Localization and completion in K-theory
As already recalled in the Introduction, localization and completion are the basic algebraic techniques for computing the algebraic K- and L-groups, by reducing the computation for a complicated ring to simpler rings (e.g. fields). The classic example of
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As already recalled in the Introduction, localization and completion are the basic algebraic techniques for computing the algebraic K- and £-groups, by reducing the computation for a complicated ring to simpler rings (e.g. fields). The classic example of localization and completion is the Hasse-Minkowski principle by which quadratic forms over Z are related to quadratic forms over Q and the finite fields 1Fp and the p-adic completions Zp, ~ of Z, Q (p prime). The localization of polynomial rings is particularly relevant to knot theory, starting with the way in which the Blanchfield form takes its values in the localization of Z[z, z- 1] inverting the Alexander polynomials. For any ring morphism f : A---t B the algebraic K -groups of A and B are related by a long exact sequence
"
... ---t Kn(A) ---t Kn(B) ---t Kn(f) ---t Kn-1(A) ---t ... with relative K -groups K * {!). In particular, K 1 {!) is the abelian group of equivalence classes of triples (P, Q, g) given by f.g. projective A-modules P, Q and a B-module isomorphism
subject to the equivalence relation defined by: (P, Q, g) "' (P', Q', g') if there exist a f.g. projective A-module R and an A-module isomorphism
h : P tB Q' tB R
e:!
P' tB Q ffi R
such that "'
T((g- 1 ffi g' ffi I,,R)h: I!{P tB Q' tB R) --=t I!(P ffi Q' ffi R))
=
0 E Kl(B) .
In the special case when f: A---tB = s- 1 A is the inclusion of A in the localization inverting a multiplicative subset S C A the relative K -groups K * {!) are identified with the K -groups of the exact category of homological dimension 1 S-torsion A-modules.
A. Ranicki, High-dimensional Knot Theory © Springer-Verlag Berlin Heidelberg 1998
20
4. Localization and completion inK-theory
4A. Commutative localization This section only deals with commutative localization, in which only central elements of a ring are inverted - see Chap. 9 below for noncommutative localization Definition 4.1 {i) A multiplicative subsetS c A is a subset of central elements which is closed under multiplication, with st E S for each s, t E S, and such that 1 E S. (ii) The localization s- 1 A is the ring obtained from A by inverting every s E S, with elements the equivalence classes a/s of pairs {a, s) E A x S subject to the equivalence relation (a, s) "' (b, t) if(at- bs)u
=
0 E A for some u E S.
Addition and multiplication are by a/s+b/t
=
0
abjst.
(at+bs)/st, (a/s)(b/t)
Proposition 4.2 The ring morphism i : A ----t
s- 1 A ; a ----t a/1
has the universal property that for any ring morphism f : A----t B with f {s) E 1 A----tB with B invertible for every s E S there is a unique morphism F:
s-
f
=
Fi : A ----t
s- 1 A ----t B
.
0
The morphism i : A----tS- 1 A is not injective in general. However, it is injective if S c A consists of non-zero divisors. Example 4.3 Given a ring A and a central element s E A define the multiplicative subset
The localization of A inverting S is written
s- 1 A
= A[1/s].
(i) If s E A is a non-zero divisor every non-zero element unique expression as x = ajsk (a E A,k ~ 0). (ii) If s = s 2 E
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