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In Introduction to Schemes, G. Ellingsrud and J. C. Ottem call the result below "The Heavenly L'Hopitals Rule".

I see absolutely no similarity between this and the usual L'Hopital rule. What do they mean by this name?

Gabriel
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    I think the idea is just that poles of finitely many rational functions can always be removed. In terms of l'Hopital's Rule, this corresponds to extending the rational functions by defining it at its troublesome points to take the value it should as given by l'Hopital (note that l'Hopital always works for algebraic functions of degree zero). If I'm right, then personally I would be more inclined to liken this theorem to Riemann Extension than l'Hopital. I don't have time to post a full answer right now but I'll try to come back later, at least if nobody else responds. – Will R Nov 24 '20 at 13:44
  • @WillR That's a nice interpretation but I agree that in this case it is closer to Riemann's extension theorem than to l'Hopital rule. – Gabriel Nov 24 '20 at 14:12
  • Typically this kind of setup is described by analogy as "taking a map defined on a punctured disc and extending over the puncture." Now one thing we learn from L'Hopital's rule is that if a rational function is bounded near a point of indeterminacy, it always has a limit. Since projective space is "compact" hence "bounded," the values of the map $\operatorname{Spec} K \to \mathbb P_K^n$ are "bounded," so the value at the closed point exists and we can fill in the dotted arrow. Again, I suppose this also sounds like Riemann Extension, but I suppose that theorem is also a generalization of L'H? – Tabes Bridges Nov 24 '20 at 21:46

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