Timeline for Picard groups of abelian étale covers
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 20, 2012 at 2:41 | vote | accept | Piotr Achinger | ||
| Aug 20, 2012 at 1:42 | comment | added | anon | For finite Galois covers f:Y ---> X with group G, the best tool I know is the Hochschild-Serre spectral sequence for etale cohomology with coefficients in G_m (i.e., H^p(G,H^q(Y,G_m)) ===> H^{p+q}(X,G_m)) and the fact that H^1(X,G_m) = Pic(X). This gives an LES involving the map Pic(X) ---> Pic(Y)^G. It doesn't suggest a good statement when Pic^0 is 0, but does lead to counterexamples to hopes of the form Pic(Y) = Pic(X)/T; we just need to find such Y and G with Pic(Y)^G strictly smaller than Pic(Y). | |
| Aug 20, 2012 at 0:52 | comment | added | Piotr Achinger | OOPS! Yes, you're right, thank you! But what if we assume that $X$ and $Y$ have trivial $\text{Pic}^0$, does it help? Is there a good description of the cokernel of $f^*: \text{Pic }X\to \text{Pic }Y$ (or just the next group in some long exact sequence extending $f^*$)? | |
| Aug 20, 2012 at 0:26 | history | answered | anon | CC BY-SA 3.0 |