence of group actions on the foundations of algebraic K-theory [13,2,16,17] when a discrete group is acting on the basic ring.

Some notations that will be used throughout the paper:

$\Gamma G$ denotes the normal subgroup of $G$ generated by the set of elements ${}^{\gamma}gg^{-1},g\in G$, $\gamma\in\Gamma$. The quotient group $G/\Gamma G$ is denoted $G_{\Gamma}$.

$\Gamma-[G,G]$ denotes the subgroup of the $\Gamma$-group $G$ generated by the commutant subgroup [$G$,$G$] and the elements of the form ${}^{\gamma}gg^{-1}$, $g\in G$, $\gamma\in\Gamma$. It is called the $\Gamma$-commutant of the group $G$.

$G^{ab}_{\Gamma}$ denotes the abelianization of the group $G_{\Gamma}$.

For an augmented simplicial group $G^{+}_{\ast}:G_{\ast}\overset{\tau}{\rightarrow}G_{-1}$, the notation $\pi_{-1}(G^{+}_{\ast})$ equal to $Coker\tau$ is used.

${\mathbf{G}}^{\Gamma}$ and ${\mathbf{R}}^{\Gamma}$ denote the category whose objects are groups and rings respectively on which a fixed group $\Gamma$ is acting, called $\Gamma$-groups and $\Gamma$-rings respectively, and morphisms are homomorphisms compatible with the action of $\Gamma$.

4. Preliminaries

We give some important examples of $\Gamma$-rings with unit.

Let $A$ be an arbitrary ring with unit and $\Gamma$ be a group. It is said that $A$ is a $\Gamma$-ring, or $\Gamma$ is acting on $A$, if there is given a group homomorphism $\Gamma\rightarrow Aut(A)$.

Rings having the identity automorphism only: the commutative rings $\mathds{Z}$, $\mathds{Q}$, $\mathds{R}$, $\mathds{Z}/p\mathds{Z}$, $\mathds{Z}/(p)$ (the ring of integers localized at p a rational prime), and $\mathds{Q}_{p}$ (the ring of p-adic numbers).

The fields are important examples of $\Gamma$-rings. For instance, the field $\mathds{C}$ of complex numbers has two automorphisms, the identity and the complex conjugation sending a + bi to a - bi; the field $F_{q}$ of prime power $q=p^{n}$ has the Frobenius automorphism $\phi:x\rightarrow x^{p}$, and the automorphism group is cyclic of order $n$ generated by $\phi$; the quadratic field $\mathds{Q}(2)$ with automorphism group of order 2; the Galois groups of field extensions.

The non-commutative ring $A$ with non-trivial group $A^{*}$ of invertible elements (called also units)(for instance the quaternion algebra). Then $A^{*}$ is acting on $A$ sending $x\rightarrow axa^{-}1$, $x\in A$, $a\in A^{*}$.

Let $RA$ be an $R$-algebra, $R$ is a commutative ring with unit. The group $\Gamma$ is acting on $RA$ if it is acting on $R$ and on $A$, and one has ${}^{\gamma}(ra)=^{\gamma}(r)^{\gamma}(a)$. For instance, the symmetric group $S_{n}$ is acting on the polynomial $R$-algebra $R[x_{1},...,x_{n}]$, acting trivially on $R$ and naturally acting on $Z[x_{1},...,x_{n}]$.

Let R(G) be a group ring and $\Gamma$ be a group acting on the group G and on the ring R.Then R(G) becomes a $\Gamma$-ring with the action ${}^{\gamma}(rg)=^{\gamma}r^{\gamma}g,r\in R,g\in G,\gamma\in\Gamma.$

Now, we recall some definitions and propositions given in [8,10] which will be used later.

Any exact sequence E of $\Gamma$-groups

[TABLE]

is called $\Gamma$-extension of the $\Gamma$-group $G$ by the $\Gamma$-group $A$. The extension $E$ is an extension with $\Gamma$-section map if there is a map $\beta:G\rightarrow X$ such that $\tau\beta=1_{G}$, and compatible with the action of $\Gamma$. In addition, if $\beta$ is a homomorphism then the extension E is called split extension.

A $\Gamma$-equivariant $G$-module $A$ is a $G$-module equipped with a $\Gamma$-module structure and the actions of $G$ and $\Gamma$ are related to each other by the equality

[TABLE]

for $g\in G$, $\sigma\in\Gamma$,$a\in A$.

The category of $\Gamma$-equivariant $G$-modules is equivalent to the category of $G\rtimes\Gamma$-modules, where $G\rtimes\Gamma$ is the semi-direct product of G and $\Gamma$.

If $E$ is an extension with $\Gamma$-section map and $A$ is a $\Gamma$-equivariant $G$-module it is called $\Gamma$-equivariant extension of $G$ by $A$. In addition, if $X$ and $G$ are $\Gamma$-equivariant G-modules it is called proper sequence of $\Gamma$-equivariant $G$-modules.

A $\Gamma$-equivariant $G$-module $F$ is called a relatively free $G\rtimes\Gamma$-module if it is a free $G$-module with basis a $\Gamma$-set, and a relatively projective $\Gamma$-equivariant $G$-module is a retract of a relatively free $\Gamma$-equivariant G-module.

The class $\mathcal{P}$ of relatively projective $\Gamma$-equivariant $G$-modules is a projective class with respect to proper sequences of $\Gamma$-equivariant $G$-modules.

For the cohomological description of the set $E^{1}_{\Gamma}(G,A)$ of equivalence classes of $\Gamma$-equivariant extensions of $G$ by $A$ the $\Gamma$-equivariant homology and cohomology of $\Gamma$-groups have been introduced as relative $Tor_{n}^{\mathcal{P}}$ and $Ext^{n}_{\mathcal{P}},n\geq 0$, in the category of $\Gamma$-equivariant $G$-modules, namely the $\Gamma$-equivariant homology and cohomology of $\Gamma$-groups are defined as follows

[TABLE]

where the functors $\bigotimes$ and $Hom$ are taken over the ring $\mathbb{Z}(G\rtimes\Gamma)$ and the groups $G$ and $\Gamma$ are trivially acting on the abelian group $\mathbb{Z}$ of integers. This cohomology is the relative group cohomology in the sense of Hochschild and Adamson [6,1].

A $\Gamma$-group is called $\Gamma$-free if it is a free group with basis a $\Gamma$-set.

Any free group $F$($G$) generated by a $\Gamma$-group $G$ becomes a $\Gamma$-free group by the following action of $\Gamma$: ${}^{\gamma}{\mid g\mid}=\mid^{\gamma}{g}\mid$, $g\in G,\gamma\in\Gamma$. The defining property of the $\Gamma$-free group $F$ with basis $E$ is that every $\Gamma$-map $E\overset{f}{\rightarrow}G$ to a $\Gamma$-group $G$ is uniquely extended to a $\Gamma$-homomorphism $F\overset{f^{\prime}}{\rightarrow}G$.

Let $\mathbb{F}$ be the projective class of $\Gamma$-free groups in the category ${\mathbf{G}}^{\Gamma}$ of $\Gamma$-groups.

There are isomorphisms

[TABLE]

for $n\geq 2$, where I(G) is the kernel of the natural homomorphism $\mathbb{Z}(G)\rightarrow\mathbb{Z}$ of $\Gamma$-equivariant G-modules, $Der(G,A)$ is the group of $\Gamma$-derivations, $L^{\mathbb{F}}_{n-1}$ and R${}^{n-1}_{\mathbb{F}}$ denote respectively the left and right derived functors with respect to the projective class $\mathbb{F}$.

Let

[TABLE]

be a short exact sequence of $\Gamma$-groups with $\Gamma$-section map and $\alpha:P\rightarrow B$ be a $\Gamma$-projective presentation of the $\Gamma$-group B. Then there is an exact sequence

[TABLE]

where V is the kernel of the $\Gamma$-homomorphism $[P,S]_{\Gamma}/[P,R]_{\Gamma}\rightarrow[B,A]_{\Gamma}$ induced by $\alpha$, R = Ker $\alpha$ and S = Ker $\tau\alpha$.

If $G$ is a $\Gamma$-group, then

(1) $H^{\Gamma}_{1}(G,A)=G/[G,G]_{\Gamma}\otimes A$, $G$ and $\Gamma$ are trivially acting on $A$.

(2) $H^{\Gamma}_{2}(G)$ is isomorphic to the group $(R\cap[P,P]_{\Gamma})/[P,R]_{\Gamma}$, where $R=Ker\alpha$ and $\alpha:P\rightarrow G$ is a $\Gamma$-projective presentation of $G$ (Hopf formula for the $\Gamma$-equivariant homology of groups).

5. $\Gamma$-algebraic K-functors

Proposition 5.1**.**

Let $\tau:G\rightarrow G^{\prime}$ be a surjective homomorphism of $\Gamma$-groups. Then one has the following exact sequence of groups:

[TABLE]

Proof.

Consider the following commutative diagram of exact columns and rows:

[TABLE]

One gets $\Gamma Ker\tau\subseteq Ker\Gamma(\tau)$, the isomorphism $Ker\tau/Ker\Gamma(\tau)\cong Ker\tau_{\Gamma}$ and the surjection $Ker\tau/\Gamma Ker\tau\rightarrow Ker\tau/Ker\Gamma(\tau)$, implying the exact sequence $e\rightarrow Ker\Gamma(\tau)/\Gamma Ker\tau\rightarrow(Ker\tau)_{\Gamma}\rightarrow Ker\tau_{\Gamma}\rightarrow e.$

According to the diagam 5.1, we finally obtain the required exact sequence. ∎

Corollary 5.2**.**

If $G\overset{\alpha}{\rightarrow}G^{\prime}\overset{\beta}{\rightarrow}G^{\prime\prime}\rightarrow e$ is an exact sequence of $\Gamma$-groups, then the sequence $G_{\Gamma}\overset{\alpha_{\Gamma}}{\rightarrow}G^{\prime}_{\Gamma}\overset{\beta_{\Gamma}}{\rightarrow}G^{\prime\prime}_{\Gamma}\rightarrow e$ is exact.

Taking into account the surjection $G_{\Gamma}\rightarrow(Ker\beta)_{\Gamma}$, follows immediately from Proposition 5.1.

Later we will need the following formula related to exact sequences of $\Gamma$-groups. Let $G$ be a $\Gamma$-group an $H$ be its normal $\Gamma$-subgroup. Consider the exact sequence of $\Gamma$-groups: $e\rightarrow H\overset{\sigma}{\rightarrow}G\rightarrow G/H\rightarrow e$. It induces the short exact sequence: $(H)_{\Gamma}\overset{\sigma_{\Gamma}}{\rightarrow}(G)_{\Gamma}\rightarrow(G/H)_{\Gamma}\rightarrow e$ implying the isomorphism $Coker\sigma_{\Gamma}\cong(G/H)_{\Gamma}$. On the other hand, one has the equality $Coker\sigma_{\Gamma}=G/H\cdot\Gamma G$, and finally one gets the needed formula

[TABLE]

We are now going to define algebraic K-functors of a ring $A$ with unit on which a group $\Gamma$ is acting and called $\Gamma$-ring.

It is easily checked that this action induces the action on the n-th general group $GL_{n}(A)$ of $n\times n$ invertible matrices over A given by ${}^{\gamma}\parallel a_{ij}\parallel=\parallel^{\gamma}a_{ij}\parallel$, and the n-th elementary group $E_{n}(A)$ of elementary $n\times n$ - matrices $e_{ij}(a),i,j\leq n,a\in A$, over $A$ becomes a $\Gamma$-subgroup of $GL_{n}(A)$. Therefore one gets a natural action of the group $\Gamma$ on the group $K_{1}(n,A)=GL_{n}(A)/E_{n}(A).$.

Definition 5.3**.**

For the unital ring A the non-stable $\Gamma$ - algebraic K-functor $K^{\Gamma}_{1}(n,A)$ is defined as $(K_{1}(n,A))_{\Gamma}$.

Consider the exact sequence $e\rightarrow E_{n}(A)\overset{\nu_{n}}{\rightarrow}GL_{n}(A)\rightarrow K_{1}(n,A)\rightarrow 0$. According to Corollary 4.2, it induces the exact sequence

$e\rightarrow(E_{n}(A))_{\Gamma}\rightarrow(GL_{n}(A))_{\Gamma}\rightarrow(K_{1}(n,A))_{\Gamma}\rightarrow 0$.

Passing to the direct limit,

$GL(A)=(GL_{n}(A))_{n\rightarrow\infty},E(A)=(E_{n}(A))_{n\rightarrow\infty},K_{1}(A)=(K_{1}(n,A))_{n\rightarrow\infty}$, one gets the action of $\Gamma$ on $GL(A)$, $E(A)$ and $K_{1}(A)$ respectively,

and the isomorphisms $(GL(A))_{\Gamma}\approx((GL_{n}(A))_{\Gamma})_{n\rightarrow\infty},(E(A))_{\Gamma}\approx((E_{n}(A))_{\Gamma})_{n\rightarrow\infty},(K_{1}(A))_{\Gamma}\approx((K_{1}(n,A))_{\Gamma})_{n\rightarrow\infty}$.

Finally, we obtain the following exact sequence

$(E(A))_{\Gamma}\overset{\nu_{\Gamma}}{\rightarrow}(GL(A))_{\Gamma}\rightarrow K^{\Gamma}_{1}(A)\rightarrow 0$.

Definition 5.4**.**

For the unital ring A the $\Gamma$ - algebraic K-functor $K^{\Gamma}_{1}(A)$ is defined as $(K_{1}(A))_{\Gamma}$.

One has isomorphisms $K^{\Gamma}_{1}(A)\cong GL(A)/(E(A)\cdot\Gamma GL(A))\cong H^{\Gamma}_{1}(GL(A))$.

Let A be a commutative $\Gamma$-ring with unit, and $A^{*}$ its multiplicative subgroup of invertible elements. There is the naturally splitting exact sequence

[TABLE]

where $det$ denotes the determinant map which is a $\Gamma$-homomorphism, SL(A) is the kernel of det, and the splitting map $A^{*}\rightarrow GL(A)$ is also a $\Gamma$-homomorphism.

This exact sequence induces the splitting exact sequence of $\Gamma$-groups,

[TABLE]

and the isomorphism of $\Gamma$-groups $K_{1}(A)\overset{\approx}{\rightarrow}SL(A)/E(A)\oplus A^{*}$, where $\Gamma$ is acting componentwise on the right side.

Finally, we obtain the following

Proposition 5.5**.**

If A is a commutative unital $\Gamma$-ring, there is an isomorphism

[TABLE]

and $K^{\Gamma}_{1}(F)=F^{*}_{\Gamma}$ for a field $F$.

For instance, if we consider the nontrivial action of the cyclic group $\mathds{Z}_{2}$ of order 2 on the field $\mathds{C}$ of complex numbers, then $K^{\mathds{Z}_{2}}_{1}(\mathds{C})=\mathds{C}^{\ast}/\mathds{Z}_{2}\mathds{C}^{\ast}$, and the subgroup $\mathds{Z}_{2}\mathds{C}^{\ast}$ is generated by the elements $(a-bi)(a+bi)^{-1},a+bi\neq 0$.

Let $St_{n}(A)$ be the $n$-th Steinberg group of the ring $A$ with generators $x^{n}_{ij}(a)$ and $i,j\leq n$, $a,b\in A$, satisfying the relations

$x^{n}_{ij}(a).x^{n}_{ij}(b)=x^{n}_{ij}(a+b),i,j\leq n,$

$[x^{n}_{ij}(a).x^{n}_{kl}(b)]=1,i\neq l,k\neq j,$

$[x^{n}_{ij}(a).x^{n}_{jk}(b)]=x^{n}_{ik}(ab),i\neq k.$

The given action of $\Gamma$ on $A$ induces the following action on the n-th Steiberg group $St_{n}(A)$ as ${}^{\gamma}(x^{n}_{ij}(a))=x^{n}_{ij}(^{\gamma}a)$ which is compatible with its defining relations.

Passing to the direct limit, $St(A)=(St_{n}(A))_{n\rightarrow\infty}$, one gets the action of $\Gamma$ on $St(A)$, and the isomorphism $(St(A))_{\Gamma}\approx((St_{n}(A))_{\Gamma})_{n\rightarrow\infty}$.

It is evident that the homomorphism $St_{n}(A)\overset{\phi_{n}}{\rightarrow}E_{n}(A)$, sending $x^{n}_{ij}(a)$ $to$ $e^{n}_{ij}(a)$, is a homomorphism of $\Gamma$-groups. and $K_{2}(n,A)$ is a $\Gamma$-subgroup of $St_{n}(A)$.

Therefore, the well-known exact sequence $0\rightarrow K_{2}(n,A)\rightarrow St_{n}(A)\rightarrow E_{n}(A)\rightarrow 0$ induces the exact sequence

$(K_{2}(n,A))_{\Gamma}\rightarrow(St_{n}(A))_{\Gamma}\rightarrow(E_{n}(A))_{\Gamma}\rightarrow 0$.

After the direct limit, we finally obtain the following exact sequence

$(K_{2}(A))_{\Gamma}\rightarrow(St(A))_{\Gamma}\overset{\phi_{\Gamma}}{\rightarrow}(E(A))_{\Gamma}\rightarrow 0$.

Definition 5.6**.**

For the unital ring A the $\Gamma$ - algebraic K-functor $K^{\Gamma}_{2}(A)$ is defined as $Ker\phi_{\Gamma}$.

It is easily checked one has the natural surjection $(K_{2}(A))_{\Gamma}\rightarrow K^{\Gamma}_{2}(A)$, and the equality $\Gamma(K_{2}(A))=K_{2}(A)/K_{2}(A)\cap\Gamma(St(A))$ holds.

It follows immediately that $K^{\Gamma}_{2}(F)=0$ for finite field F.

Let A be an unital ring and $Sym(A)$ be the symbol group of the ring $A$ generated by the elements $\{u,v\}$, $u,v\in A^{*}$, satisfying the following relations

$(S_{0})$ $\langle u,1-u\rangle=1,u\neq 1,1-u\in A^{*},$

$(S_{1})$ $\langle uu^{\prime},v\rangle=\langle u,v\rangle\langle u^{\prime},v\rangle$,

$(S_{2})$ $\langle u,vv^{\prime}\rangle=\langle u,v\rangle\langle u,v^{\prime}\rangle$.

where $A^{*}$ denotes the multiplicative group of invertible elements of the ring A [5]. By Matsumoto’s theorem, the groups $Sym(A)$ and $K_{2}(A)$ are isomorphic when A is a field [11].

If $A$ is a $\Gamma$-ring, the action of $\Gamma$ induces a natural action on the group $Sym(A)$ given by ${}^{\gamma}\langle u,v\rangle=\langle^{\gamma}u,^{\gamma}v\rangle,u,v\in A^{\star}$. The $\Gamma$-equivariant symbol group $Sym^{\Gamma}(A)$ is defined as $Sym^{\Gamma}(A)=(Sym(A))_{\Gamma}$.

In particular, if $\Gamma$ is the group $A^{*}$ of invertible elements of $A$, the action of $A^{*}$ on $A$ by conjugation induces an action on $Sym(A)$ given by ${}^{\gamma}\langle u,v\rangle=\langle\gamma u\gamma^{-1},\gamma v\gamma^{-1}\rangle,\gamma,u,v\in A^{\star}$. One gets the $A^{*}$-equivariant symbol group $Sym^{A^{*}}(A)=(Sym(A))_{A^{*}}$.

Now, Matsumoto’s theorem for $\Gamma$-fields will be given. Namely,

Theorem 5.7**.**

If F is a $\Gamma$-field, then the sequence

[TABLE]

is exact, where $\phi$ denotes the canonical surjection $St(F)\rightarrow E(F)$.

$Proof$. As we see, the action of the group $\Gamma$ on the field $F$ induces its action on $Sym(A)$ and also on $K_{2}(F)$ as a subgroup of the Steinberg group $St(F)$. It is easily checked that the Matsumoto’s isomorphism $Sym(F)\cong K_{2}(F)$ is compatible with the action of $\Gamma$. Therefore, this implies the isomorphism $(Sym(F))_{\Gamma}\cong(K_{2}(F))_{\Gamma}$. It remains to apply Proposition 5.1 for the homomorphism $\phi$. This completes the proof.

Let $A$ be a non-commutative local ring such that $A/Rad(A)\neq F_{2}$. Consider A as a $\Gamma$-ring, where $\Gamma$ is the group $A^{\ast}$ of units acting on A by conjugation.

There is an exact sequence [5,10]

[TABLE]

where the group $U(A)$ is generated by the elements $\langle u,v\rangle$,$u,v\in A^{*}$, satisfying the following relations

$(U_{0})$ $\langle u,1-u\rangle=1,u\neq 1,1-u\in A^{*},$

$(U_{1})$ $\langle uv,w\rangle=^{u}\langle v,w\rangle\langle u,w\rangle$,

$(U_{2})$ $\langle u,vw\rangle\langle v,wu\rangle\langle w,uv\rangle=1$,

and ${}^{u}\langle v,w\rangle=\langle uvu^{-1},uwu^{-1}\rangle$ [5].

Generalization of Matsumoto’s theorem for non-commutative local $\Gamma$-rings.

Theorem 5.8**.**

If A is a non-commutative local $A^{*}$-ring such that $A/Rad(A)\neq F_{2}$, there is an exact sequence

[TABLE]

[TABLE]

where $A^{*}$ is the group of units of $A$ acting on $A$ by conjugation.

$Proof$. It is easily checked that the group $U(A)$ becomes $A^{*}$-group with respect to the action ${}^{\gamma}\langle v,w\rangle=\langle\gamma v\gamma^{-1},\gamma w\gamma^{-1}\rangle$ and by results of [5,10] there is a surjective $A^{*}$-homomorphism $U(A)\rightarrow Sym(A)$. Moreover, it is proven that the group $(Sym(A))_{A\star}$ is abelian and isomorphic to $(U(A))_{A^{\star}}$. One gets the exact sequence $(K_{2}(A))_{\Gamma}\rightarrow(U(A))_{\Gamma}\overset{\tau_{\Gamma}}{\rightarrow}[A^{*},A^{*}]/\Gamma([A^{*},A^{*}])$. By applying Proposion 5.1 again for the homomorphism $\tau$, we obtain the needed exact sequence.

Definition 5.9**.**

Let $\mathds{A}$ be an arbitrary category. It is said a group $\Gamma$ is acting on $\mathds{A}$, if for any pair (A,B) of objects of $\mathds{A}$ the corresponding set H(A,B) of morphisms is a $\Gamma$-set such that for any $f:A\rightarrow B$ and $g:B\rightarrow C$ the following equality hold: ${}^{\gamma}(gf)=^{\gamma}(g)^{\gamma}(f)$, and ${}^{\gamma}(1_{A})=1_{A}$ for any unit morphism $1_{A}:A\rightarrow A$ of the category $\mathds{A}$, and any element $\gamma\in\Gamma$. This category will be denoted $\mathds{A}^{\Gamma}$ and called $\Gamma$-category.

Examples.

The category $\mathcal{G}r^{\Gamma}$ of $\Gamma$-groups is the category of groups on which the group $\Gamma$ is acting. The category $\mathcal{R}^{\Gamma}$ of $\Gamma$-rings is the category of rings on which the group $\Gamma$ is acting.

Definition 5.10**.**

It is said that a covariant functor $T:\mathds{A}^{\Gamma}\rightarrow\mathcal{G}r^{\Gamma}$ is given, if T is a covariant functor from the category $\mathds{A}$ to the category $\mathcal{G}r$ of groups such that $T(^{\gamma}(f))=^{\gamma}(T(f))$ for any morphism f of $\mathds{A}$ and any element $\gamma$ of $\Gamma$.

Definition 5.11**.**

It is said that $(\mathrm{F},\tau,\delta)$ is a cotriple of the $\Gamma$-category $\mathds{A}^{\Gamma}$ if it is a cotriple of the category $\mathds{A}$, the functor $F$ is compatible with the action of $\Gamma$, and the morphisms $\tau,\delta$ are $\Gamma$-morphisms.

Let $(\mathrm{F},\tau,\delta)$ be a cotriple on the category $\mathds{A}^{\Gamma}$, where $\tau:F(A)\rightarrow A,\delta:F(A)\rightarrow F(F(A))$. It induces the augmented simplicial object

[TABLE]

$\lambda^{n}_{i}=F_{i}\tau F_{n-i},s^{n}_{i}=F_{i}\delta F_{n-i}$ for $0\leq i\leq n$.

Let $L_{\ast}=(L_{n},\lambda^{n}_{i},s^{n}_{i},n\geq 0,n\in\mathds{Z})$ be a simplicial group with bord and degeneracy operators $\lambda^{n}_{i}$ and $s^{n}_{i}$ respectively.

Definition 5.12**.**

It is said that the group $\Gamma$ is acting on the simplicial group $L_{\ast}$, if it is acting on the groups $L_{n},n\geq 0,n\in\mathds{Z}$, such that the homomorphisms $\lambda^{n}_{i},s^{n}_{i},n\geq 0,n\in\mathds{Z}$, are $\Gamma$-homomorphisms. The quotient simplicial group $L^{\Gamma}_{\ast}$ of $L_{\ast}$, $L^{\Gamma}_{\ast}=((L_{n})_{\Gamma},\lambda^{n}_{i,\Gamma},s^{n}_{i,\Gamma},n\geq 0,n\in\mathds{Z})$, where the homomorphisms $\lambda^{n}_{i,\Gamma}$ and $s^{n}_{i,\Gamma}$ are induced by $\lambda^{n}_{i}$ and $s^{n}_{i}$ respectively, is called Connes simplicial group of $L_{\ast}$ with respect to the action of $\Gamma$.

The action of $\Gamma$ on the simplicial group $L_{\ast}$ induces its action on the homotopy groups of $L_{\ast}$, and the homotopy groups $\pi_{n}(L^{\Gamma}_{\ast}),n\geq 0,n\in\mathds{Z}$, of the Connes simplicial group $L^{\Gamma}_{\ast}$ of $L_{\ast}$, are called $\Gamma$-equivariant homotopy groups of the simplicial group $L_{\ast}$.

It is evident that the action of the group $\Gamma$ on pseudo-simplicial groups [7] is defined analogously.

A similar definition of the group action on a chain complex of modules was given in [10]. Its quotient chain complex could be called Connes chain complex. The homology groups of the quotient chain complex were called $\Gamma$-equivariant homology groups of the chain complex.

Remark 5.13*.*

Connes complex firstly appear for Hochschild complex under the cyclic group action [11] which is the action of the group of integers on the Hochschild complex [10].

One has the exact sequence

[TABLE]

inducing the exact sequence

[TABLE]

The action of the group $\Gamma$ on the simplicial group $L_{\ast}$ induces an action of $\Gamma$ on the first sequence implying the second exact sequence. It is evident the sequence similar to the sequenc 5.3 holds for the augmented $\Gamma$-simplicial group $L^{+}_{\ast}$.

Definition 5.14**.**

Let $T$ be a covariant functor from the $\Gamma$-category $\mathds{A}^{\Gamma}$ to the category $\mathcal{G}r^{\Gamma}$ of $\Gamma$-groups. The non-abelian left $\Gamma$-derived functors $L^{\Gamma}T_{n}(A)$ of $T$ with respect to the cotriple $\mathds{F}=(\mathrm{F},\tau,\delta)$ are defined as follows:

[TABLE]

Non-abelian $\Gamma$-derived functors of the functor $T$ could also be defined with respect to the projective class of the category $\mathds{A}^{\Gamma}$ in the sense of Eilenberg - Moore by using $\Gamma$-pseudo-simplicial groups. All these definitions of non-abelian $\Gamma$-derived functors generalized the well-known non-abelian derived functors of covariant functors when the group $\Gamma$ is acting trivially [7].

Let $G$ be a $\Gamma$-group and $A$ be a $\Gamma$-ring, and let $F(G)$ and $F(A)$ be the free group and the free ring, generated by the set $(|g|),g\in G$, and by the set $(|a|),a\in A$, respectively. Define the action of the group $\Gamma$ on $F(G)$ by ${}^{\gamma}|g|=|^{\gamma}g|$, and on $F(A)$ by ${}^{\gamma}|a|=|^{\gamma}a|,\gamma\in\Gamma$. Then the free cotriples induced by $G$ and $A$ become cotriples of the categories $\mathcal{G}r^{\Gamma}$ and $\mathcal{R}^{\Gamma}$ respectively.

Let $A$ be a unital $\Gamma$-ring and $GL(A)$ the induced $\Gamma$-group. We are going to define Swan’s $\Gamma$-algebraic K-functors as $\Gamma$-derived functors of the functor $Gl(-)$ with respect to the free cotriple, namely

Definition 5.15**.**

For a $\Gamma$-ring $A$ the Swan’s $\Gamma$-algebraic K-functors $K_{n}^{S,\Gamma}(A)$ are defined as follows

$K_{n}^{S,\Gamma}(A)=\pi_{n-2}(GLF_{\ast}(A))^{\Gamma}$ for $n\succeq 3$,

and the groups $K_{1}^{S,\Gamma}(A)$, $K_{2}^{S,\Gamma}(A)$ are defined by the exact sequence $0\rightarrow K_{2}^{S,\Gamma}(A)\rightarrow\pi_{0}(GLF_{\ast}(A))^{\Gamma}\rightarrow(GL(A))_{\Gamma}\rightarrow K_{1}^{S,\Gamma}(A)\rightarrow 0,$

where $(GLF_{\ast}(A))^{\Gamma}$ is the Connes simplicial group of the $\Gamma$-simplicial group $GLF_{\ast}(A)$ induced by the $\Gamma$-ring $A$.

To define Swan’s relative $\Gamma$-algebraic K-functors consider the augmented simplicial ring $F^{+}_{\ast}(A)=F_{\ast}(A)\overset{\tau}{\rightarrow}A$ for the $\Gamma$-ring $A$.

Let $f:A\rightarrow B$ be a surjective homomorphism of $\Gamma$-rings with unit. It induces a surjective morphism of augmented $\Gamma$-simplicial groups $GL(F^{+}_{\ast}(f):GL(F^{+}_{\ast}(A))\rightarrow GL(F^{+}_{\ast}(B))$, and therefore the morphism $(GL(F^{+}_{\ast}(f))_{\Gamma}:(GL(F^{+}_{\ast}(A)))^{\Gamma}\rightarrow(GL(F^{+}_{\ast}(B)))^{\Gamma}$.

Define $K^{S,\Gamma}_{n}(A,I)=\pi_{n}(Ker((GL(F^{+}_{\ast}(f))_{\Gamma})),n>2$, where $I=Kerf$. The groups $K_{1}^{S,\Gamma}(A,I)$ and $K_{2}^{S,\Gamma}(A,I)$ are defined by the exact sequence: $0\rightarrow K_{2}^{S,\Gamma}(A,I)\rightarrow\pi_{0}Ker(GLF_{\ast}(f))^{\Gamma}\rightarrow Ker(GL(f))_{\Gamma}\rightarrow K_{1}^{S,\Gamma}(A,I)\rightarrow 0$. When the group $\Gamma$ is trivially acting on $A$, we recover the Swan’s algebraic K-functors [17].

We also propose the definition of Quillen’s $\Gamma$-algebraic K-functors by the following way.

Consider the free resolution $F_{\ast}(A)\overset{\tau}{\rightarrow}A$ of the unital $\Gamma$-ring $A$. The map $\tau$ induces a map of simplicial rings $\varphi_{\ast}:F_{\ast}\rightarrow(A)_{\ast}$, where $(A)_{\ast}$ is the constant simplicial ring, $A_{n}=A$ for all $n\geq 0$, and $\varphi_{n}=\tau\tilde{\lambda}^{n-1}_{0},\tilde{\lambda}^{n-1}_{0}=\lambda^{1}_{0}\cdot\cdot\cdot\lambda^{n-1}_{0}$ for $n\geq 1$, and $\varphi_{0}=\tau$. Denote $I_{\ast}$ the kernel of $\varphi_{\ast}$ which is an acyclic ideal of $F_{\ast}$.

Let $A$ be a unital $\Gamma$-ring. The action of $\Gamma$ induces the action on the simplicial ring $I_{\ast}$, then on the simplicial group $GL(I_{\ast})$ and therefore the action on the space $BGL(I_{\ast})$ which is a homeomorphim, and if $f:[0,1]\rightarrow BGL(I_{\ast})$ is a continuous map, $(^{\gamma}f)(t)=^{\gamma}(f(t)),t\in[0,1]$. This space could be used for the definition of Quillen’s K-groups, since we have $K^{Q}_{n}(A)\cong\pi_{n-1}(BGL(I_{\ast})),n>1$. The action of $\Gamma$ on $BGL(I_{\ast})$ induces its action on the homotopy groups of $BGL(I_{\ast})$, therefore on $K^{Q}_{n}(A)$, and one has the isomorphism $(K^{Q}_{n}(A))_{\Gamma}\cong(\pi_{n-1}(BGL(I_{\ast})))_{\Gamma},n>1$.

Finally, we arrive to the definition of Quillen’s algebraic $\Gamma$-functors $K^{Q,\Gamma}_{n}(A),n>1$, as equal to the homotopy group $\pi_{n-1}(BGL(I_{\ast})/\Gamma)$ of the quotient space $BGL(I_{\ast})/\Gamma$ by the action of the discrete group $\Gamma$.

Theorem 5.16**.**

1. For a homomorphism $f:A\rightarrow B$ of $\Gamma$-rings with unit there is a long exact sequence

$\cdot\cdot\cdot\rightarrow K^{S,\Gamma}_{n+1}(B)\rightarrow K^{S,\Gamma}_{n}(A,I)\rightarrow K^{S,\Gamma}_{n}(A)\rightarrow K^{S,\Gamma}_{n}(B)\rightarrow\cdot\cdot\cdot\rightarrow K^{S,\Gamma}_{2}(B)\rightarrow K^{S,\Gamma}_{1}(A,I)\rightarrow K^{S,\Gamma}_{1}(A)\rightarrow K^{S,\Gamma}_{1}(B)$,

where $I=Kerf$, and one has a natural homomorphism $K^{S,\Gamma}_{n}(I)\rightarrow K^{S,\Gamma}_{n}(A,I),n\geq 1$;

2. If A is a free ring on which a group $\Gamma$ is acting, then $K^{S,\Gamma}_{n}(A)=o$ for $n\geq 1$;

3. The relation between Swan’s and Quillen’s $\Gamma$-algebraic K-functors is expressed by the following exact sequence

$\pi_{n-2}(\Gamma GLF^{+}_{\ast}(A))\rightarrow(K^{Q}_{n}(A))_{\Gamma}\rightarrow K^{S,\Gamma}_{n}(A)\rightarrow\pi_{n-3}(\Gamma GLF^{+}_{\ast}(A))$,

$n>1$, $F^{+}_{\ast}(A):F_{\ast}(A)\rightarrow A$ is the free resolution of the $\Gamma$-ring A;

4. There is the central exact sequence $0\rightarrow K^{\Gamma}_{2}(A)\rightarrow K^{S,\Gamma}_{2}(A)\rightarrow Ker\nu_{\Gamma}\rightarrow e$, where $\nu_{\Gamma}$ is induced by the inclusion $\nu:E(A)\rightarrow GL(A)$.

$Proof$. 1. Induced by the short exact sequence of augmented $\Gamma$-simplicial groups $e\rightarrow Ker((GL(F^{+}_{\ast}(f))_{\Gamma})\rightarrow(GL(F^{+}_{\ast}(A)))_{\Gamma}\rightarrow(GL(F^{+}_{\ast}(B)))_{\Gamma}\rightarrow e$; the homomorphism $K^{S,\Gamma}_{n}(I)\rightarrow K^{S,\Gamma}_{n}(A,I)$ is induced by the composite of morphisms $(GL(F^{+}_{\ast}(I)))_{\Gamma}\rightarrow(GL(F^{+}_{\ast}(f))_{\Gamma}\rightarrow(GL(F^{+}_{\ast}(A)))_{\Gamma}$ for $n\geq 1$.

2. First it will be shown when A is $\Gamma$-free ring. In that case the augmented $\Gamma$-simplicial group $GLF^{+}_{\ast}(A)$ is left contractible and therefore all its homotopy groups are trivial. Let $F(X)$ be a free group with basis the subset $X$ on which the group $\Gamma$ is acting, then $F(F(X))$ is a $\Gamma$-free group, and consider the $\Gamma$-homomorphisms $F(X)\overset{\sigma}{\rightarrow}F(F(X))\overset{\tau}{\rightarrow}F(X)$ induced by the injection $X\rightarrow F(X)$ and the identity map $F(X)\rightarrow F(X)$ respectively. Therefore $K^{S,\Gamma}_{n}(F(X))$ is isomorphic to a subgroup of $K^{S,\Gamma}_{n}(F(F(X))$ which is trivial for $n\geq 1$.

3. By using the sequence 5.4 for the augmented $\Gamma$-simplicial group $GL(F^{+}_{\ast}(A))$ we obtain the following exact sequence $\pi_{n-2}(\Gamma GL(F^{+}_{\ast}(A))\rightarrow(\pi_{n-2}(GL(F^{+}_{\ast}(A)))_{\Gamma}\rightarrow\pi_{n-2}((GL(F^{+}_{\ast}(A)))_{\Gamma})\rightarrow\pi_{n-3}(\Gamma GL(F^{+}_{\ast}(A)))$, $n>1$, implying the exact sequence $\pi_{n-2}(\Gamma GL(F^{+}_{\ast}(A))\rightarrow(K^{S}_{n}(A))_{\Gamma}\rightarrow K^{S,\Gamma}_{n}(A)\rightarrow\pi_{n-3}(\Gamma GL(F^{+}_{\ast}(A)))$, $n>1$. One has the isomorphisms $(K^{Q}_{n})_{\Gamma}\cong(\pi_{n-1}BGL(I_{\ast}))_{\Gamma}\cong(\pi_{n-2}GL(I_{\ast}))_{\Gamma}\cong(\pi_{n-2}GLF^{+}_{\ast}(A)_{\Gamma}\cong(K^{S}_{n}(A))_{\Gamma},n>1$, and we get the required exact sequence.

4. It is easily checked $\pi_{0}((GL(F^{+}_{\ast}(A)))_{\Gamma})$ is isomorphic to $(St(A))_{\Gamma}$. Therefore $K^{S}_{2}(A)$ is isomorphic to the kernel of the homomorhism $(St(A))_{\Gamma}\rightarrow(GL(A))_{\Gamma}$, implying the required exact sequence which is central, since $K_{2}(A)$ is the center of $St(A)$. This completes the proof of the theorem.

Remark 5.17*.*

Let A be a $\Gamma$-ring without unit and $A+\mathds{Z}$ be the unitization of A. The action of $\Gamma$ on A induces its action on $A+\mathds{Z}$ by ${}^{\gamma}(a,z)=(^{\gamma}a,z),a\in A,\gamma\in\Gamma$. Then there is the short exact sequence $0\rightarrow(K^{Q}_{n}(A))_{\Gamma}\rightarrow(K^{Q}_{n}(A+\mathds{Z}))_{\Gamma}\rightarrow(K^{Q}_{n}(\mathds{Z}))_{\Gamma}\rightarrow 0$ induced by the canonical $\Gamma$-homomorphisms $i:A\rightarrow A+\mathds{Z}$ and $p:A+\mathds{Z}\rightarrow\mathds{Z}$.

For a commutative $\Gamma$-ring A with unit the product operation

[TABLE]

is defined. For this aim we will follow Milnor’s construction of the map $GL(A)\times GL(A)\rightarrow K_{2}(A)$ by using the symbol $[x,y]$, $x,y\in GL(A)$ given in [13]. Consider the canonical homomorphism $\tau:St(A)\rightarrow E(A)$ and let $x,y\in E(A)$ such that $xy=yx$. There is a correctly defined element of $K_{2}(A)$ denoted $x\ast y$ by taking $v,z\in St(A)$ such that $\tau(v)=x,\tau(z)=y$. Then $x\ast y=vzv^{-1}z^{-1}$ and $\tau(vzv^{-1}z^{-1})=e$.

Now the symbol $[x,y]$ is defined as follows (see [13]). For $x\in GL_{m}(A)$ and $y\in GL_{n}(A)$ denote $x\otimes y$ the corresponding matrix to the induced automorphism of $A^{m}\otimes A^{n}$. Let I denote the $m\times m$ identity matrix and $I^{\prime}$ be the $n\times n$ identity matrix. The matrices $diag(x\otimes I^{\prime};x^{-1}\otimes I^{\prime};I\otimes I^{\prime})$ and $diag(I\otimes y,I\otimes I^{\prime};I\otimes y^{-1})$ of $E_{3mn}(A))$ commute with each other, and the symbol $[x,y]$ is equal to

[TABLE]

The map $GL(A)\times GL(A)\rightarrow K_{2}(A)$ induced by the symbol $[x,y]$ is skew-symmetric, bimultiplicative, and gives rise to a homomorphism $K_{1}(A)\times K_{1}(A)\rightarrow K_{2}(A)$ [13]. Moreover, the following properties hold:

If u and $1-u$ are units, then $[u,1-u]=1$ and $[u,-u]=1$.

We are able now to show that the defined map $GL(A)\times GL(A)\rightarrow K_{2}(A)$ is a $\Gamma$-map if the ring A is a $\Gamma$-ring. If $\gamma\in\Gamma$, one has ${}^{\gamma}(x,y)=(^{\gamma}x,^{\gamma}y)$ and ${}^{\gamma}(x\ast y)=^{\gamma}(vzv^{-1}z^{-1})=^{\gamma}{v}^{\gamma}{z}(^{\gamma}{v})^{-1}(^{\gamma}{z})^{-1}=(^{\gamma}x\ast^{\gamma}y)$. Therefore,

$diag(^{\gamma}x\otimes I^{\prime};(^{\gamma}x)^{-1}\otimes I^{\prime};I\otimes I^{\prime})\ast diag(I\otimes^{\gamma}y,I\otimes I^{\prime};I\otimes(^{\gamma}y)^{-1})=^{\gamma}(diag(x\otimes I^{\prime};x^{-1}\otimes I^{\prime};I\otimes I^{\prime}))\ast^{\gamma}(diag(I\otimes y,I\otimes I^{\prime};I\otimes y^{-1}))=^{\gamma}(diag(x\otimes I^{\prime};x^{-1}\otimes I^{\prime};I\otimes I^{\prime})\ast diag(I\otimes y,I\otimes I^{\prime};I\otimes y^{-1}))$ showing that the homomorphism $K_{1}(A)\otimes K_{1}(A)\rightarrow K_{2}(A)$ is a $\Gamma$-homomorphism. This implies a homomorphism $(K_{1}(A)\otimes K_{1}(A))_{\Gamma}\rightarrow(K_{2}(A))_{\Gamma}$. Taking into account the isomorphism $K^{\Gamma}_{1}(A)\otimes K^{\Gamma}_{1}(A)\cong(K_{1}(A)\otimes K_{1}(A))_{\Gamma}$ and the canonical surjection $(K_{2}(A))_{\Gamma}\rightarrow K^{\Gamma}_{2}(A)$, we obtain the required homomorphism $K^{\Gamma}_{1}(A)\otimes K^{\Gamma}_{1}(A)\rightarrow K^{\Gamma}_{2}(A)$.

The symbol $[x,y]$ is closely related to the Milnor’s algebraic K-theory which we will define now.

Let A be a ring with unit. The n-th Milnor algebraic K-group $K^{M}_{n}(A)$ is defined as

[TABLE]

where $a_{i}\in A^{\star}$, $n\geq 2$ and $\{a_{1}\otimes a_{2}...\otimes a_{n}\}$ is the Abelian subgroup generated by the elements $a_{1}\otimes a_{2}...\otimes a_{n}$ satisfying the relation $a_{i}+a_{i+1}=1$ called the Steinberg relation [14].

If a group $\Gamma$ is acting on the ring A, it induces a natural action on $K^{M}_{n}(A)$ given by its action on $(A^{\star})^{\otimes n}$ componentwise. Therefore, if A is a $\Gamma$-ring, the nth Milnor’s $\Gamma$-algebraic K-functor $K^{M,\Gamma}_{n}(A)$ is defined as

[TABLE]

where $a_{i}\in A^{\star}$ and $n\geq 2$.

Matsumoto proved that the homomorphism $A^{\star}\times A^{\star}/a\otimes(1-a)\rightarrow K_{2}(A)$, $a\neq 0,1$, induced by the symbol $[u,v]$, is an isomorphism, if A is a field and as we have shown, it is a $\Gamma$-map. As a consequence, we obtain

Theorem 5.18**.**

Matsumoto’s theorem for $\Gamma$-fields.

If F is a $\Gamma$-field, one has the isomorphism

$K^{M,\Gamma}_{2}(F)\cong(K_{2}(F))_{\Gamma}$* and the exact sequence*

$0\rightarrow(K_{2}(F)\cap\Gamma St(F))/\Gamma K_{2}(F)\rightarrow K^{M,\Gamma}_{2}(F)\rightarrow K^{\Gamma}_{2}(F)\rightarrow 0$.

It is well-known (14[,18]) that the Milnor algebraic K-theory is closely related to the Witt ring of quadratic forms and to the Bloch higher Chow groups. Now we will attempt to show these relations for the case of $\Gamma$-fields by considering the Milnor’s conjectures.

Let $F$ be a field of characteristic different from 2. The Witt ring $W(F)$ is consisting of equivalence classes of non-degenerate quadratic forms. It has a filtration

[TABLE]

given by the powers $I^{n}(F)$ of the fundamental ideal $I(F)$ of even-dimensional quadratic forms. The quotients $I^{n}(F)/I^{n+1}(F)$ play an important role in the study of the Witt ring $W(F)$.

Milnor conjectures the isomorphism $\kappa^{M}_{n}(F)\cong I^{n}/I^{n+1}$, where $\kappa^{M}_{n}(F)$ is the n-th Milnor K-group modulo 2 of the field $F$, proved in [18].

For the Milnor conjecture about Chow groups, first we will give the definition of the Chow group $CH^{k}(F,n)$ of the field $F$.

Let $V$ be a vector space over the field $F$. The projective space $P(V)$ is the set of equivalence classes of $V\backslash 0$ under the relation $\sim$ defined by $x\sim y$ if there is non zero element $\lambda\in F$ such that $x=\lambda y$. If $V=F^{n+1}$, the relevant projective space is denoted $P^{n}_{F}$,

[TABLE]

not all $a_{i}=0$. and $(a_{1},...,a_{n+1})\sim(a^{\prime}_{1},...,a^{\prime}_{n+1})$ if there is a non zero element $\lambda\in F$ such that $a^{\prime}_{i}=\lambda a_{i},i=1,...,n+1.$

Let $Z_{k}(F,n)$ be the free abelian group on the set of $k$- dimensional subvarieties of $P^{n}_{F}$ called the group of $k$- dimensional algebraic cycles on $P^{n}_{F}$. Denote $B_{k}(F,n)$ the subgroup of $k$-cycles rationally equivalent to zero. Then the Chow group $CH^{k}(F,n)$ is defined as $Z_{k}(F,n)/B_{k}(F,n)$. It is clear the Chow group $CH^{k}(F,n)$ is trivial for $k\succ n$.

Milnor conjectures the isomorphism $K^{M}_{n}(F)\cong CH^{n}(F,n)$, proved in [15,19,20].

Let $F$ be a $\Gamma$-field. Denote $\kappa^{M,\Gamma}_{n}(F)$ the n-th Milnor K-group modulo 2 and $CH^{n,\Gamma}(F,n)$ the n-th Chow group of the $\Gamma$-field $F$ which will be defined a bit later.

The Milnor conjectures for the $\Gamma$-field $F$ take the form:

Theorem 5.19**.**

If $F$ is a $\Gamma$-field, there are isomorphisms

1. $\kappa^{M,\Gamma}_{n}(F)\cong(I^{n}/I^{n+1})_{\Gamma},$

2. $K^{M,\Gamma}_{n}(F)\cong CH^{n,\Gamma}(F,n).$

$Proof.$ For the first isomorphism we have to define the group $\kappa^{M,\Gamma}_{n}(F)$ and the action of $\Gamma$ on $I^{n}/I^{n+1}.$ The action of $\Gamma$ on $K^{M}_{n}(F)$ induces its action on $\kappa^{M}_{n}(F)=K^{M}_{n}(F)/2K^{M}_{n}(F)$, and define $\kappa^{M,\Gamma}_{n}(F)$ as equal to $(\kappa^{M}_{n}(F))_{\Gamma}.$

The ideal $I^{n}(F)$ is additively generated by the $n$-fold Pfister forms $<<a_{1},...,a_{n}>>=\bigotimes^{n}_{i=1}<1,-a_{i}>$ which is a quadratic form of dimension $2^{n}$, $a_{i}\in F^{\star}$. For instance, the 1-fold and 2-fold Pfister forms are respectively $<<a>>\cong<1,-a>=x_{1}^{2}-ax_{2}^{2}$ and $<<a,b>>\cong<1,-a,-b-,ab>=x_{1}^{2}-ax_{2}^{2}-bx_{3}^{2}+abx_{4}^{2}$.

If $F$ is a $\Gamma$-ring, the action of the group $\Gamma$ on the field $F$ induces a natural action of $\Gamma$ on the generators of the power $I^{n}(F)$ given by ${}^{\gamma}(<<a_{1},...,a_{n}>>)=<<^{\gamma}a_{1},...,^{\gamma}a_{n}>>$, $\gamma\in\Gamma$. The above given filtration of the powers of the fundamental ideal $I(F)$ becomes a filtration of $\Gamma$-groups. It is easily checked that this action is compatible with the equivalence of quadratic forms.

For $\Gamma$-fields the quotient corresponding to the quotient $I^{n}/I^{n+1}$ would be its equivariant form $(I^{n}/I^{n+1})_{\Gamma}$ which is equal to $I^{n}/I^{n+1}\cdot\Gamma I^{n}$.

There is a homomorphism $\alpha^{M}_{n}$ from $\kappa^{M}_{n}(F)$ to the quotient $I^{n}/I^{n+1}$ defined by Milnor and given by $(a_{1},...,a_{n})\rightarrow<<a_{1},...,a_{n}>>$.

We need to show that the homomorphism $\alpha^{M}_{n}$ is a $\Gamma$-map. In effect, it suffices to show this fact on the set of generators $\{(a_{1},...,a_{n}),a_{i}\in F^{\star}\}$. One has ${}^{\gamma}(a_{1},...,a_{n})=(^{\gamma}a_{1},...,^{\gamma}a_{n})=<<^{\gamma}a_{1},...,^{\gamma}a_{n}>>=^{\gamma}(<<a_{1},...,a_{n}>>)$. This implies the isomorphism

[TABLE]

and the proof of the first Milnor conjecture.

Regarding the second Milnor conjecture, we will define the action of the group $\Gamma$ on the Chow group $CH^{k}(F,n)$ induced by its action on the field $F$. It is clear that it induces an action on the projective space $P^{n}_{F}$ defined by ${}^{\gamma}(a_{1},...,a_{n})=(^{\gamma}a_{1},...,^{\gamma}a_{n})$. This action induces an action on the set of $k$-dimensional subvarieties and therefore, on the groups $Z_{k}(F,n)$ and $B_{k}(F,n)$. Finally, the action of $\Gamma$ on the Chow group $CH^{k}(F,n)$ is induced by its action on $Z_{k}(F,n)$. The $\Gamma$-Chow group $CH^{k,\Gamma}(F,n)$ is defined as equal to $(CH^{k}(F,n))_{\Gamma}$.

There is a well-known homomorphism $\beta^{M}_{n}:K^{M}_{n}(F)\rightarrow CH^{n}(F,n)$ and Milnor raised the homomorphism $\beta^{M}_{n}$ is an isomorphism [14]. For its definition we need the product map of Chow groups. The product map $CH^{p}(F,q)\times CH^{r}(F,s)\rightarrow CH^{p+r}(F,q+s)$ is given by intersecting algebraic cycles, namely the product $|y||z|$ in $CH^{p+r}(F,q+s)$ is the sum of the varieties of the intersection $Y\bigcap Z$ which all have codimension $q+s$, where $Y$ and $Z$ are subvarieties of $(F,q)$ and $(F,s)$ respectively. It is easily checked that the product map is a $\Gamma$-map with componentwise action of $\Gamma$ on the product.

The homomorphism $\beta^{M}_{n}$ is induced by the homomorphism $\beta^{M}_{1}:F^{\ast}\rightarrow CH^{1}(F,1)$ of abelian groups, sending the unit $e$ of $F^{\ast}$ to $0\in CH^{1}(F,1)$, and the element $a,a\neq e,$ to the class of the element $(a,1)$ of the projective space $P^{1}_{F}$. Since $\beta^{M}_{1}$ and the product map are $\Gamma$-maps, it follows that the induced homomorphism $\beta^{M}_{n}$ is a $\Gamma$-homomorphism. This implies the isomorphism

[TABLE]

and completes the proof of Milnor’s conjectures for $\Gamma$-fields.

Corollary 5.20**.**

For $\Gamma$-fields the relationship between Chow groups and Milnor K-functor $K_{2}$ takes the following form:

[TABLE]

Follows immediately from Theorems 5.18 and 5.19.

6. Acknowledgment

I would like to thank the referee for carefully reading the manuscript and for giving such constructive remarks which substantially helped improving the quality of the presentation of the paper.