arXiv subjects

Zuhong Zhang

Publications and source records attributed to Zuhong Zhang.

16 recordsLinked to original sources

arXiv2020

Relative centralisers of relative subgroups

Let $R$ be an associative ring with 1, $G=GL(n, R)$ be the general linear group of degree $n\ge 3$ over $R$. In this paper we calculate the relative centralisers of the relative elementary subgroups or the principal congruence subgroups, corresponding to an ideal $A\unlhd R$ modulo the relative elementary subgroups or the principal congruence subgroups, corresponding to another ideal $B\unlhd R$. Modulo congruence subgroups the results are essentially easy exercises in linear algebra. But modulo the elementary subgroups they turned out to be quite tricky, and we could get definitive answers only over commutative rings, or, in some cases, only over Dedekind rings. We discuss also some further related problems, such as the interrelations of various birelative commutator subgroups, etc., and state several unsolved questions.

math.RA↗

arXiv2020

Commutators of elementary subgroups: curiouser and curiouser

Let $R$ be any associative ring with $1$, $n\ge 3$, and let $A,B$ be two-sided ideals of $R$. In our previous joint works with Roozbeh Hazrat [17,15] we have found a generating set for the mixed commutator subgroup $[E(n,R,A),E(n,R,B)]$. Later in [29,34] we noticed that our previous results can be drastically improved and that $[E(n,R,A),E(n,R,B)]$ is generated by 1) the elementary conjugates $z_{ij}(ab,c)=t_{ij}(c)t_{ji}(ab)t_{ij}(-c)$ and $z_{ij}(ba,c)$, 2) the elementary commutators $[t_{ij}(a),t_{ji}(b)]$, where $1\le i\neq j\le n$, $a\in A$, $b\in B$, $c\in R$. Later in [33,35] we noticed that for the second type of generators, it even suffices to fix one pair of indices $(i,j)$. Here we improve the above result in yet another completely unexpected direction and prove that $[E(n,R,A),E(n,R,B)]$ is generated by the elementary commutators $[t_{ij}(a),t_{hk}(b)]$ alone, where $1\le i\neq j\le n$, $1\le h\neq k\le n$, $a\in A$, $b\in B$. This allows us to revise the technology of relative localisation, and, in particular, to give very short proofs for a number of recent results, such as the generation of partially relativised elementary groups $E(n,A)^{E(n,B)}$, %% normality of $E(n,AB+BA)$ inside $[E(n,R,A),E(n,R,B)]$, multiple commutator formulas, commutator width, and the like.

math.RA↗

arXiv2020

Commutators of relative and unrelative elementary unitary groups

In the present paper we find generators of the mixed commutator subgroups of relative elementary groups and obtain unrelativised versions of commutator formulas in the setting of Bak's unitary groups. It is a direct sequel of our similar results were obtained for $GL(n,R)$ and for Chevalley groups over a commutative ring with 1, respectively. Namely, let $(A,Λ)$ be any form ring and $n\ge 3$. We consider Bak's hyperbolic unitary group $GU(2n,A,Λ)$. Further, let $(I,Γ)$ be a form ideal of $(A,Λ)$. One can associate with $(I,Γ)$ the corresponding elementary subgroup $FU(2n,I,Γ)$ and the relative elementary subgroup $EU(2n,I,Γ)$ of $GU(2n,A,Λ)$. Let $(J,Δ)$ be another form ideal of $(A,Λ)$. In the present paper we prove an unexpected result that the non-obvious type of generators for $\big[EU(2n,I,Γ),EU(2n,J,Δ)\big]$, as constructed in our previous papers with Hazrat, are redundant and can be expressed as products of the obvious generators, the elementary conjugates $Z_{ij}(ab,c)=T_{ji}(c)T_{ij}(ab)T_{ji}(-c)$ and $Z_{ij}(ba,c)$, and the elementary commutators $Y_{ij}(a,b)=[T_{ji}(a),T_{ij}(b)]$, where $a\in(I,Γ)$, $b\in(J,Δ)$, $c\in(A,Λ)$. It follows that $\big[FU(2n,I,Γ),FU(2n,J,Δ)\big]= \big[EU(2n,I,Γ),EU(2n,J,Δ)\big]$. In fact, we establish much more precise generation results. In particular, even the elementary commutators $Y_{ij}(a,b)$ should be taken for one long root position and one short root position. Moreover, $Y_{ij}(a,b)$ are central modulo $EU(2n,(I,Γ)\circ(J,Δ))$ and behave as symbols. This allows us to generalise and unify many previous results,including the multiple elementary commutator formula, and dramatically simplify their proofs.

math.RA↗

arXiv2020

Commutators of relative and unrelative elementary subgroups in Chevalley groups

In the present paper, which is a direct sequel of our papers [10,11,35] joint with Roozbeh Hazrat, we achieve a further dramatic reduction of the generating sets for commutators of relative elementary subgroups in Chevalley groups. Namely, let $Φ$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a commutative ring and let $A,B$ be two ideals of $R$. We consider subgroups of the Chevalley group $G(Φ,R)$ of type $Φ$ over $R$. The unrelative elementary subgroup $E(Φ,A)$ of level $A$ is generated (as a group) by the elementary unipotents $x_α(a)$, $α\inΦ$, $a\in A$, of level $A$. Its normal closure in the absolute elementary subgroup $E(Φ,R)$ is denoted by $E(Φ,R,A)$ and is called the relative elementary subgroup of level $A$. The main results of [11,35] consisted in construction of economic generator sets for the mutual commutator subgroups $[E(Φ,R,A),E(Φ,R,B)]$, where $A$ and $B$ are two ideals of $R$. It turned out that one can take Stein---Tits---Vaserstein generators of $E(Φ,R,AB)$, plus elementary commutators of the form $y_α(a,b)=[x_α(a),x_{-α}(b)]$, where $a\in A$, $b\in B$. Here we improve these results even further, by showing that in fact it suffices to engage only elementary commutators corresponding to {\it one\/} long root, and that modulo $E(Φ,R,AB)$ the commutators $y_α(a,b)$ behave as symbols. We discuss also some further variations and applications of these results.

math.RA↗

arXiv2019

Inclusion among commutators of elementary subgroups

In the present paper we continue the study of the elementary commutator subgroups $[E(n,A),E(n,B)]$, where $A$ and $B$ are two-sided ideals of an associative ring $R$, $n\ge 3$. First, we refine and expand a number of the auxiliary results, both classical ones, due to Bass, Stein, Mason, Stothers, Tits, Vaserstein, van der Kallen, Stepanov, as also some of the intermediate results in our joint works with Hazrat, and our own recent papers [40,41]. The gimmick of the present paper is an explicit triple congruence for elementary commutators $[t_{ij}(ab),t_{ji}(c)]$, where $a,b,c$ belong to three ideals $A,B,C$ of $R$. In particular, it provides a sharper counterpart of the three subgroups lemma at the level of ideals. We derive some further striking corollaries thereof, such as a complete description of generic lattice of commutator subgroups $[E(n,I^r),E(n,I^s)]$, new inclusions among multiple elementary commutator subgroups, etc.

math.RA↗

arXiv2019

Multiple commutators of elementary subgroups: end of the line

In our previous joint papers with Roozbeh Hazrat and Alexei Stepanov we established commutator formulas for relative elementary subgroups in $GL(n,R)$, $n\ge 3$, and other similar groups, such as Bak's unitary groups, or Chevalley groups. In particular, there it was shown that multiple commutators of elementary subgroups can be reduced to double such commutators. However, since the proofs of these results depended on the standard commutator formulas, it was assumed that the ground ring $R$ is quasi-finite. Here we propose a different approach which allows to lift any such assumptions and establish almost definitive results. In particular, we prove multiple commutator formulas, and other related facts for $GL(n,R)$ over an {\it arbitrary} associative ring $R$.

math.RA↗

arXiv2019

Relative and unrelative elementary groups, revisited

Let $R$ be any associative ring with $1$, $n\ge 3$, and let $A,B$ be two-sided ideals of $R$. In the present paper we show that the mixed commutator subgroup $[E(n,R,A),E(n,R,B)]$ is generated as a group by the elements of the two following forms: 1) $z_{ij}(ab,c)$ and $z_{ij}(ba,c)$, 2) $[t_{ij}(a),t_{ji}(b)]$, where $1\le i\neq j\le n$, $a\in A$, $b\in B$, $c\in R$. Moreover, for the second type of generators, it suffices to fix one pair of indices $(i,j)$. This result is both stronger and more general than the previous results by Roozbeh Hazrat and the authors. In particular, it implies that for all associative rings one has the equality $\big[E(n,R,A),E(n,R,B)\big]=\big[E(n,A),E(n,B)\big]$ and many further corollaries can be derived for rings subject to commutativity conditions.

math.RA↗

arXiv2019

An elementary description of $K_1(R)$ without elementary matrices

Let $R$ be a ring with unit. Passing to the colimit with respect to the standard inclusions $GL(n,R) \to GL(n+1,R)$ (which add a unit vector as new last row and column) yields, by definition, the stable linear group $GL(R)$; the same result is obtained, up to isomorphism, when using the "opposite" inclusions (which add a unit vector as new first row and column). In this note it is shown that passing to the colimit along both these families of inclusions simultaneously recovers the algebraic $K$-group $K_1(R) = GL(R)/E(R)$ of~$R$, giving an elementary description that does not involve elementary matrices explicitly.

math.KT↗

arXiv2018

Jump loci for the rank of matrices and Betti numbers of chain complexes over Laurent polynomial rings

Let $K$ be a non-empty set of ideals of the commutative ring $R$, closed under taking smaller ideals. A subset $X$ of the group ring $R[\mathbb{Z}^s]$ is called a $K$-set if the ideal generated by the coefficients of the elements of $X$ is in $K$. For $X$ not a $K$-set we investigate the set of those homomorphisms $p \colon \mathbb{Z}^s \to \mathbb{Z}^t$ such that $p_*(X)$ is a $K$-set. We also consider corresponding notions of rank of matrices and Betti numbers of chain complexes; this includes an analysis of the case of McCoy rank. Our setup also recovers results on jump loci obtained by Kohno and Pajitnov as a special case.

math.AC↗

arXiv2018

Generation of relative commutator subgroups in Chevalley groups. II

In the present paper, which is a direct sequel of our paper [12] joint with Roozbeh Hazrat, we prove unrelativised version of the standard commutator formula in the setting of Chevalley groups. Namely, let $Φ$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a commutative ring and let $I,J$ be two ideals of $R$. We consider subgroups of the Chevalley group $G(Φ,R)$ of type $Φ$ over $R$. The unrelativised elementary subgroup $E(Φ,I)$ of level $I$ is generated (as a group) by the elementary unipotents $x_α(ξ)$, $α\inΦ$, $ξ\in I$, of level $I$. Obviously, in general $E(Φ,I)$ has no chances to be normal in $E(Φ,R)$, its normal closure in the absolute elementary subgroup $E(Φ,R)$ is denoted by $E(Φ,R,I)$. The main results of [12] implied that the commutator $\big[E(Φ,I),E(Φ,J)]$ is in fact normal in $E(Φ,R)$. In the present paper we prove an unexpected result that in fact $\big[E(Φ,I),E(Φ,J)]=\big[E(Φ,R,I),E(Φ,R,J)\big]$. It follows that the standard commutator formula also holds in the unrelativised form, namely $\big[E(Φ,I),C(Φ,R,J)]=\big[E(Φ,I),E(Φ,J)\big]$, where $C(Φ,R,I)$ is the full congruence subgroup of level $I$. In particular, $E(Φ,I)$ is normal in $C(Φ,R,I)$.

math.GR↗

arXiv2015

Triangular objects and systematic K-theory

We investigate modules over "systematic" rings. Such rings are "almost graded" and have appeared under various names in the literature; they are special cases of the G-systems of Grzeszczuk. We analyse their K-theory in the presence of conditions on the support, and explain how this generalises and unifies calculations of graded and filtered K-theory scattered in the literature. Our treatment makes systematic use of the formalism of idempotent completion and a theory of triangular objects in additive categories, leading to elementary and transparent proofs throughout.

math.KT↗

arXiv2014

A matrix description for $K_1$ of graded rings

The current paper is dedicated to the study of the classical $K_1$ groups of graded rings. Let $A$ be a $Γ$ graded ring with identity $1$, where the grading $Γ$ is an abelian group. We associate a category with suspension to the $Γ$ graded ring $A$. This allows us to construct the group valued functor $K_1$ of graded rings. It will be denoted by $K_1^{gr}$. It is not only an abelian group but also a $\mathbb Z[Γ]$-module. From the construction, it follows that there exists "locally" a matrix description of $K_1^{gr}$ of graded rings. The matrix description makes it possible to compute $K_1^{gr}$ of various types of graded rings. The $K_1^{gr}$ satisfies the well known $K$-theory exact sequence $$ K_{1}^{gr}(A,I)\to K_1^{gr}(A)\to K_1^{gr}(A/I) $$ for any graded ideal $I$ of $A$. The above is used to compute $K^{gr}_1$ of cross products.

math.KT↗

arXiv2012

Generation of relative commutator subgroups in Chevalley groups

Let $Φ$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a commutative ring and let $I,J$ be two ideals of $R$. In the present paper we describe generators of the commutator groups of relative elementary subgroups $\big[E(Φ,R,I),E(Φ,R,J)\big]$ both as normal subgroups of the elementary Chevalley group $E(Φ,R)$, and as groups. Namely, let $x_{\a}(ξ)$, $\a\inΦ$, $ξ\in R$, be an elementary generator of $E(Φ,R)$. As a normal subgroup of the absolute elementary group $E(Φ,R)$, the relative elementary subgroup is generated by $x_{\a}(ξ)$, $\a\inΦ$, $ξ\in I$. Classical results due to Michael Stein, Jacques Tits and Leonid Vaserstein assert that as a group $E(Φ,R,I)$ is generated by $z_{\a}(ξ,η)$, where $\a\inΦ$, $ξ\in I$, $η\in R$. In the present paper, we prove the following birelative analogues of these results. As a normal subgroup of $E(Φ,R)$ the relative commutator subgroup $\big[E(Φ,R,I),E(Φ,R,J)\big]$ is generated by the following three types of generators: i) $\big[x_α(ξ),z_α(ζ,η)\big]$, ii) $\big[x_α(ξ),x_{-α}(ζ)\big]$, and iii) $x_α(ξζ)$, where $α\inΦ$, $ξ\in I$, $ζ\in J$, $η\in R$. As a group, the generators are essentially the same, only that type iii) should be enlarged to iv) $z_α(ξζ,η)$. For classical groups, these results, with much more computational proofs, were established in previous papers by the authors. There is already an amazing application of these results, namely in the recent work of Alexei Stepanov on relative commutator width.

math.RA↗

arXiv2012

Relative commutator calculus in Chevalley groups

We revisit localisation and patching method in the setting of Chevalley groups. Introducing certain subgroups of relative elementary Chevalley groups, we develop relative versions of the conjugation calculus and the commutator calculus in Chevalley groups $G(Φ,R)$, $\rk(Φ)\geq 2$, which are both more general, and substantially easier than the ones available in the literature. For classical groups such relative commutator calculus has been recently developed by the authors in \cite{RZ,RNZ}. As an application we prove the mixed commutator formula, \[ \big [E(Φ,R,\ma),C(Φ,R,\mb)\big ]=\big [E(Φ,R,\ma),E(Φ,R,\mb)\big], \] for two ideals $\ma,\mb\unlhd R$. This answers a problem posed in a paper by Alexei Stepanov and the second author.

math.RA↗

arXiv2012

Commutator width in Chevalley groups

The present paper is the [slightly expanded] text of our talk at the Conference "Advances in Group Theory and Applications" at Porto Cesareo in June 2011. Our main results assert that [elementary] Chevalley groups very rarely have finite commutator width. The reason is that they have very few commutators, in fact, commutators have finite width in elementary generators. We discuss also the background, bounded elementary generation, methods of proof, relative analogues of these results, some positive results, and possible generalisations.

math.RA↗

arXiv2012

Multiple Commutator Formulas for Unitary Groups

Let $(\FormR)$ be a form ring such that $A$ is quasi-finite $R$-algebra (i.e., a direct limit of module finite algebras) with identity. We consider the hyperbolic Bak's unitary groups $\GU(2n,\FormR)$, $n\ge 3$. For a form ideal $(I,Γ)$ of the form ring $(\FormR)$ we denote by $\EU(2n,I,Γ)$ and $\GU(2n,I,Γ)$ the relative elementary group and the principal congruence subgroup of level $(I,Γ)$, respectively. Now, let $(I_i,Γ_i) $, $i=0,...,m$, be form ideals of the form ring $(A,Λ)$. The main result of the present paper is the following multiple commutator formula [\big[\EU(2n,I_0,Γ_0),&\GU(2n,I_1,Γ_1),\GU(2n, I_2,Γ_2),..., \GU(2n,I_m,Γ_m)\big]= &\big[\EU(2n,I_0,Γ_0),\EU(2n,I_1,Γ_1),\EU(2n,I_2,Γ_2),..., \EU(2n, I_m, Γ_m)\big],] which is a broad generalization of the standard commutator formulas. This result contains all previous results on commutator formulas for classical like-groups over commutative and finite-dimensional rings.

math.RA↗