The localization exact sequence

6 this will be extended to the range n -1, and these identifications will be used together with Proposition 3.2.1 to obtain Mayer-Vietoris exact sequences for the L-qroups of the rings with involution appearing in the cartesian square 

Proposition 3.2.2 i) Ttie liomotopy equivalence classes of e-symmetric -quadratic 

Proof I i) Immediate from Proposition 3.1.2. ii) Immediate from i). 

Proof i) It follows from Proposition 3.1.1 1) that the maps 

The pair (A,S) is m-dimensional if every f.g S-torsion A-fnodule M has a f.g. projective A-module resolution of length m+1 

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Our role model here is the localization exact sequence of algebraic K-theory, which identifies the relative K-groups K (A-... 

In S3.6 we shall apply the localization exact sequence ( e-symmetric... 

The localization exact sequence and the Mayer-Vietoris exact sequence associated to a localization-completion squat are key tools in the computations of the surgery obstruction groups L (Z(7t]) of finite groups ti due to Wall [91, Bak [2], Pardon [51, Carlsson and Milgram (11,(21, Kolster (11, Bak a Roister (11, Hambleton and Milgram (21. 

In the first instance we define some subquotient groups of S A, which are needed to define the various types of linking form that arise in the localization exact sequences of Witt groups. 

Proposition 3.4.7 i) The localization exact sequence of algebraic Poincar4 cobordism groups... 

The localization exact sequence of S3.6 will now be extended to the intermediate L-invariant subgroups xCKj (A,S) and one which is indexed by the -invariant subgroups XCk A) (m = 0,1). The generalizations may be proved in the same way as the original sequence, or else may be deduced from it using the comparison exact sequences of Sl-lO. 

We shall now apply the localization exact sequence of S3 to the L-theory of rings with involution which ate algebras over a Dedekind ring. As usual, we start with some K-theory. 

It is possible to give an alternative proof of the L-theory Mayer-Vietoris exact sequence of Proposition 6.3.1 il) (the localization-completion case) which avoids the localization exact sequence of 3, and is closer in spirit to the proof of the Mayer-Vietoris sequences of Proposition 6.3.1 i)... 

Even prior to the theory of Wall [4 it was clear from the work of Kervaire and Milnor (1 and Wall (2) that quadratic linking forms over (2 (n), ZZ- 0)) play an important role in surgery obstruction theory, in the first instance as a computational tool for finite groups n. Later, Passman and Petrie (1) and Connolly (1) obtained special cases of the localization exact sequence... 

In S5 the localization exact sequence of 3 is applied to obtain splitting theorems for the L-groups of the a-twisted polynomial extensions A (x), A (x,x of a ring with... 

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