Book contents
- Frontmatter
- Contents
- Introduction
- 1 Projective class and torsion
- 2 Graded and bounded categories
- 3 End invariants
- 4 Excision and transversality in K-theory
- 5 Isomorphism torsion
- 6 Open cones
- 7 K-theory of C1(A)
- 8 The Laurent polynomial extension category A[z, z–1]
- 9 Nilpotent class
- 10 K-theory of A[z, z–1]
- 11 Lower K-theory
- 12 Transfer in K-theory
- 13 Quadratic L-theory
- 14 Excision and transversality in L-theory
- 15 L-theory of C1(A)
- 16 L-theory of A[z, z–1]
- 17 Lower L-theory
- 18 Transfer in L-theory
- 19 Symmetric L-theory
- 20 The algebraic fibering obstruction
- References
- Index
10 - K-theory of A[z, z–1]
Published online by Cambridge University Press: 10 October 2009
- Frontmatter
- Contents
- Introduction
- 1 Projective class and torsion
- 2 Graded and bounded categories
- 3 End invariants
- 4 Excision and transversality in K-theory
- 5 Isomorphism torsion
- 6 Open cones
- 7 K-theory of C1(A)
- 8 The Laurent polynomial extension category A[z, z–1]
- 9 Nilpotent class
- 10 K-theory of A[z, z–1]
- 11 Lower K-theory
- 12 Transfer in K-theory
- 13 Quadratic L-theory
- 14 Excision and transversality in L-theory
- 15 L-theory of C1(A)
- 16 L-theory of A[z, z–1]
- 17 Lower L-theory
- 18 Transfer in L-theory
- 19 Symmetric L-theory
- 20 The algebraic fibering obstruction
- References
- Index
Summary
The splitting theorem of Bass, Heller and Swan [8] and Bass [7] for the torsion group of the Laurent polynomial extension A[z, z–1] of a ring A
will now be generalized to the torsion group of the finite Laurent extension A[z, z–1] of a filtered additive category A
The proof makes use the Mayer-Vietoris presentations of §8 and the nilpotent objects of §9 to obtain a split exact sequence
and the analogue with K1iso replace by K1
as well as a version for the Whitehead torsion and reduced class groups
Given an object L in A and j ∈ Z define objects in G1(A)
The projections onto ζjL+ and ζjL– define morphisms in G1(A)
definition 10.1 The split surjection
sends the torsion of an isomorphism in A[z, z–1]
with inverse
to the end invariant
and the reduced nilpotent classes
of the objects (P±, v±) in Nil(P0(A)) given by
There are two distinct ways of splitting K1iso (A[z, z–1]), as the algebraically significant direct sum system
and the geometrically significant direct sum system
Similarly for K1 instead of K1iso. The algebraically significant splitting j! of i! is induced by the functor
definition 10.2 (i) The algebraically significant injection
is the splitting of B ⊕ N+ ⊕ N– with components
- Type
- Chapter
- Information
- Lower K- and L-theory , pp. 86 - 100Publisher: Cambridge University PressPrint publication year: 1992