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Regarding this article, I noticed a few possible discrepancies that I was hoping someone could help me understand.
The page says that a closed immersion of schemes determines a homeomorphism of the underlying spaces. But Lurie instead says that it a homeomorphism onto a closed subspace. Could this be a typo?
This page states that the comorphism, oriented as $\mathcal{O}_Y\to f_*\mathcal{O}_X$, is an epimorphism; this agrees with the definition from classical scheme theory as far as I know. But Lurie instead requires $f^*\mathcal{O}_Y\to\mathcal{O}_X$ to be an epimorphism. The classic condition obviously implies Lurie’s condition (using the fact that the inverse image functor preserves epis and the counit of the adjunction is an iso), but I did not see how to show the converse.
I wonder what’s going on here? Thanks in advance!
I think for 1 Lurie is surely right. Rather than a typo, though, it might be attributable to an execrable old tradition whereby words like “isomorphism” always mean “isomorphism onto their image” rather than “isomorphism onto their codomain”. I’ve heard this most commonly for “isometry” but I think sometimes it was used for “homeomorphism” and “diffeomorphism” too.
I don’t know enough algebraic geometry to speak to 2.
Hi Mike, thanks! I’ll edit (1) to clarify.
Regarding (2), I guess Lurie has developed quite a bit of the geometry of closed immersions of structured topoi in DAG IX so I would be extremely surprised if his definition was wrong — though it is hard for me to understand how it restricts to the classic definition in the case of locally ringed spaces.
There appears to be some cultural difference between Lurie’s work (which most often speaks of properties of the comorphism phrased using inverse image) and traditional algebraic geometry, which speaks of properties of the comorphism transposed in terms of the direct image. For instance, Lurie has the beautiful account of local morphisms of locally ringed spaces in terms of certain naturality squares involving the inverse image phrasing being cartesian. But I would really benefit from a dictionary between these two styles, especially in cases where they seem (to me?) to not agree.
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