Timeline for base change of adele rings
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 30, 2021 at 22:32 | vote | accept | Alice | ||
| Oct 29, 2021 at 16:44 | history | edited | GH from MO |
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| Oct 29, 2021 at 16:44 | answer | added | GH from MO | timeline score: 4 | |
| Oct 29, 2021 at 4:22 | comment | added | KConrad | Do you see how $\mathbb A_K \otimes_K L$ and $\mathbb A_L$ are both $\mathbb A_K$-modules, and that both of these are free $\mathbb A_K$-modules of (finite) rank $[L:K]$? If that is too general, think first about a concrete example like $K = \mathbf Q$ and $L = \mathbf Q(i)$. The point is that to prove a ring homomorphism $A \to B$ is an isomorphism where $A$ and $B$ are not just rings but also $R$-modules for a ring $R$, think linearly and try to show the mapping is an $R$-module isomorphism. | |
| Oct 29, 2021 at 3:50 | comment | added | Alice | I have an agreement to the fact that for each place $v$ there is the isomorphism $K_v \otimes_K L \to \prod_{w|v} L_w$, however the difficulty is still existing (in my thought). The map is described by the usage of restricted products. Does tensor product commute to restricted product? | |
| Oct 29, 2021 at 3:31 | comment | added | KConrad | Can you say what you have done so far? For example, can you think of a candidate mapping from the tensor product to the adele ring of the larger field? Are you familiar with the isomorphism from $K_v \otimes_K L$ to $\prod_{w\mid v} L_w$ for each place $v$ of $K$? | |
| Oct 29, 2021 at 1:36 | history | asked | Alice | CC BY-SA 4.0 |