arXiv subjects
Publications and source records attributed to Behrooz Mirzaii.
In this article we study the first, the second and the third homology groups of the elementary group $\textrm{E}_2(A)$, where $A$ is a commutative ring. In particular, we prove a refined Bloch-Wigner type exact sequence over a semilocal ring (with some mild restriction on its residue fields) such that $-1\in (A^{\times})^2$ or $|A^{\times}/(A^{\times})^2|\leq 4$.
In this article, we explore the second integral homology, or Schur multiplier, of the special linear group ${\rm SL}_2(\mathbb{Z}[1/n])$ for a positive integer $n$. We definitively calculate the group structure of $H_2({\rm SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})$ when $n$ is divisible by one of the primes $2$, $3$, $5$, $7$ or $13$. For a general $n > 1$, we offer a partial description by placing the homology group within an exact sequence, and we investigate its rank. Finally, we propose a conjectural structure for $H_2({\rm SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})$ when $n$ is not divisible by any of those specific primes.
We determine the exact group structure of the abelianization of $\text{SL}_2(A)$, where $A$ is a Dedekind domain of arithmetic type with infinitely many units. In particular, our results show that $\text{SL}_2(A)^\text{ab}$ is finite, with exponent dividing $12$ when $\text{char}(A)=0$, and dividing $6$ when $\text{char}(A)>0$. As illustrative cases, we compute $\text{SL}_2(A)^\text{ab}$ explicitly for instances where $A$ is the ring of integers of a real quadratic field or a cyclotomic extension.
In this article, we investigate the Schur multiplier of the special linear group $\mathrm{SL}_2(A)$ over finite commutative local rings $A$. We prove that the Schur multiplier of these groups is isomorphic to the $K$-group $K_2(A)$ whenever the residue field $A/\mathfrak{m}_A$ has odd characteristic and satisfies $|A/\mathfrak{m}_A| \neq 3,5,9$. As an application, we show that if $A$ is either the Galois ring $\mathrm{GR}(p^l,m)$ or the quasi-Galois ring $A(p^m,n)$ with residue field of odd characteristic and $|A/\mathfrak{m}_A| \neq 3,5,9$, then the Schur multiplier of $\mathrm{SL}_2(A)$ is trivial.
In this paper we investigate the third homology of the projective special linear group ${\rm PSL}_2(A)$. As a result of our investigation we prove a projective refined Bloch-Wigner exact sequence over certain class of rings. The projective Bloch-Wigner exact sequence over an algebraically closed field of characteristic zero is a classical result and has many application in algebra, number theory and geometry.
In this article we study the low dimensional homology of the projective linear group $\textrm{PGL}_2(A)$ over a commutative ring $A$. In particular, we prove a Bloch-Wigner type exact sequence over local domains. As applications we prove that $H_2(\textrm{PGL}_2(A),\mathbb{Z}\left[\frac{1}{2}\right])\simeq K_2(A)\left[\frac{1}{2}\right]$ and $H_3(\textrm{PGL}_2(A),\mathbb{Z}\left[\frac{1}{2}\right])\simeq K_3^{\textrm{ind}}(A)\left[\frac{1}{2}\right]$ provided $|A/\mathcal{m}_A|\neq 2,3,4,8$.
For an arbitrary ring $A$, we study the abelianization of the elementary group $\textrm{E}_2(A)$. In particular, we show that for a commutative ring $A$ there exists an exact sequence \[ K_2(2,A)/C(2,A) \to A/M \to \textrm{E}_2(A)^\textrm{ab} \to 1, \] where $C(2,A)$ is the central subgroup of the Steinberg group $\textrm{St}(2,A)$ generated by the Steinberg symbols and $M$ is the additive subgroup of $A$ generated by $x(a^2-1)$ and $3(b+1)(c+1)$, with $x\in A$, $a,b,c \in A^{\times}$.
In this article we study the low dimensional homology groups of the special linear group $\textrm{SL}_2(A)$ and the projective special linear group $\textrm{PSL}_2(A)$, $A$ a domain, through the natural surjective map $\textrm{SL}_2(A) \to \textrm{PSL}_2(A)$. In particular, we study the connection of the first, the second and the third homology groups of these groups over euclidean domains $\mathbb{Z}[\frac{1}{m}]$, $m$ a square free integer, and local domains.
There is a natural connection between the third homology of $\textrm{SL}_2(A)$ and the refined Bloch group $\mathcal{RB}(A)$ of a commutative ring $A$. In this article we investigate this connection and as the main result we show that if $A$ is a universal $\textrm{GE}_2$-domain such that $-1 \in A^{\times 2}$, then we have the exact sequence $H_3(\textrm{SM}_2(A),\mathbb{Z}) \to H_3(\textrm{SL}_2(A),\mathbb{Z}) \to \mathcal{RB}(A) \to 0$, where $\textrm{SM}_2(A)$ is the group of monomial matrices in $\textrm{SL}_2(A)$. Moreover we show that $\mathcal{RP}_1(A)$, the refined scissors congruence group of $A$, naturally is isomorph with the relative homology group $H_3(\textrm{SL}_2(A), \textrm{SM}_2(A),\mathbb{Z})$.
Let $A$ be a local domain of characteristic $2$ such that its residue field has more than $64$ elements. Then we find an exact relation between the third integral homology of the group $\mathrm{SL}_2(A)$ and Hutchinson's refined Bloch group $\mathcal{RB}(A)$.
In this article we study the homology of nilpotent groups. In particular a certain vanishing result for the homology and cohomology of nilpotent groups is proved.
Let $A \rightarrowtail G\twoheadrightarrow Q$ be a stem-extension and let $ρ: A\times G\to G$ be the multiplication map. We show that there is a natural map $φ: H_1(Σ_2^ε, {\rm Tor}_1^{\mathbb{Z}}({}_{2^\infty}A,{}_{2^\infty}A))\to H_3(G,\mathbb{Z})/ρ_\ast(A \otimes_{\mathbb{Z}} H_2(G,\mathbb{Z}))$ such that, the image of $φ$ coincides with the image of the natural map $H_3(A,\mathbb{Z})\to H_3(G,\mathbb{Z})/ρ_\ast(A \otimes_{\mathbb{Z}} H_2(G,\mathbb{Z}))$. An important tool used here is Whitehead's quadratic functor $Γ$. As part of our proof of the main result, we give a precise homological description of the kernel of the natural map $Γ(A) \to A\otimes_{\mathbb{z}} A$, $γ(a)\mapsto a\otimes a$.
For an infinite field $F$, we study the kernel of the map $H_{n}(\mathrm{GL}_{n-1}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big]) \to H_{n}(\mathrm{GL}_{n}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big])$ and the cokernel of $H_{n+1}\Big(\mathrm{GL}_{n-1}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big]\Big) \to H_{n+1}\Big(\mathrm{GL}_{n}(F),\mathbb{Z}\Big[\frac{1}{(m-2)!}\Big]\Big)$. We give conjectural estimates of these kernels and cokernels and prove our conjectures for $n\leq 4$.
For a central perfect extension of groups $A \rightarrowtail G\twoheadrightarrow Q$, first we study the natural image of $H_3(A,\mathbb{Z})$ in $H_3(G, \mathbb{Z})$. As a particular case, we show that if the extension is universal this image is 2-torsion. Moreover when the plus-construction of the classifying space of $Q$ is an $H$-space, we also study the kernel of the surjective homomorphism $H_3(G,\mathbb{Z}) \to H_3(Q, \mathbb{Z})$.
Let $A$ be a discrete valuation ring with field of fractions $F$ and (sufficiently large) residue field $k$. We prove that there is a natural exact sequence $H_3(\mathrm{SL}_2(A),\mathbb{Z}[\frac{1}{2}]) \to H_3(\mathrm{SL}_2(F),\mathbb{Z}[\frac{1}{2}])\to \mathcal{RP}_1(k)[\frac{1}{2}]\to 0$, where $\mathcal{RP}_1(k)$ is the refined scissors congruence group of $k$. Let $Γ_0(\mathfrak{m}_A)$ denote the congruence subgroup consisting of matrices in $\mathrm{SL}_2(A)$ whose lower off-diagonal entry lies in the maximal ideal $\mathfrak{m}_A$. We also prove that there is an exact sequence $0\to \overline{\mathcal{P}}(k)[\frac{1}{2}]\to H_2(Γ_0(\mathfrak{m}_A),\mathbb{Z}[\frac{1}{2}])\to H_2(\mathrm{SL}_2(A),\mathbb{Z}[\frac{1}{2}])\to I^2(k)[\frac{1}{2}]\to 0$, where $I^2(k)$ is the second power of the fundamental ideal of the Grothendieck-Witt ring $\mathrm{GW}(k)$ and $\overline{\mathcal{P}}(k)$ is a certain quotient of the scissors congruence group (in the sense of Dupont-Sah) $\mathcal{P}(k)$ of $k$.
In this paper we study virtual rational Betti numbers of a nilpotent-by-abelian group $G$, where the abelianization $N/N'$ of its nilpotent part $N$ satisfies certain tameness property. More precisely, we prove that if $N/N'$ is $2(c(n-1)-1)$-tame as a $G/N$-module, $c$ the nilpotency class of $N$, then $\mathrm{vb}_j(G):=\sup_{M\in\mathcal{A}_G}\dim_\mathbb{Q} H_j(M,\mathbb{Q})$ is finite for all $0\leq j\leq n$, where $\mathcal{A}_G$ is the set of all finite index subgroups of $G$.
In this article we extend the Bloch-Wigner exact sequence over local rings, where their residue fields have more than nine elements. Moreover, we prove Van der Kallen's theorem on the presentation of the second $K$-group of local rings such that their residue fields have more than four elements. Note that Van der Kallen proved this result when the residue fields have more than five elements. Although we prove our results over local rings, all our proofs also work over semilocal rings where all their residue fields have similar properties as the residue field of local rings.
In this article we prove a generalization of the Bloch-Wigner exact sequence over commutative rings with many units. When the ring is a domain, we get a generalization of Suslin's Bloch-Wigner exact sequence over infinite fields. Our proof is different and is easier, even in its general form. But nevertheless we use some of Suslin's results which relates the Bloch group of the ring to the third homology group of the general linear group of the ring. From there we take an easier path.