Notes on the filtration of the K-theory for abelian p-groups

TL;DR

This paper corrects previous errors and extends the understanding of gamma filtrations in the K-theory of classifying spaces for non-elementary abelian p-groups, including odd primes.

Contribution

It corrects earlier inaccuracies and generalizes Chetard's results from 2-groups to p-groups for odd primes in the context of gamma filtrations.

Findings

Corrected previous errors in gamma filtration analysis

Extended results to odd prime p-groups

Enhanced understanding of K-theory for non-elementary abelian p-groups

Abstract

In this paper, I correct errors in my paper (Kodai Math. J. (2015)) about gamma filtrations for classifying spaces for abelian p-groups which are not elementary. We also extend Chetard's results for such 2-groups to p-groups for odd prime.

Equations

$$g{r}_{top}^{\ast }(BG)\cong \mathbb{Z}[y_1,...,y_n]/(q{y}_i,y_i^q{y}_j-y_i{y}_j^q|1\le i,j\le n),|y_i|=2.$$

$$g{r}_{\gamma }^{\ast }(B(\mathbb{Z}/4\times \mathbb{Z}/4))\cong \mathbb{Z}[y_1,y_2]/(4y_1,4y_2,2y_1^2{y}_2+2y_1{y}_2^2,y_1^4{y}_2^2-y_1^2{y}_2^4).$$

$$g{r}_{\gamma }^{\ast }(B(\mathbb{Z}/2^r\times \mathbb{Z}/2))\cong \mathbb{Z}[y_1,y_2]/(2^r{y}_1,2y_2,y_1{y}_2^{r+1}+y_1^2{y}_2^r).$$

$$g{r}_{top}(BG)\cong \mathbb{Z}[y_1,y_2]/(p^2{y}_1,p^2{y}_2,py(1),y(1{)}^p).$$

$$g{r}_{top}^{\ast }(BG)\cong \mathbb{Z}[y_1,y_2]/(p^r{y}_1,p{y}_2,s_r)$$

$$g{r}_{\gamma }^{\ast }(BG)\cong H^{even}(BG)/(Q_1{H}^{odd}(BG)).$$

$$...\to H^{\ast -1}(X;\mathbb{Z}/q^{\prime })\stackrel{\delta }{\to }{H}^{\ast }(X)\stackrel{q^{\prime }}{\to }{H}^{\ast }(X)\to H^{\ast }(X;\mathbb{Z}/q^{\prime })\to ...,$$

$$H^{\ast }(X;\mathbb{Z}/q)\cong H^{\ast }(X)/q\{1,x\},x={\delta }^{-1}y$$

$$H^{\ast }(X;\mathbb{Z}/p)\cong H^{\ast }(X)/p\{1,x^{\prime }\}{x}^{\prime }={\delta }^{-1}(p^{r-1}y).$$

$$E_2^{\ast ,{\ast }^{\prime }}\cong H^{\ast }(X;H^{{\ast }^{\prime }}(X))⟹H^{\ast }(X\times X)$$

$$where{E}_2^{\ast .{\ast }^{\prime }}\cong \{\begin{array}{l}\mathbb{Z}[y_1]/(q{y}_1){\ast }^{\prime }=0\\ \mathbb{Z}/q[y_1]\{1,x\}\otimes y_2^{{\ast }^{\prime }}{\ast }^{\prime }>0.\end{array}$$

$$E_{\infty }^{\ast ,{\ast }^{\prime }}\cong \mathbb{Z}/q[y_1,y_2]\{1,y_{2}x\}.$$

$$H^{\ast }(X\times X)\cong \mathbb{Z}[y_1,y_2]\{1,\alpha \}/(q{y}_i,q\alpha ),|\alpha |=3.$$

$$E_2^{\ast ,{\ast }^{\prime }}\cong H^{\ast }(X;H^{{\ast }^{\prime }}(B\mathbb{Z}/p))⟹H^{\ast }(X\times B\mathbb{Z}/p)$$

$$where{E}_2^{\ast .{\ast }^{\prime }}\cong \{\begin{array}{l}\mathbb{Z}/(q)[y_1]{\ast }^{\prime }=0\\ \mathbb{Z}/p[y_1]\{1,x^{\prime }\}\otimes y_2^{{\ast }^{\prime }}{\ast }^{\prime }>0.\end{array}$$

$$H^{\ast }(X\times B\mathbb{Z}/p)\cong \mathbb{Z}[y_1,y_2]\{1,{\alpha }^{\prime }\}/(q{y}_1,p{y}_2,p{\alpha }^{\prime }),|{\alpha }^{\prime }|=3.$$

$$g{r}_{top}^{\ast }(BG)\cong \mathbb{Z}[y_1,y_2]/(p^r{y}_1,p{y}_2,s_r)$$

$$K^{\ast }(B\mathbb{Z}/p)\cong K^{\ast }[[y]]/([p](y))|y|=2.$$

$$[p](y)=py+v_1{y}^p.$$

$$[p^r](y)=[p^{r-1}]([p](y))=p^{r}y+p^{r-1}{v}_1{y}^p+...$$

$$K^{\ast }(BG)\cong K^{\ast }(X){\otimes }_{K^{\ast }}{K}^{\ast }(B\mathbb{Z}/p)\cong K^{\ast }[[y_1,y_2]]/([p^r](y_1),[p](y_2))$$

$$where\{\begin{array}{l}[p^r](y_1)=p^r{y}_1+p^{r-1}{v}_1{y}_1^p+...,\\ [p](y_2)=p{y}_2+v_1{y}_2^p.\end{array}$$

$$p^r{y}_2=-p^{r-1}{v}_1{y}_2^p=p^{r-2}{v}_1^2{y}_2^{2p-1}=...=(-1{)}^r{v}_1^r{y}_2^{r(p-1)+1}.$$

$$y_2[p^r](y_1)=p^r{y}_1{y}_2+p^{r-1}{v}_1{y}_1^p{y}_2+...$$

$$=(-1{)}^r{v}_1^r{y}_1{y}_2^{r(p-1)+1}+(-1{)}^{r-1}{v}_1^r{y}_1^p{y}_2^{(r-1)(p-1)+1}+...=(-1{)}^r{v}_1^r{s}_n.$$

$$K^{\ast }(BG)(y_2)\cong K^{\ast }/p[[y_1,y_2]]\{y_2\}/(s_r).$$

$$E_2^{\ast ,{\ast }^{\prime }}\cong H^{\ast }(X\times B\mathbb{Z}/p)\otimes K^{{\ast }^{\prime }}⟹K^{\ast }(X\times B\mathbb{Z}/p).$$

$$H^{\ast }(X\times B\mathbb{Z}/p)\cong \mathbb{Z}[y_1,y_2]\{1,{\alpha }^{\prime }\}/(q{y}_1,p{y}_2,p{\alpha }^{\prime })with|{\alpha }^{\prime }|=3.$$

$$v_1^{-t^{\prime }}\otimes d_t:H^{odd}(BG)\cong \mathbb{Z}/p[y_1,y_2]\{{\alpha }^{\prime }\}\to Z/p[y_1,y_2]\{s^{\prime }\}\subset \mathbb{Z}/p[y_1,y_2]\{y_2\}$$

$$E_{t+1}^{\ast .0}\cong \mathbb{Z}/p[y_1{]}^{+}/(q{y}_1)\oplus \mathbb{Z}/p[y_1,y_2]\{y_2\}/(s^{\prime }).$$

Full text

1. Introduction

Let $p$ be a prime number. For a given finite group $G$, let $gr_{top}^{*}(BG)$ (resp. $gr_{\gamma}^{*}(BG)$) be the associated ring of the topological (resp. gamma) filtration of the $K$-theory $K^{0}(BG)$ for the classifying space $BG$. In this paper, we correct errors in the paper [Ya] and extend results by B.Chetard.

In Theorem 4.1 in [Ya], I wrote

Theorem 1.1**.**

($p=2,r=1$ case by Atiyah [At]) Let $q=p^{r}$ and $G=\oplus^{n}{\mathbb{Z}}/q$. Then

[TABLE]

Hence the filtrations are the same.

This theorem was error for $r\geq 2$, indeed, arguments for the higher Bokstein $Q_{0}^{\prime}$ in its proof were errors. However the statement holds still for $p=q$, i.e., for an elementary abelian $p$-group $G$. (The second statement holds for all abelian $p$ groups [At].)

Beatrice Chetard pointed out this fact [Ch]. Indeed, she shows the following isomorphism by using the definition of the gamma filtration of the representation ring

[TABLE]

She also computes $gr_{\gamma}^{*}(B({\mathbb{Z}}/4\times{\mathbb{Z}}/2))$, and conjectured

[TABLE]

In this note, we see that the above Chetard results can be extended to odd prime cases. Let us write $y(1)=y_{1}^{p}y_{2}-y_{1}y_{2}^{p}$. Then we have

Theorem 1.2**.**

For each prime $p$, let $G={\mathbb{Z}}/p^{2}\times{\mathbb{Z}}/p^{2}$. Then

[TABLE]

Theorem 1.3**.**

For each prime $p$, let $G=(Z/p^{r}\times Z/p)$, $r\geq 1$. Then

[TABLE]

where $s_{r}=y_{1}y_{2}^{r(p-1)+1}-y_{1}^{p}y_{2}^{(r-1)(p-1)+1}$ $mod(y_{1}^{2}y_{2}^{(r-1)(p-1)+2})$.

Here note that $gr_{\gamma}^{*}(BG)$ are known for many nonabelian $p$-groups $G$ by using $gr_{top}^{*}(BG)$ and the Atiyah-Hirzebruch spectral sequence, while the direct computations of $gr_{\gamma}^{*}(BG)$ by using representations theory of $G$ are not so many.

For example, when $|G|=p^{3}$ and nonabelian, we know [Ya]

[TABLE]

Here $H^{odd}(BG)$ is just $p$-torsion and we can define the Milnor $Q_{1}$-operation on $H^{odd}(BG)$ (see the proof of Theorem 4.2 in [Ya]). In particular, when $G=Q_{8}$, it is known $H^{odd}(BG)=0$. Atiyah [At] used representation arguments to get the multiplicative structure of $H^{*}(BQ_{8})\cong gr_{\gamma}^{*}(BG)$.

I thank Beatrice Chetard who pointed out my error in [Ya].

2. $H^{*}(B({\mathbb{Z}}/q\times{\mathbb{Z}}/q))$ and $H^{*}(B{\mathbb{Z}}/q\times B{\mathbb{Z}}/p)$

Let $X=B{\mathbb{Z}}/q$ with $q=p^{r}$, $r\geq 2$. Its integral cohomology is $H^{*}(X)\cong{\mathbb{Z}}[y]/(qy)$ with the degree $|y|=2.$ Considering the long exact sequence for $q^{\prime}=q$ or $q^{\prime}=p$

[TABLE]

we have

[TABLE]

[TABLE]

Here for a ring $A$, the notation $A\{x,...,z\}$ means the $A$-free module generated by $x,...,z$.

We consider the Serre spectral sequence

[TABLE]

[TABLE]

Since $H^{*}(X)\subset H^{*}(X\times X)$, element $y_{1}^{3}$ is permanent , and so is $x$. Hence we have

[TABLE]

Writing by $\alpha\in H^{*}(X\times X)$ which represents $y_{2}x\in E_{\infty}^{1,2}$, we have

Lemma 2.1**.**

For $X=B{\mathbb{Z}}/q$, $q=p^{r}$, we have

[TABLE]

Next, we consider $X\times B{\mathbb{Z}}/p$. Consider the spectrall sequence

[TABLE]

[TABLE]

Lemma 2.2**.**

For $X=B{\mathbb{Z}}/q$, $q=p^{r}$, we have

[TABLE]

3. $gr_{top}^{*}(X\times B{\mathbb{Z}}/p)$

In this section, we will prove

Theorem 3.1**.**

Let $G=({\mathbb{Z}}/p^{r}\times{\mathbb{Z}}/p)$. Then

[TABLE]

where $s_{r}=y_{1}y_{2}^{r(p-1)+1}-y_{1}^{p}y_{2}^{(r-1)(p-1)+1}$ $mod(y_{1}^{2}y_{2}^{(r-1)(p-1)+2})$.

At first, we study relations in $K^{*}(BG)$. Recall that $[p](y)$ is the $p$-th product of the formal group law of the localized (Morava) $K$-theory ([Ha], [Ra]) so that

[TABLE]

We can identify $K^{*}={\mathbb{Z}}_{(p)}[v_{1},v_{1}^{-1}],$ with $|v_{1}|=-2(p-1)$, and write

[TABLE]

We know $K^{*}(B{\mathbb{Z}}/q)\cong K^{*}[[y]]/([q](y))$ and

[TABLE]

Hence we have

[TABLE]

[TABLE]

The second equation implies

[TABLE]

Applying this equation to $y_{2}[p^{r}](y_{1})=0$ in $K^{*}(BG)$, we get

[TABLE]

[TABLE]

Hence $s_{n}=0$ in $K^{*}(G)$.

Lemma 3.2**.**

Let $K^{*}(BG)(y_{2})$ be the ideal generated by $y_{2}$ in $K^{*}(BG)$. Then we have

[TABLE]

We study the Atiyah-Hirzebruch spectral sequence

[TABLE]

Here we recall

[TABLE]

Since $K^{*}(X\times B{\mathbb{Z}}/p)$ is generated by even dimensional elements, there are $t,t^{\prime}>1$ and $s^{\prime}\not=0$ in $H^{even}(BG)/p\{y_{2}\}$ such that $d_{t}(\alpha^{\prime})=v_{1}^{t^{\prime}}\otimes s^{\prime}$.

The map

[TABLE]

(via $\alpha^{\prime}\mapsto s^{\prime}$) is injective. Hence we get

[TABLE]

This term is generated by even dimensional elements, and isomorphic to

[TABLE]

Hence $s^{\prime}$ divides $s_{r}$. Moreover we can take $s^{\prime}=s_{r}$ since $s^{\prime}$ $mod(v_{1})$ must be in the ideal $(s_{r})$ from lemma.

4. $gr_{top}^{*}(B{\mathbb{Z}}/p^{2}\times B{\mathbb{Z}}/p^{2})$

Let $X=B{\mathbb{Z}}/p^{2}$. We study the Atiyah-Hirzebruch spectral sequence

[TABLE]

Here we recall $H^{*}(X\times X)\cong{\mathbb{Z}}[y_{1},y_{2}]\{1,\alpha\}/(p^{2}y_{i},p^{2}\alpha)$ with $|\alpha|=3.$ We will prove

[TABLE]

for $y(1)=y_{1}^{p}y_{2}-y_{1}y_{2}^{p}$. Then we see that

Theorem 4.1**.**

Let $X=B{\mathbb{Z}}/p^{2}$. Then we have the isomorphism

[TABLE]

Recall that the $p$-product of the formal group law for $K^{*}$-theory is given by $[p](y)=py+v_{1}y^{p}$. Then

[TABLE]

[TABLE]

in $K^{*}(X)$. Consider in $K^{*}(X\times X)$, and we have

[TABLE]

[TABLE]

Let us write $y(i)=y_{1}^{p^{i}}y_{2}-y_{1}y_{2}^{p^{i}}$ so that

[TABLE]

Here we recall the connected $K$-theory $k^{*}(X)$ such that

[TABLE]

Then we can write the above equation $(*)$

[TABLE]

Hence $py(1)$ is zero in $gr_{top}^{*}(G)$ but it is nonzero in $K^{*}(X\times X)$ (hence $y(2)\not=0$) since $K^{*}(X\times X)$ is torsion free. Therefore we see

[TABLE]

Since $K^{*}(BG)$ is generated by even dimensional elements, we see $\alpha^{\prime\prime}=p\alpha$ is not a permanent cycle, i.e. there are $r>2p,t^{\prime\prime}>1$, $d\in E_{r}^{*,0}$ such that $d_{r}(\alpha^{\prime\prime})=v_{1}^{t^{\prime\prime}}\otimes d\not=0$.

We study this $d$. At first $d$ is invariant $mod(p)$ under the action of $GL_{2}({\mathbb{Z}}/p)$, namely $d=b$ or $pb$ with $b\in{\mathbb{Z}}/p[y_{1},y_{2}]^{GL_{2}({\mathbb{Z}}/p)}$. The invariant ring is known as the Dickson algebra

[TABLE]

[TABLE]

Consider the restriction to $K^{*}(X)$

[TABLE]

Hence we can not take $d=y(2)/y(1)$ neither $d=py(2)/y(1)$.

Moreover we still see that $y(2)$ is nonzero.

Therefore when $|d|\leq 2(p^{2}+p)$, we see $d=y(1)^{i}$ for $i\leq p$. Here we consider the restriction to

[TABLE]

Hence $d$ is in the $Ideal(y_{1}^{p^{2}},y_{2}^{p^{2}})$. Thus we see that the possibility of the smallest degree element for $d$ is $y(1)^{p}$.

We see $y(1)^{p}=0$ in $gr_{top}^{*}(X\times X)$ as follows. Consider in $k^{*}(X\times X)$

[TABLE]

[TABLE]

Here we can see the following equations with $mod(p^{2}v_{1}y(1))$ in $k^{*}(X\times X)$

[TABLE]

[TABLE]

Continuing this arguments, we see

[TABLE]

Thus we can take $d=y(1)^{p}$.

We see that the map

[TABLE]

by $\alpha^{\prime}\mapsto y(1)^{p}$ is injective. Hence $E_{2p^{2}+2p-3}^{*,*^{\prime}}$ is generated by even dimensional elements, and is isomorphic to the infinite term $E_{\infty}^{*,*^{\prime}}$.