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Register a new userBloch-Wigner theorem over rings with many units II
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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 Suslins 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 Suslins 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.
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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 Kallens 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.
For a commutative ring R with many units, we describe the kernel of H_3(inc): H_3(GL_2(R), Z) --> H_3(GL_3(R), Z). Moreover we show that the elements of this kernel are of order at most two. As an application we study the indecomposable part of K_3(R).
A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups $G$ for which the integral group ring $mathbb{Z}G$ has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that $mathbb{Z}G$ has SFC provided at most one copy of the quaternions $mathbb{H}$ occurs in the Wedderburn decomposition of the real group ring $mathbb{R}G$. This generalises the Eichler condition in the case of integral group rings.
We study five different types of the homology of a Lie algebra over a commutative ring which are naturally isomorphic over fields. We show that they are not isomorphic over commutative rings, even over $mathbb Z,$ and study connections between them. In particular, we show that they are naturally isomorphic in the case of a Lie algebra which is flat as a module. As an auxiliary result we prove that the Koszul complex of a module $M$ over a principal ideal domain that connects the exterior and the symmetric powers $0to Lambda^n Mto M otimes Lambda^{n-1} M to dots to S^{n-1}M otimes M to S^nMto 0 $ is purely acyclic.
The goal of the present paper is to push forward the frontiers of computations on Farrell-Tate cohomology for arithmetic groups. The conjugacy classification of cyclic subgroups is reduced to the classification of modules of group rings over suitable rings of integers which are principal ideal domains, generalizing an old result of Reiner. As an example of the number-theoretic input required for the Farrell-Tate cohomology computations, we discuss in detail the homological torsion in PGL(3) over principal ideal rings of quadratic integers, accompanied by machine computations in the imaginary quadratic case.
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