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 …
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 $\mathcal{RB}(A)$.
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 …
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 $H_3(\textrm{SL}_2(A), \textrm{SM}_2(A),\mathbb{Z})$.
In this paper we investigate the third homology of the projective special linear group $\textrm{PSL}_2(A)$. As a result of our investigation we prove a projective refined Bloch-Wigner exact sequence over …
In this paper we investigate the third homology of the projective special linear group $\textrm{PSL}_2(A)$. 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 algebraically closed fields is a classical result.
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 \[ …
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 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 third homology groups of the special linear group $\textrm{SL}_2(A)$ and the projective special linear group $\textrm{PSL}_2(A)$, through the natural surjective map $\textrm{SL}_2(A) \to \textrm{PSL}_2(A)$. …
In this article we study the third homology groups of the special linear group $\textrm{SL}_2(A)$ and the projective special linear group $\textrm{PSL}_2(A)$, through the natural surjective map $\textrm{SL}_2(A) \to \textrm{PSL}_2(A)$. As an application we prove a refined Bloch-Wigner exact sequence over certain local domains.
In this article we study the low dimensional homology of the projective linear group $\textrm{PGL}_2(A)$ over a $\textrm{GE}_2$-ring $A$. In particular, we prove a Bloch-Wigner type exact sequence over local …
In this article we study the low dimensional homology of the projective linear group $\textrm{PGL}_2(A)$ over a $\textrm{GE}_2$-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 $H_3(\textrm{PGL}_2(A),\mathbb{Z}\left[\frac{1}{2}\right])\simeq K_3^{\textrm{ind}}(A)\left[\frac{1}{2}\right]$.
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 …
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 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 …
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 study the low dimensional homology of the projective linear group $\textrm{PGL}_2(A)$ over a $\textrm{GE}_2$-ring $A$. In particular, we prove a Bloch-Wigner type exact sequence over local …
In this article we study the low dimensional homology of the projective linear group $\textrm{PGL}_2(A)$ over a $\textrm{GE}_2$-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 $H_3(\textrm{PGL}_2(A),\mathbb{Z}\left[\frac{1}{2}\right])\simeq K_3^{\textrm{ind}}(A)\left[\frac{1}{2}\right]$.
In this article we study the third homology groups of the special linear group $\textrm{SL}_2(A)$ and the projective special linear group $\textrm{PSL}_2(A)$, through the natural surjective map $\textrm{SL}_2(A) \to \textrm{PSL}_2(A)$. …
In this article we study the third homology groups of the special linear group $\textrm{SL}_2(A)$ and the projective special linear group $\textrm{PSL}_2(A)$, through the natural surjective map $\textrm{SL}_2(A) \to \textrm{PSL}_2(A)$. As an application we prove a refined Bloch-Wigner exact sequence over certain local domains.
In this paper we investigate the third homology of the projective special linear group $\textrm{PSL}_2(A)$. As a result of our investigation we prove a projective refined Bloch-Wigner exact sequence over …
In this paper we investigate the third homology of the projective special linear group $\textrm{PSL}_2(A)$. 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 algebraically closed fields is a classical result.
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 \[ …
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 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 $x\in A$, $a,b,c \in A^{\times}$.
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 …
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 $\mathcal{RB}(A)$.
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 …
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 $H_3(\textrm{SL}_2(A), \textrm{SM}_2(A),\mathbb{Z})$.
Abstract We introduce a refinement of the Bloch-Wigner complex of a field F . This refinement is complex of modules over the multiplicative group of the field. Instead of computing …
Abstract We introduce a refinement of the Bloch-Wigner complex of a field F . This refinement is complex of modules over the multiplicative group of the field. Instead of computing K 2 (F) and K ind 3 (F) - as the classical Bloch-Wigner complex does - it calculates the second and third integral homology of SL 2 (F) . On passing to F × -coinvariants we recover the classical Bloch-Wigner complex. We include the case of finite fields throughout the article.
One way to study the cohomology of a group T of finite virtual cohomological dimension is to find a finite dimensional contractible space X on which F acts properly (such …
One way to study the cohomology of a group T of finite virtual cohomological dimension is to find a finite dimensional contractible space X on which F acts properly (such a space X always exists by [18], 1-7), and to then analyze the action, In this paper we want to consider the arithmetic groups SL2(0) and PSL2(0) = SL2(O)/±I where 0 is the ring of integers in an imaginary quadratic number field fc; the classical choice of X in this case is hyperbolic three-space H, i.e., the associated symmetric space SL2(C)/SU(2). As early as 1892 Bianchi [2] exhibited fundamental domains for the action of PSL2(0) on H for some small values of the discriminant. The space H has also turned out to be very useful in studying the relation between automorphic forms associated to SL2(0) and the cohomology of SL2(O) (cf. [12], [10]), and in studying the topology of certain hyperbolic 3-manifolds (cf, [25]). However, this choice of X is inconvenient for actual explicit computations of the cohomology of r = (P)SL2(0) with integral coefficients because the dimension of H is three, whereas the virtual cohomological dimension of F is two, indicating that it may be possible for F to act properly on a contractible space of dimension two; in addition, the quotient F\H is not compact. A more useful space X for our purposes is given by work of Mendoza [14], which we recall in §3; using Minkowski's reduction theory (cf. §2), he constructs a f-invariant 2-dimensional deformation retract I(fe) of H such that the quotient of I(fc) by any subgroup of F of finite index is compact; I(k) is endowed with a natural CW structure such that the action of F is cellular and the quotient F\I(k) is a finite CW-complex. The main object of this paper is to show how this construction can be used to completely determine the integral homology groups of PSL2(0). This is done by analyzing a spectral sequence which relates the homology of PSL2(0) to the homology of the quotient space PSL2(0)\I(k) and the homology of the stabilizers of the cells (cf. [5], VII). We will confine our computations to the cases where 0 is a euclidean ring, i.e., @-@_d is the ring of integers in fc = Q(V-d) for d = 1,2,3,7 and 11. We will write out in detail the case d = 2 (cf. §5), which contains
Foundations Introduction to Witt rings Quaternion algebras and their norm forms The Brauer-Wall group Clifford algebras Local fields and global fields Quadratic forms under algebraic extensions Formally real fields, real-closed …
Foundations Introduction to Witt rings Quaternion algebras and their norm forms The Brauer-Wall group Clifford algebras Local fields and global fields Quadratic forms under algebraic extensions Formally real fields, real-closed fields, and pythagorean fields Quadratic forms under transcendental extensions Pfister forms and function fields Field invariants Special topics in quadratic forms Special topics on invariants Bibliography Index.
For a commutative ring A we consider a related graph, Γ(A), whose vertices are the unimodular rows of length 2 up to multiplication by units. We prove that Γ(A) is …
For a commutative ring A we consider a related graph, Γ(A), whose vertices are the unimodular rows of length 2 up to multiplication by units. We prove that Γ(A) is path-connected if and only if A is a GE2-ring, in the terminology of P. M. Cohn. Furthermore, if Y(A) denotes the clique complex of Γ(A), we prove that Y(A) is simply connected if and only if A is universal for GE2. More precisely, our main theorem is that for any commutative ring A the fundamental group of Y(A) is isomorphic to the group K2(2,A) modulo the subgroup generated by symbols.
We calculate the structure of H3(SL2(Q),Z[12]). Let H3(SL2(Q),Z)0 denote the kernel of the (split) surjective homomorphism H3(SL2(Q),Z)→K3ind(Q). Each prime number p determines an operator 〈p〉 on H3(SL2(Q),Z) with square the …
We calculate the structure of H3(SL2(Q),Z[12]). Let H3(SL2(Q),Z)0 denote the kernel of the (split) surjective homomorphism H3(SL2(Q),Z)→K3ind(Q). Each prime number p determines an operator 〈p〉 on H3(SL2(Q),Z) with square the identity. We prove that H3(SL2(Q),Z[12])0 is the direct sum of the (−1)-eigenspaces of these operators. The (−1)-eigenspace of 〈p〉 is the scissors congruence group, over Z[12], of the field Fp, which is a cyclic group whose order is the odd part of p+1.
Often perceived as dry and abstract, homological algebra nonetheless has important applications in a number of important areas, including ring theory, group theory, representation theory, and algebraic topology and geometry. …
Often perceived as dry and abstract, homological algebra nonetheless has important applications in a number of important areas, including ring theory, group theory, representation theory, and algebraic topology and geometry. Although the area of study developed almost 50 years ago, a textbook at this level has never before been available. An Elementary Approach to Homological Algebra fills that void. Designed to meet the needs of beginning graduate students, the author presents the material in a clear, easy-to-understand manner with many examples and exercises. The book's level of detail, while not exhaustive, also makes it useful for self-study and as a reference for researchers.
Abstract Research mathematicians in algebraic topology will be interested in this new attempt to classify homotopy types of simply connected CW-complexes. This book provides a modern treatment of a long …
Abstract Research mathematicians in algebraic topology will be interested in this new attempt to classify homotopy types of simply connected CW-complexes. This book provides a modern treatment of a long established set of questions in algebraic topology. The author is a leading figure in this important research area.
For rings with a large number of units the authors prove a strengthened theorem on homological stabilization: the homomorphism Hk(GLn(A)) → Hk(GL(A)) is surjective for n ≥ k + sr …
For rings with a large number of units the authors prove a strengthened theorem on homological stabilization: the homomorphism Hk(GLn(A)) → Hk(GL(A)) is surjective for n ≥ k + sr A – 1 and bijective for n ≥ k + sr A. If A is a local ring with an infinite residue field, then this result admits further refinement: the homomorphism Hn(GLn(A)) → Hn(GL(A)) is bijective and the factor group Hn(GL(A)) / Hn(GLn -1(A)) is canonically isomorphic to Milnor's nth K-group of the ring A. The results are applied to compute the Chow groups of algebraic varieties. Bibliography: 16 titles.
This paper gives a description of the torsion and cotorsion in the Milnor groups and for an arbitrary field . The main result is that, for any natural number with …
This paper gives a description of the torsion and cotorsion in the Milnor groups and for an arbitrary field . The main result is that, for any natural number with , and the group is uniquely -divisible if . This theorem is a consequence of an analogue of Hilbert's Theorem 90 for relative -groups of extensions of semilocal principal ideal domains. Among consequences of the main result we obtain an affirmative solution of the Milnor conjecture on the bijectivity of the homomorphism , where is the ideal of classes of even-dimensional forms in the Witt ring of the field , as well as a more complete description of the group for all global fields.
It is proved that the group of matrices of order two with determinant 1 over a Dedekind ring of arithmetic type is generated by elementary matrices if there are infinitely …
It is proved that the group of matrices of order two with determinant 1 over a Dedekind ring of arithmetic type is generated by elementary matrices if there are infinitely many invertible elements in this ring. We also obtain a more general result, describing the group generated by elementary matrices belonging to a congruence subgroup. Bibliography: 6 items.
In this note we compute the integral cohomology groups of the subgroups $\Gamma _0(n)$ of $SL(2, \mathbf {Z})$ and the corresponding subgroups $P\Gamma _0(n)$ of its quotient, the classical modular …
In this note we compute the integral cohomology groups of the subgroups $\Gamma _0(n)$ of $SL(2, \mathbf {Z})$ and the corresponding subgroups $P\Gamma _0(n)$ of its quotient, the classical modular group, $PSL(2, \mathbf {Z}).$
Abstract The present paper studies the group homology of the groups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>[</m:mo> <m:mi>C</m:mi> <m:mo>]</m:mo> </m:mrow> </m:mrow> …
Abstract The present paper studies the group homology of the groups <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>[</m:mo> <m:mi>C</m:mi> <m:mo>]</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\operatorname{SL}_{2}(k[C])} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>PGL</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>[</m:mo> <m:mi>C</m:mi> <m:mo>]</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\operatorname{PGL}_{2}(k[C])} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>C</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mover> <m:mi>C</m:mi> <m:mo>¯</m:mo> </m:mover> <m:mo>∖</m:mo> <m:mrow> <m:mo>{</m:mo> <m:msub> <m:mi>P</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mi>…</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>P</m:mi> <m:mi>s</m:mi> </m:msub> <m:mo>}</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {C=\overline{C}\setminus\{P_{1},\dots,P_{s}\}} is a smooth affine curve over an algebraically closed field k . It is well known that these groups act on a product of trees and the quotients can be described in terms of certain equivalence classes of rank two vector bundles on the curve <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mover> <m:mi>C</m:mi> <m:mo>¯</m:mo> </m:mover> </m:math> {\overline{C}} . There is a natural subcomplex consisting of cells with suitably non-trivial isotropy group. The paper provides explicit formulas for the equivariant homology of this “parabolic subcomplex”. These formulas also describe group homology of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>SL</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo></m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo></m:mo> <m:mrow> <m:mo>[</m:mo> <m:mi>C</m:mi> <m:mo>]</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\operatorname{SL}_{2}(k[C])} above degree s , generalizing a result of Suslin in the case <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>s</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {s=1} .