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Mathematische Zeitschrift

, Volume 275, Issue 1–2, pp 653–655 | Cite as

Erratum to: Bloch–Wigner theorem over rings with many units

Erratum
First Online:

1 Erratum to: Math. Z. (2011) 268:329–346 DOI 10.1007/s00209-010-0674-9

2 Introduction

In this erratum, we correct a mistake that I made in [1]. The formulation of our main theorem [1, Theorem 5.1] is not correct. The correct form of the theorem, which is sufficient for our applications, is as follows:

Let \(R\) be a commutative ring with many units and define
$$\begin{aligned} \tilde{H}_3(\mathrm{SL}_2(R), \mathbb Z ):=H_3(\mathrm{GL}_2(R), \mathbb Z )/(H_3(\mathrm{GL}_1(R), \mathbb Z ) + {R^{*}}\cup H_2(\mathrm{GL}_1(R), \mathbb Z ), \end{aligned}$$
which is induced by the diagonal inclusion of \(R^*\times \mathrm{GL}_1(R)\) in \(\mathrm{GL}_2(R)\). Then, there is a quotient \(M\) of \(H_1(\Sigma _2, {R^{*}}\otimes {R^{*}})\) which fits into exact sequences
$$\begin{aligned}&0 \longrightarrow T_R \longrightarrow \tilde{H}_3(\mathrm{SL}_2(R), \mathbb Z ) \longrightarrow \mathfrak p (R) \longrightarrow (R^*\otimes _\mathbb Z R^*)_\sigma \longrightarrow K_2(R) \longrightarrow 0,\\&\quad {\mathrm{Tor}_1^\mathbb{Z }}(\mu (R),\mu (R)) \longrightarrow T_R \longrightarrow M \longrightarrow 0. \end{aligned}$$
When \(R\) is an integral domain, the left-hand side map in the second exact sequence is injective.

3 The main theorem

Lemma 3.1 in [1] is not correct. In fact, in the proof of the lemma, the map
$$\begin{aligned} F_q \otimes _{\mathrm{Stab}_{\mathrm{GL}_2}(\infty )} \mathbb Z \rightarrow F_q \otimes _{\mathrm{GL}_2} C_1(R^2), \quad s\otimes 1 \mapsto s \otimes (\infty ,0) \end{aligned}$$
is not well defined. Because of this mistake, we made few uncorrect claims in [1]. Here, we give a correct formulations of the lemma and these claims that are sufficient for our applications. We follow the notations in [1].

Lemma 3.1

The groups \(E_{1,0}^2\) and \(E_{1,1}^2\) are trivial. Also, there is a surjective map \(H_1(\Sigma _2, {R^{*}}\otimes {R^{*}}) \twoheadrightarrow E_{1,2}^2\), where the action of \(\Sigma _2=\{1, \sigma \}\) on \({R^{*}}\otimes {R^{*}}\) is defined by \(\sigma (a \otimes b)=-b \otimes a\). In particular, \(E_{1,2}^2\) is a \(2\)-torsion group.

Proof

In [1] on page 335, we have shown that \(H_0(\mathrm{GL}_2,H_1(X))\simeq \mathbb Z \) and \(d_{2,0}^1=\mathrm{id}_\mathbb Z \). Consider the differential \(d_{2,1}^1:E_{2,1}^1=H_1(\mathrm{GL}_2,H_1(X)) \rightarrow \mathrm{ker}(d_{1,1}^1)\simeq {R^{*}}\). Let \(\varphi : {R^{*}}\simeq H_1(\mathrm{GL}_2,C_2(R^2)) \rightarrow H_1(\mathrm{GL}_2,H_1(X))\). It is easy to see that \(d_{2,1}^1\circ \varphi =\mathrm{id}_{R^{*}}\). These facts immediately imply the triviality of \(E_{1,0}^2\) and \(E_{1,1}^2\). To compute \(E_{1,2}^2\), first note that \(\mathrm{ker}(d_{1,2}^1)\simeq H_2({R^{*}}) \oplus ({R^{*}}\otimes {R^{*}})^\sigma \). Again, one can easily see that the composition
$$\begin{aligned} H_2({R^{*}})\!\simeq \! H_2\left( \mathrm{GL}_2,C_1\left( R^2\right) \right) \!\rightarrow \! H_2(\mathrm{GL}_2,H_1(X))\! \rightarrow \! H_2({R^{*}}) \!\oplus \!({R^{*}}\!\otimes \!{R^{*}})^\sigma \end{aligned}$$
is given by \(x \mapsto (x, 0)\). Thus, \(E_{1,2}^2\simeq ({R^{*}}\otimes {R^{*}})^\sigma /A\). By an easy analysis of our main spectral sequence, we have
$$\begin{aligned} E_{1,2}^2=({R^{*}}\otimes {R^{*}})^\sigma /A \hookrightarrow H_3(\mathrm{GL}_2)/H_3({R^{*}}\times {R^{*}}). \end{aligned}$$
We call this map \(\beta \). We know that
$$\begin{aligned} H_1(\Sigma _2, {R^{*}}\otimes {R^{*}})= \frac{({R^{*}}\otimes {R^{*}})^\sigma }{(1+\sigma )({R^{*}}\otimes {R^{*}})}= \frac{({R^{*}}\otimes {R^{*}})^\sigma }{\langle a \otimes b-b\otimes a: a, b \in {R^{*}}\rangle }. \end{aligned}$$
Let \(x_{a,b}=a \otimes b-b\otimes a\). Then,
$$\begin{aligned} h=[(a,1)|(1,b)]-[(1,b)|(a,1)] -[(b,1)|(1,a)]+[(1,a)|(b,1)] \end{aligned}$$
is the cycle that represents the element \(x_{a,b} \in ({R^{*}}\otimes {R^{*}})^\sigma \subseteq H_2({R^{*}}\times {R^{*}})^\sigma \). Let \(\tau \) be the automorphism of transposition of terms. Then, \(\tau (h)-h=0\). Now, by [2, Lemma 2.5], the image of the class \(\overline{h}\) under \(\beta \) is given by \(-\overline{\rho _s(h)}\), where \(s=\left( \begin{array}{c@{\quad }c} 0 &{} 1 \\ 1 &{} 0 \end{array} \right) \) and
$$\begin{aligned} \rho _s(h){:=}&+[s|(1,a)|(b,1)]-[(a,1)|s|(b,1)]+[(a,1)|(1,b)|s]\\&-[s|(b,1)|(1,a)]+[(1,b)|s|(1,a)]-[(1,b)|(1,1)|s]\\&-[s|(1,b)|(a,1)]+[(b,1)|s|(a,1)]-[(b,1)|(1,a)|s]\\&+[s|(a,1)|(1,b)]-[(1,a)|s|(1,b)]+[(1,a)|(b,1)|s]. \end{aligned}$$
Now, by a direct computation, one can see that \(\overline{\rho _s(h)}=0\). Thus, \(x_{a,b} \in A\) and therefore \((1+\sigma )({R^{*}}\otimes {R^{*}}) \subseteq A\). This implies the surjectivity that we are looking for. \(\square \)

Now, we are ready to correct Theorem 5.1 in [1].

Theorem 5.1

Let \(R\) be a commutative ring with many units and define \(\tilde{H}_3(\mathrm{SL}_2(R)):=H_3(\mathrm{GL}_2)/(H_3(\mathrm{GL}_1) + {R^{*}}\cup H_2(\mathrm{GL}_1))\). There is a quotient \(M\) of \(H_1(\Sigma _2, {R^{*}}\otimes {R^{*}})\) which fits into exact sequences
$$\begin{aligned}&0 \longrightarrow T_R \longrightarrow \tilde{H}_3(\mathrm{SL}_2(R)) \longrightarrow B(R) \longrightarrow 0,\\&\quad {\mathrm{Tor}_1^\mathbb{Z }}(\mu (R),\mu (R)) \longrightarrow T_R \longrightarrow M \longrightarrow 0. \end{aligned}$$
When \(R\) is an integral domain, the left-hand side map in the second exact sequence is injective.

Proof

The proof is very similar to the proof of Theorem 5.1 in [1]. \(\square \)

Remark 0.1

Here, we further make some minor corrections.
  1. (i)

    In Proposition 2.1 and Remark 5.2 in [1], we have to assume that either the coefficient group is \(\mathbb Z [1/2]\) or the ring \(R\) has the property \({R^{*}}={R^{*}}^2\) (\(K_1(R)=K_1(R)^2\) for Remark 5.2).

     
  2. (ii)

    The claim made in Remark 2.2 in [1] is not correct, and Suslin’s claim in [2, Remark 2.2] remains true.

     

The rest of our claims in [1] remains true.

References

  1. 1.
    Mirzaii, B.: Bloch–Wigner theorem over rings with many units. Math. Z. 268, 329–346 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Suslin, A.A.: K \(_{3}\) of a field and the Bloch group. Proc. Steklov Inst. Math. 183(4), 217–239 (1991) Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsInstitute for Advanced Studies in Basic SciencesZanjanIran

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