Timeline for A form of cohomology and base change
Current License: CC BY-SA 2.5
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 19, 2010 at 18:32 | vote | accept | Andrea Ferretti | ||
| Nov 19, 2010 at 17:32 | comment | added | t3suji | Funnily, I started expanding it before your comment :) | |
| Nov 19, 2010 at 17:31 | answer | added | t3suji | timeline score: 4 | |
| Nov 19, 2010 at 17:28 | comment | added | Andrea Ferretti | @t3suji: If you expanded your second comment into an answer, I would be glad to accept it. | |
| Nov 19, 2010 at 17:21 | comment | added | Andrea Ferretti | Thank you! I forgot the hypothesis of flatness of the sheaf for the cohomology and base change! :-/ | |
| Nov 19, 2010 at 16:38 | comment | added | BCnrd | Dear Andrea: The standard base change results don't apply since there's no flatness around the interesting part of the map. | |
| Nov 19, 2010 at 16:23 | comment | added | Andrea Ferretti | As for your first example, I actually do not want to assume anything on $Z$. The interesting case for me is when $Z$ is reduced, and if I'm not wrong if the result holds in this case, it also holds on the non-reduced case. But it would be interesting to assume that $Y$ is reduced. | |
| Nov 19, 2010 at 16:20 | comment | added | Andrea Ferretti | @t3suji: I don't understand your second example. It seems to me that in this case the two sides agree by the standard cohomology and base change results, since there is no second cohomology on any fiber. | |
| Nov 19, 2010 at 15:57 | comment | added | t3suji | In fact, even in your particular case, it seems that the statement does not hold. Let $Y$ be a plane, $X$ be its blow-up at a point $Z$, and let $F$ be the line bundle corresponding to a high power of the exceptional divisor on $X$. Then the right-hand side and the left-hand side have different length. Right? | |
| Nov 19, 2010 at 15:48 | comment | added | t3suji | In general case, the conditions seem too weak. Why would the direct image $R^kf_*F$ be supported by $Z$ in the scheme-theoretic sense? (Note that you are not assuming anything about the scheme structure of $Z$.) For an easy counterexample, suppose $Y$ is non-reduced, and $Z$ is its underlying reduced subscheme. Now let $X$ be the product of $Y$ with a proper $k$-dimensional scheme, and let $F$ be a sheaf on $X$ whose top direct image is say a line bundle. | |
| Nov 19, 2010 at 14:30 | history | asked | Andrea Ferretti | CC BY-SA 2.5 |