Timeline for Direct image of flat module along quasi-affine morphism
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 2 at 10:50 | history | became hot network question | |||
| May 2 at 7:14 | vote | accept | Lin Chen | ||
| May 2 at 4:49 | answer | added | abx | timeline score: 5 | |
| May 2 at 3:05 | comment | added | Lin Chen | Thank you! My main interest is in the smooth case. I knew in the general case there are stupid counterexamples when $X$ is not irreducible, but didn’t realize this good normalization construction! | |
| May 2 at 3:01 | comment | added | Alex Youcis | (cont.) I will have to think whether $X$ being smooth saves you somehow. | |
| May 2 at 3:00 | comment | added | Alex Youcis | No, this is false, even with codimension assumptions. A good place to look for counterexamples is $U$ is the normal locus in $X$. If $X-U$ has codimension at least $2$ then $j_\ast\mathcal{O}_U=f_\ast\mathcal{O}_{X^N}$, where $X^N$ is the normalization of $X$ (e.g., use the fact that the structure sheaf doesn't change values on a normal scheme if you remove a subset of codimension at least 2). But, this won't be flat if $X$ is normal, else $f$ would be finite flat and so degree makes sense, but then it would have to be degree $1$ (as it's birational), but that implies it's an isomorphism. | |
| May 2 at 2:57 | history | edited | Lin Chen | CC BY-SA 4.0 |
added 21 characters in body
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| S May 2 at 2:49 | review | First questions | |||
| May 2 at 2:57 | |||||
| S May 2 at 2:49 | history | asked | Lin Chen | CC BY-SA 4.0 |