张亚坤
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, 唐国平
中国科学院大学数学科学学院, 北京 100049
摘要:
G为阶数小于6的非平凡群,
p为整除群
G的阶数的素数,确定
k≥2时
K2(
$\mathbb{Z}$G/
pk$\mathbb{Z}$G)的结构。
关键词:
K2-群Dennis-Stein符号群环
Let
G be a finite abelian
p-group, and Γ the maximal
$\mathbb{Z}$-order of
$\mathbb{Z}$G in
$\mathbb{Q}$G. By Propo sition 2.2 in Ref.[
1], Γ is the integral clo sure of
$\mathbb{Z}$G in
$\mathbb{Q}$G, and |
G|Γ
$\subseteq $$\mathbb{Z}$G$\subseteq $Γ. When 1 < |
G| < 6, we find that calculation of
K2(
$\mathbb{Z}$G/
pk$\mathbb{Z}$G)(
k≥2) is equivalent to calcula tion of
K2 of
K2(
$\mathbb{Z}$G) and
K2(Γ), which are listed in
Table 1. A brief in troduction to the explicit structures of
K2(
$\mathbb{Z}$G) and
K2(Γ) can be found in Ref.[
2].
Table 1
Table 1 Structures of K2($\mathbb{Z}$G) and K2(Γ) G | Γ | K2($\mathbb{Z}$G) | K2(Γ) | C2 | $\mathbb{Z}$2 | (C2)2 | (C2)2 | C3 | $\mathbb{Z}$⊕$\mathbb{Z}$[ζ3] | C2 | C2 | C4 | $\mathbb{Z}$2⊕$\mathbb{Z}$[i] | (C2)2 | (C2)2 | C5 | $\mathbb{Z}$⊕$\mathbb{Z}$[ζ5] | C2 | C2 | (C2)2 | $\mathbb{Z}$4 | (C2)6 | (C2)4 |
| Table 1 Structures of K2($\mathbb{Z}$G) and K2(Γ) |
For simplicity, for any positive integer
k≥2, we define
$\mu \left(k \right)=\left\{ \begin{align} & {{\left({{C}_{2}} \right)}^{2}}, \ \ \ \ \ \ \ \ \ \ \ \ k=2, \\ & {{\left({{C}_{2}} \right)}^{2}}\oplus {{C}_{4}}, \ \ \ \ \ k\ge 3. \\ \end{align} \right.$ |
We summarize our results in
Table 2.
Table 2
Table 2 Main results (k≥2) G | K2($\mathbb{Z}$G/pkΓ) | K2($\mathbb{Z}$G/pk$\mathbb{Z}$G) | C2 | (C2)2 | (C2)2 | C3 | C3 | C3 | C4 | μ(k) | μ(k) | C5 | C5 | C5 | (C2)2 | * | (C2)6 |
| Table 2 Main results (k≥2) |
1 Preliminaries
Lemma 1.1??Let
$\mathscr{O}$ be a totally imaginary Dedekind domain of arithmetic type with field of function
$\mathbb{F}$. Let
ps=|
μp(
$\mathbb{F}$)|, where
μp(
$\mathbb{F}$) is the
p-th power roots unity in the field
$\mathbb{F}$. Then for any integer
m > 1,
SK1(
$\mathscr{O}$,
m$\mathscr{O}$) is a cyclic subgroup of
μ(
$\mathbb{F}$) of order
l, where
$\text{or}{{\text{d}}_{p}}\left( l \right)={{\left[ \text{or}{{\text{d}}_{p}}\left( m \right)-\frac{1}{p-1} \right]}_{\left[ 0, s \right]}}, $ |
i.e., ord
p(
l) is in the interval [0,
s], and is the greatest integer≤
$\left( \text{or}{{\text{d}}_{p}}\left( m \right)-\frac{1}{p-1} \right)$.
Proof??This follows directly form Corollary 4.3 in Ref.[
3].
Lemma 1.2??Let 1 < |
Cpr| < 6. Then for any integer
k≥
r≥1.
${{K}_{2}}\left( \mathbb{Z}{{C}_{{{p}^{r}}}}/{{p}^{k}}\Gamma \right)\cong \underset{i=0}{\overset{r}{\mathop{\oplus }}}\, {{K}_{2}}\left( \mathbb{Z}\left[ {{\zeta }_{{{p}^{i}}}} \right]/\left( {{p}^{k}} \right) \right).$ |
Proof??Let
R=
$\mathbb{Z}$Cpr,
I=
pkΓ such that
k≥
r. By Proposition 2.2.(b) in Ref.[
1], we have
I$\subseteq $R. Then the Cartesian square
$\begin{matrix} R & \to & R\text{/}I \\ \downarrow & {} & \downarrow \\ \Gamma & \to & \Gamma /I \\\end{matrix}$ |
gives rise to a nature commuative diagram with exact rows
By Corrollary 1.6 in Ref.[
4],
K2(
R) maps onto
K2(Γ). Hence, it is indicated in
Table 1 that
φ2 is an isomorphism induced by inclusion. According to Theorem 1.3 in Ref.[
1],
φ1 is a surjection and
φ4 is an isomorphism. Since both of
SK1(
R) and
SK1(Γ) are trivial,
φ5 is an isomorphism. A diagram chasing shows that
φ3 is an isomorphism. Hence,
${{K}_{2}}\left( \mathbb{Z}{{C}_{{{p}^{r}}}}\text{/}{{p}^{k}}\Gamma \right)\cong {{K}_{2}}\left( \Gamma \text{/}{{p}^{k}}\Gamma \right).$ |
According to Corollary 2.10 in Ref.[
5],
$\mathbb{Q}$Cpr is isomorphic to a direct sum of cyclotomic field
$\mathbb{Q}{{C}_{p^r}}\cong \underset{i=0}{\overset{r}{\mathop{\oplus }}}\, \mathbb{Q}\left( {{\zeta }_{{{p}^{i}}}} \right).$ |
By Proposition 2.2 in Ref.[
1],
$\Gamma \cong \underset{i=0}{\overset{r}{\mathop{\oplus }}}\, \mathbb{Z}\left( {{\zeta }_{{{p}^{i}}}} \right)$. Thus the result follows.
2 Main resultsIt is well known that
K2(
$\mathbb{Z}$/(2
k))=
C2(
k≥2). If
$\mathbb{Z}$[
ζpi] is totally imaginary, by Lemma 3.4, Theorem 3.8 and 5.1 in Ref.[
6], and Corollary 4.3 in Ref.[
3],
${{K}_{2}}\left( \mathbb{Z}\left[ {{\zeta }_{{{p}^{i}}}} \right]/\left( {{p}^{k}} \right) \right)\cong S{{K}_{1}}\left( \mathbb{Z}\left[ {{\zeta }_{{{p}^{i}}}} \right], \left( {{p}^{k}} \right) \right).$ |
$\mathbb{F}$or
k≥2, using Lemma 1.1 and Lemma 1.2, we deduce that
K2(
$\mathbb{Z}$C2/2
kΓ)=(
C2)
2,
K2(
$\mathbb{Z}$C3/3
kΓ)=
C3,
K2(
$\mathbb{Z}$C5/5
kΓ)=
C5,
K2(
$\mathbb{Z}$C4/2
kΓ)=(
C2)
2⊕
C4(
k≥3), and
K2(
$\mathbb{Z}$C4/4Γ)=(
C2)
3.
Note that, for any integer
s≥
r,
pr+sΓ
$\subseteq $ps$\mathbb{Z}$Cpr$\subseteq $psΓ. Hence, according to Proposition 1.1 in Ref.[
7],
K2(
$\mathbb{Z}$Cpr/
pr+sΓ) maps onto
K2(
$\mathbb{Z}$Cpr/
ps$\mathbb{Z}$Cpr) and
K2(
$\mathbb{Z}$Cpr/
ps$\mathbb{Z}$Cpr) maps onto
K2(
$\mathbb{Z}$Cpr/
psΓ). Therefore, we deduce that for
k≥2,
K2(
$\mathbb{Z}$C2/(2
k))=(
C2)
2,
K2(
$\mathbb{Z}$C3/(3
k))=
C3,
K2(
$\mathbb{Z}$C5/(5
k))=
C5,
K2(
$\mathbb{Z}$C4/(2
k))=(
C2)
2⊕
C4(
k≥3), and
K2(
$\mathbb{Z}$C4/(4))=(
C2)
3 since, by Lemma 4.1.(a) in Ref.[
7],
K2(
$\mathbb{Z}$C4/(4)) has an exponent of 2.Summarizing, we have proven the theorems as follows.
Theorem 2.1??
K2(
$\mathbb{Z}$C2/(2
k))(
k≥2) is an elementary abelian 2-group of rank 2.
Theorem 2.2??
$\mathbb{F}$or any integer
k≥2,
${{K}_{2}}\left( \mathbb{Z}{{C}_{4}}/\left( {{2}^{k}} \right) \right)=\left\{ \begin{align} & {{\left( {{C}_{2}} \right)}^{3}}, \ \ \ \ \ \ \ \ \ \ \ k=2, \\ & {{\left( {{C}_{2}} \right)}^{2}}\oplus {{C}_{4}}, \ \ \ \ k\ge 3. \\ \end{align} \right.$ |
Theorem 2.3??
$\mathbb{F}$or any integer
k≥2 and
p=3, 5,
K2(
$\mathbb{Z}$Cp/(
pk)) is a cyclic group of order
p.
The method mentioned above is not suitable for the case when
G=(
C2)
2. However, we have the result as follows.
Theorem 2.4??Let (
C2)
2=〈
σ〉×〈
τ〉 be an elementary abelian 2-group of rank 2. Then
K2(
$\mathbb{Z}$(
C2)
2/(2
k))(
k≥2) is an elementary abelian 2-group of rank 6 with generators
$\begin{align} & \left\{ -1, -1 \right\}, \left\{ \sigma , -1 \right\}, \left\{ \tau , -1 \right\}, \left\{ \sigma , \tau \right\}, \\ & \left\langle \sigma -1, \tau \left( \sigma +1 \right) \right\rangle \ \text{and}\ \left\langle \tau \text{-}1, \sigma \left( \tau +1 \right) \right\rangle . \\ \end{align}$ |
Proof??According to Theorem 1.10 in Ref.[
7],
SK1(
$\mathbb{Z}$(
C2)
2, (2
k)) is trivial.
$\mathbb{F}$rom the relative
K-theory exact sequence associated to the pair (
$\mathbb{Z}$(
C2)
2, (2
k)),
K2(
$\mathbb{Z}$(
C2)
2) maps onto
K2(
$\mathbb{Z}$(
C2)
2/(2
k)). We deduce from the result of Ref.[
2] that
K2(
$\mathbb{Z}$(
C2)
2/(2
k)) is an elementary abelian 2-group of rank at most 6.
Besides, from Proposition 1.1 in Ref.[
1],
K2(
$\mathbb{Z}$(
C2)
2/(2
k)) maps onto
K2(
$\mathbb{F}$2(
C2)
2). Obviously, the kernel contains {-1, -1}, {
σ, -1}, and {
τ, -1}, while by the result on page 272 in Ref.[
7], they are nontrivial and linearly independent. Based on Theorem 4 and Example 3 in Ref.[
8],
K2(
$\mathbb{F}$2(
C2)
2) is an elementary abelian 2-group of rank 3 with generators {1+
σ+
τ,
σ}, {1+
σ+
τ,
τ}, and {
σ,
τ}. Therefore, the 2-rank of
K2(
$\mathbb{F}$2(
C2)
2) is at least 6.
Hence,
K2(
$\mathbb{Z}$(
C2)
2/(2
k)) is isomorphic to
K2(
$\mathbb{Z}$(
C2)
2). In Ref.[
2], the explicit structure of
K2(
$\mathbb{Z}$(
C2)
2) is given. The result follows.
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