Coupling invariants
In Chap. 41 A is a Dedekind ring with involution, and F = S−1 A is the quotient field, with S = A\{0} ⊂ A. The undecorated L-groups L*, L* are the projective L-groups Lp*, L* p.
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In Chap. 41 A is a Dedekind ring with involution, and F = s- 1 A is the quotient field, with S = A\{0} CA. The undecorated £-groups L*, L. are the projective L-groups L;,
a.
The results of Chap. 39 for the endomorphism £-groups of a field F are now extended to the endomorphism £-groups of a Dedekind ring A, using the ordinary £-theory of the appropriate localizations of the polynomial extensions: (i) A[x], with x = x, (ii) A[s], with s = 1 - s, {iii) A[z, z- 1], with z = z- 1 . Note that if A is not a field these polynomial extensions have global dimension 2, so that they are not themselves Dedekind rings {although some of the localizations may be so, e.g. if S = A\{0} C A is inverted). The case most directly relevant to the computation of the high-dimensional knot cobordism groups C2 .+1 is {ii) with A = Z, as will be discussed in Chap. 42 below. However, the other cases are also of interest, for example in open books and the bordism of automorphisms of manifolds. The endomorphism class group of A splits as Endo(A)
=
ED En~(z)co (A)
p(z)
with the sum running over all the monic irreducible polynomials p(x) E A[x] (14.16), and p(z)co { Nilo(A) = Ko(A) if p(x) En) over A together with an endomorphism f : M----+ M such that
=
J*c/> for some N submodule
~
c/>f : M----+ M* , p(f)N
1. If N
~
=
0 : M----+ M
2 reduce to the case N = 1, as follows. Consider the
K = p(f)N- 1(M)
C
M.
The A-submodule
L
=
{x EM IaxE K for some a E A\{0}}
~
M
contains K, with L I K the torsion submodule of M I K. The quotient Amodule MIL is projective, so that L is a direct summand of M. For any x1, x2 E L there exist a1, a2 E A\ {0}, Y1, Y2 E M such that
a1x1
=
p(/)N-l(Yl) , a2x2
a2c/>(x1, x2)a1
=
=
p(/)N-l(Y2) E K ,
cp{p(f)N-l (yl),p(f)N-l(y2))
= cp(yl,p(/)2N-2(y2))
= 0EA
(since 2N- 2 ~ N),
cp(x1,x2) = 0 EA. Thus Lis a sublagrangian of (M, 4>) and
(M, c/>, f)
{iii) Let
=
(LJ. I L, [c/>], f) E im(LEnd~(z) (A, f)----+LEnd~(z)oo (A, f)) .
41A. Endomorphism £-theory
p(x)
=
d
La; xi E Mz(A) (ad i=O
589
= 1)
with reduction p(x)
=
xdp(x- 1 )
d
=
L:ad-ixi E A[x]. i=O
The t:-symmetric £-groups of A[x,p(x)- 1 ] are such that
= = by the special caseS= {xip(x)k there is defined an isomorphism
-n
Ln(A, t:) E9 LEndp(z)(A, t:)
LEnd;(z)(A,t:)
li,k
~ 0} C A[x] of 35.4. In particular,
LEnd~(z) (A, t:) ~ L~o(A) (A[x,p(x)- 1 ], t:)
;
(M, ¢,f) ----+ (M[x,p(x)- 1], ¢(1- xf)) . The stated localization exact sequence is just the localization exact sequence of 25.4
... ----+ LRo(A) (A[x,p(x)- 1 ], t:) ----+ Ln(F[x,p(x)- 1 ], t:) ----+ Ln(A[x,p(x)- 1 ],S,t:) ----+ L'!..- 1 (A[x,p(x)- 1 ],t:) ----+ Ko(A)
The ring R
=
A[x]/(p(x))
is an integral domain which is an order with quotient field E
=
F[x]/(p(x)) .
As in 41.3 choose a non-zero involution-preserving F-linear map h: E---+F, e.g. the trace map trE/F if p(x) is separable. The inverse different (alias codifferent) of R Ll = {v E Eih(Rv) fA} is equipped with a pairing
Ll x R ----+ A ; (u, v) ----+ h(uv) such that the adjoint A-module morphisms
L
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