Timeline for Why are flat morphisms "flat?"
Current License: CC BY-SA 2.5
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| Mar 21, 2021 at 16:52 | comment | added | Nathan Lowry | "Neighbors" should indeed be those primes $Q$ contained in $P$. Then we can localize the module $M_P$ at (the image of) $Q$ and recover $M_Q$. If $M_P$ is a free $A_P$ module, then $M_Q$ will in fact be a free $A_Q$ module (of the same rank!) I'm sure this is an exercise in Hartshorne, but I can't quite find it. At the very least exercise II.5.8 is of a very similar nature. | |
| Feb 12, 2019 at 1:20 | comment | added | user267839 | We fix firstly a prime $P$ and consider the fiber $M(P) = M_P/PA_P =M_P \otimes_A A_P/P$. Which fibers do you interpret as "neighbours" of $M(P)$? The $M(Q)$ with $Q \subset P$? And why it is reasonable to say that if the fibers vary "continuously" then the lifting provides the freeness of $A_P$-module $M_P$? How "translates" this kind of "continuity" intuitively to this algebraic statement about freeness of the stalk? | |
| Feb 12, 2019 at 1:19 | comment | added | user267839 | This seems indeed to provide a nice geometric approximation. One point seems still unclear to me: what do you concretely mean by "continuous" variation of the nearby fibers? | |
| Nov 25, 2009 at 14:53 | history | answered | Marc Nieper-Wißkirchen | CC BY-SA 2.5 |