Timeline for Second summand to make projective module free
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 21, 2020 at 3:19 | vote | accept | zjs | ||
| Jun 21, 2020 at 2:21 | comment | added | Steven Landsburg | In a Dedekind ring $R$ the direct sum of two ideals is related to their product by the formula $P\oplus Q\approx PQ\oplus R$. This is quite standard material, and any commutative algebra textbook should give you all the details. | |
| Jun 21, 2020 at 2:15 | comment | added | zjs | Thank you! Do you have advice on how to show that that $M$ suffices? e.g. in $K=\mathbb{Q}[\sqrt{-5}]$ where $\frak{a}=\langle 2,1+\sqrt{-5}\rangle$ gives a nontrivial element of the class group $Cl(\mathcal{O}_K)\cong\mathbb{Z}/2\mathbb{Z}$, I'm having some trouble seeing the way to view $\frak{a}\oplus\frak{a}$ as free (either as a $\mathcal{O}_K$-module or $\mathbb{Z}$-module), and more generally how to see direct-summing of ideals as being equivalent in some way to their product. | |
| Jun 21, 2020 at 2:08 | history | answered | Steven Landsburg | CC BY-SA 4.0 |