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I am currently reading Husemoller's wonderful book on fibre bundles, specifically the section on Clifford algebras. He defines these groups $L_k$ as follows. Let $M_k$ denote the free abelian groups generated by the irreducible $\mathbb{Z}_2$ graded $C_k$ modules. The inclusion $C_k \hookrightarrow C_{k+1}$ induces a map $i^* : M_{k+1} \rightarrow M_k$, and $L_k$ is the cokernel of this map. Then he proves that the groups are periodic of period $8$, and the sequence is $\mathbb{Z}_2, \mathbb{Z}_2, 0, \mathbb{Z}, 0, 0, 0, \mathbb{Z}$. So we have somehow recovered Bott periodicity, using the Clifford algebras corresponding to the "negative" of the standard inner product on $\mathbb{R}^n$. There is a similar result in the complex case.

We can define similar Clifford algebras for any other field, say $\mathbb{F}_p$, by taking $\mathbb{F}_p^n$ equipped with the quadratic form $-(x,y)$, where $(x,y)$ is the standard inner product, and define these groups $L_k$. What is known about these groups? Are they periodic? If so, is there some other notion of characteristic $p$ Bott periodicity which gives the same sequence of groups as $L_k$? I am aware that in some cases there is a periodicity known for the $p$-adic fields.

More generally, how well-understood are Clifford algebras for other fields? I am aware of Lam's book on quadratic forms which I very briefly skimmed before posting this question, and I couldn't find any results about these $L_k$. I also spent some time googling around, it seems like most people are interested in Clifford algbebras over $\mathbb{R}$ and $\mathbb{C}$.

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  • $\begingroup$ Would you want your modules in the definition of $M_k$ to be real or complex, or over $\mathbb F_p$ or its algebraic closure? $\endgroup$
    – LSpice
    Feb 19, 2022 at 22:21
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    $\begingroup$ @LSpice As I understand it, any Clifford algebra is an algebra over the base ring corresponding to the quadratic form you started with. So any Clifford module should also be a module over the base ring. In the case of $\mathbb{F}_p$, the modules defining M_k would then be $\mathbb{F}_p$ vector spaces. Can you have "mixed characteristic" for a Clifford algebra and a module over than algebra? $\endgroup$ Feb 19, 2022 at 22:49
  • $\begingroup$ I'm sorry, I was thinking of representations of groups, where it makes sense to have them act on vector spaces over any field. You are right that it probably doesn't make much sense to speak of a $K$-algebra acting on an $L$-vector space unless $K$ is equal to, or at least contained in, $L$. (Of course one could forget the algebra structure and just consider them as $\mathbb Z$-algebras and $\mathbb Z$-modules; but that would be unlikely to be helpful.) $\endgroup$
    – LSpice
    Feb 19, 2022 at 23:15
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    $\begingroup$ I thought I'd seen this question before (unanswered), but the closest I can find are this and this. The first points out that $K(F_q)$ is not periodic, although mod $\ell$ it is (and mod $p$, where it is zero). But is often larger than 8 or the order of the Witt group. The larger $q$, the shorter the period, so maybe something about quadratic forms works for large fields and subsumes a more complicated definition that also works for small fields. $\endgroup$ Feb 20, 2022 at 19:37

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