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Math 208  Problem Set 3

2022

Projective modules

For both problems use only R-valued functions.

1. (Try to do this problem before looking at the next prob- lem.) Let

A = {f 2 C(R) : f(t +1) = f(t),t 2 R}.

So A can be thought of as the algebra of continuous func- tions on the circle T = R/Z. For any p 2 N let

⌅p(+) = {⇠ 2 C(R) : ⇠(t +p) = +⇠(t),t 2 R}. and

= {⇠ 2 C(R) : ⇠(t +p) = −⇠(t),t 2 R}.       These are modules over A for the evident pointwise actions. Prove that these are projective modules. Towards this end:

a) Deﬁne in a natural way A-valued inner products on these modules.

b) Deﬁne standard module frames for these modules. (For this notice what the ﬁber is over each point of T. Find a convenient open cover of T, with the smallest possible number of subsets, over each of which the given bundle is trivial. You may ﬁnd pieces of the function sine useful.)

2. Let O(n) be the group of orthogonal n ⇥n matrices. For any R 2 O(n) let

VR  = {⇠ 2 C(R,Rn ) : ⇠(t +1) = R(⇠(t)),t 2 R}.

(This is sometimes called a“clutching construction”.) These are modules over A for the evident pointwise actions. Prove that these are projective modules. Towards this end:     a) Show that if R1 ,R2  2 O(n) are connected by a contin- uous path in O(n) then VR1        VR2 . (One says they are “homotopic” .)

b) Notice how to express VR1  ⊕ VR2   as VR  for a bigger-sized

.

c) Let A be any unital algebra over R. Use the fact that the matrix R for rotation by 90 degrees is path-connected to the identity matrix, and use conjugation by R, to prove that for any a,b 2 A the matrix (0(ab) 1(0) ) is path connected to the matrix (0(a) b(0) ). [And so, path connected to the matrix ( 0(ba) 1(0) ). This is a key calculation in the proof that the group K1 (A) is commutative.]

d) From part c) it follows that for any invertible element a 2 A the matrix  0(a) a 1     is path connected to the matrix (0(1) 1(0) ). Use this to show that each module VR  is projective.

e) For each ⌅p(+)  and ⌅ of problem 1 ﬁnd an R 2 O(n) for some n such that they are isomorphic to VR .

f) Any R 2 O(n) can be“diagonalized”into blocks of size 2 ⇥ 2 and 1 ⇥ 1 on the diagonal. Every vector bundle over T has cross-section module isomorphic to some VR . (I won’t ask you to prove this, but I expect that you could.) With this information and the parts above, give a simple description of the semigroup S(T) of isomorphism classes of vector bundles over T.