Timeline for How to define a principal divisor on general complex spaces?
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| Aug 23, 2022 at 16:50 | comment | added | red_trumpet | @SándorKovács Do you have a reference for dealing with generic points of complex analytic spaces? References I know (eg Grauert's and Remmert's CAS) don't mention them, and I currently have some problem to define them: They seem to be incompatible with the (relatively fine) euclidean topology, and if I consider the analytic Zariski topology, I don't know if the local rings $\mathcal O_{X,x}$ change. See also this question of mine. | |
| Aug 19, 2022 at 23:42 | comment | added | Sándor Kovács | @map, I believe so. $X$ being normal implies that this ring is integrally closed, the fact that D has codimension 1 implies that it is 1-dimensional, so it is indeed a DVR and in particular its maximal ideal (which is the ideal defining D) is principal. | |
| Aug 19, 2022 at 6:26 | comment | added | map | @SándorKovács I see, thank you very much! Does a generic point $x$ of $D$ satisfy following condition? → Some prime elements $f$ defines $D$ around $x$, i.e. $\mathcal{O}_{D,x} =\mathcal{O}_{X,x}/(f_x)$ (Sorry if I'm not making sense.) | |
| Aug 19, 2022 at 5:56 | comment | added | map | @JasonStarr What I want to know is the latter approach. I mainly deal with compact, connected, and normal complex spaces, so I would like to know what can be handled under these conditions. | |
| Aug 18, 2022 at 23:03 | comment | added | Sándor Kovács | @map: You are right, $D$ does not need to be regular, but you don't need that. What you need is that $X$ is regular at the generic point of $D$. Also, it is not $\mathscr O_{D,x}$ that should be 1-dimensional, but $\mathscr O_{X,\delta}$ where $\delta$ is the generic point of $D$. The fact that $D$ has codimension 1 should mean that that ring is 1-dimensional. | |
| Aug 18, 2022 at 14:48 | comment | added | Jason Starr | Welcome new contributor. I do not fully understand what you are asking. Do you want an alternative definition of principal divisor that manifestly works for complex analytic spaces (e.g., the pushforward via the projection to $X$ of the pullback from the projective line of the "tautological" principal divisor with respect to the closure of the graph of the meromorphic function in the product of the projective line and $X$)? Or are you asking how to extend the algebraic approach to also work for complex analytic spaces? | |
| Aug 18, 2022 at 1:33 | comment | added | map | I'm also unclear that such $\mathcal{O}_{D,x}$ is 1-dim(in the sense of Krull dimension). | |
| Aug 18, 2022 at 1:26 | comment | added | map | $X$ is normal when every $\mathcal{O}_{X,x}$ is normal ring(integral and integrally closed in its fraction field). $X$ is regular in codim 1 is true. But I thought it doesn't mean every $D$ is always regular. | |
| Aug 17, 2022 at 17:58 | comment | added | Sándor Kovács | What's your definition of normal? Doesn't that imply that $X$ is regular in codimension 1? | |
| S Aug 17, 2022 at 7:41 | review | First questions | |||
| Aug 18, 2022 at 1:30 | |||||
| S Aug 17, 2022 at 7:41 | history | asked | map | CC BY-SA 4.0 |