topology on tensor product

Stream: maths

[Kevin Buzzard]

If $D$ is a quaternion algebra over a number field $K$ (for example the ring of 2x2 matrices over $K$), if $\widehat{\Z}$ denotes the profinite completion of $\Z$ and if $\mathbb{A}^f_F$ denotes the finite adeles of the number field $F$, then I believe that all of $D\otimes_{\Z}\widehat{\Z}$, $D\otimes_{\mathbb{Q}}\mathbb{A}_{\mathbb{Q}}^f$ and $D\otimes_{\mathbb{F}}\mathbb{A}_F^f$ are "canonically" isomorphic, and I need to choose one for the "finite adelification" of $D$. I propose the last one: $D\otimes_{\mathbb{F}}\mathbb{A}_F^f$. I would now like to put a topology on this, but my understanding of tensor products is that in general the tensor product of two topological rings (or even topological vector spaces) admits more than one topology.

I would imagine that in my situation this is not an issue, because $D$ is four-dimensional over the field $F$, so $D\otimes_{\mathbb{F}}\mathbb{A}_F^f$ is isomorphic to $(\mathbb{A}_F^f)^4$ and we can give it the induced topology. But what's the correct way of setting all this up in general?

[Antoine Chambert-Loir]

Here, I would say you are looking at the adelic points of the “quaternion algebra schemes”, and this is what it topology is.

[Kevin Buzzard]

I am unclear about how the Z-hat story fits into this. I am also concerned that the topologies on (D/K)(A_K) and (Res D/Q)(A_Q) are not going to be defeq.

[Antoine Chambert-Loir]

1) Good books (or a paper by Conrad, etc.) explain carefully how to define the topology on points of schemes when the topological ring has some property. That should work for Z hat.
Regarding your second sentence, I feel you're too greedy :
2) You need a theorem to identify (D/Z)(A_K) with D(A_K)/Z(A_K)
3) You need another theorem to prove that (Res D) (A_Q) = D(A_K) is a topological isomorphism
4) And I think you need a further theorem to pass to the quotient.

[Kevin Buzzard]

Oh sorry I didn't mean the quotient, I just meant "A_K-points of D, considered as scheme over K" = "A_Q-points of D, considered as a scheme over Q". Yes I've seen the paper of Conrad: he would define the topology on D(A_K) by embedding it into the set-theoretic maps from K[e1,e2,e3,e4] to A_K where e_i are a K-basis of D, and he would define the topology on the "identical" set D_Q(A_Q) by embedding it into the totally different space of functions from Q[f_1,f_2,f_3,...,f_{4d}] to A_Q. So up to propeq it seems to really matter which of these definitions I choose.

Re Z-hat: what I am unclear about here is whether $D\otimes_{\Z}{\widehat{Z}}$ has a natural topology, because I'm now viewing D as a scheme not of finite type over Spec(Z) or something :-/

[Antoine Chambert-Loir]

Re Z-hat, you're right, what I wrote more or less assumed that one talks about schemes of finite type, although the basic definition is simple: if X is scheme over a topological ring R, endow X(R) with the smallest (coarsest) topology such that for every Zariski open set U, U(R) is open in X(R), and every regular function f on U, f_R: U(R)->R is continuous. The question, then, becomes about commuting products and product topology, identifying the topology of X(R) with the induced topology of R^n when X is a closed subscheme of A^n.