Question

I think this question should already be abound in literature but the only place I find is from this article:

http://math.uchicago.edu/~lxiao/files/notes/Fundamental%20Groups.pdf

which seems to be elaborating this definition:

http://en.wikipedia.org/wiki/%C3%89tale_fundamental_group

but unfortunately as I do not understand much algebraic geometry I do not how to make use of this definition.

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I am thinking about extending classical Bott periodicity to arbitrary rings that is good enough (UFD, for example). By extending I mean that I want to measure infinite matrices of entires in a ring $R$ with determinant 1 by the "one point compactification" of $R^{n}$ via introducing some topology. Hence in the classical case we can measure $U$ by $S^{n}$. I want to ask:

1): Is this possible? ( I thought about it over a bus trip but do not know how to establish universal bundles if the base ring is discrete, so I am stuck in here).

2): Is there any previous such constructions? What are their properties?

I feel there must be something well-known because Bott-periodicity theorem is a very old theorem. I do not know whether this is more appropriate for MO or for stack exchange, but I decided to put in here.

Answer 1

Perhaps an easier approach if you wanted to avoid going into the world of schemes would be to define orthogonal, symplectic, and unitary matrices over $R$ in analogy with the case when $R=\mathbb{C}$. We should be able to at least find conditions on $R$ for which this could work (defining your ``good ring’’ of the title). For example, to define determinant and $GL(R)$, it's sufficient for $R$ to be commutative. To get the infinite unitary group $U(R)$ you need $R$ to have an involution (because of conjugate transpose). A good place to look for more info on this would be The Book of Involutions: http://www.amazon.com/Book-Involutions-Colloquium-Publications-Mathematical/dp/0821809040

We know that when quaternion algebras have involutions. This theory is developed in depth in the book. The book also looks at more general central simple algebras and includes a great example for endomorphism algebras on page 23. They prove that a central simple algebra $A$ over field $F$ has an involution of the first kind (fixing the center elementwise) iff $A\otimes_F A$ splits. If $B$ is a central simple algebra over $K$, a separable quadratic extension of $F$, then $B$ has an involution of the second kind (acts as an order 2 automorphism on the center) which fixes $F$ pointwise iff $N_{K/F}(B)$ splits.

This book also discusses the connection between Clifford algebras and orthogonal involutions, between the discriminant algebra and unitary involutions, and between Tits algebras and irreducible representations of the classical groups. So you might be able to use these connections to at least get some hands-on examples to see if Bott Periodicity extends (although you of course already have it for Clifford algebras).

Anyway, I’d also be very curious to see if the other approach (via schemes and algebraic homotopy groups) would work. But it seems much harder to me.