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 $\mathbb{C}$.
