?? ok, so ... we've got some conflicting if vague ideas about what the underlying module of a higher codimension subvariety is typically like ... so it seems like we should (??re- ... ???)visit the example of the ideal in k[x,y] ... as somewhat ultratypical ... ?? ....

??so ... we've got two generators x,y ... but i guess that we should give them different names to try to avoid too much confusion with the original x,y ... say a,b, i guess ...

so then ... a*y = b*x .... and that looks like pretty much it for the relators ... ?? ...

?? so for example let's think about what this says about "fibers" ... ??? ....

?? well, so away from the origin it seems to be saying about what you'd expect, that we're putting in 1 relator, cancelling out one of the 2 generators ... leaving fiber of net dimension 1 ...

?? but at the origin, the relator seems to become trivial ... ???so the fiber seems to be higher(namely 2)-dimensional ... ??? ... ??which i think fits our vague memories, but conflicts with recent idle imagining about ... ideal being trivially braided module ... conflict being that it seems pretty safe that AG morphisms preserve trivially braided objects, whereas a 2d vector space is not trivially braided ... ?? so presumably the idea that trivial-braiding is inherited by submodules (and that ideals would thus be trivially braided) is wrong ... but it would probably be good to see the non-trivial braiding here nice and concretely ... ??? ...

??so ... tensor square ... 4 generators aa,ab,ba,bb ... 4 relators aa y = ba x, ab y = bb x, aa y = ab x, ba y = bb x ... ???? then perhaps we're expecting the cokernel of the switch map here to be skyscraper at the origin ??? ...


??? so ... idea about ... it being interesting to develop doctrine of "symmetric monoidal (??V-)categories where every object is trivially braided" as seeming much iffier now ?? (?? ... ??functorial operations preserving property of being trivially braided ... ??? tensor product ??? .... ?? cokernel ???? .... ??? ??kernel ???? ...... ????? ..... ???? ideal _as_ kernel of map between trivially braided objects??? .... ..... ??? .... ??? being trivially braided as "smallness" condition ??? ....)

?? and also ... ?? idea of universally compelling underlying module of ideal to become line object seems somewhat trickier now, as we probably have to actually consciously think about the aspect of compelling it to become trivially braided, as opposed to merely invertible ... ???? .....

?? vague feeling about ... ?? we were already expecting that "blowing-up of subvariety" relates to doing certain stuff with corresponding ideal ... (?? universally compelling its underlying module to become invertible ... ??? ... "rees ..." .... ???? ...) ... ???but .... ???vague feeling that the underlying module of the ideal already embodies idea of "blowing up" in some sense .... ????? ..... ??? the rees (??? ...) stuff (??which i seem to be trying to shoe-horn in now as ess just usual way of "turning vector space into space (or algebraic counterpart of space)" ... ??) then simply transports the already blown-up situation from living in the module world to living in the commutative algebra world .... ????? ....

??idea that from module perspective, what's being "blown-up" is the unit module, and the inclusion of the ideal into the unit module is the algebraic (=?= contravariant ... ???) manifestation of a geometric "blowing-back-down" comparison map ?? ... ??or is this completely backwards / screwed up ?? ....

??maybe not screwed up ??? .... ???really true that included submodule (= ideal) here really does correspond to something geometrically (co-)bigger ... and the process of bringing this out is ess just the process of looking at the blowing-back-down comparison map contravariantly induced by the submodule inclusion map wrt certain hopefully obvious contravariant functor _module_ -> _relative variety_ ...

?? so ... ??? this seems like a pretty nice, almost lowbrow perspective on blowing-up ... ?? but good to develop those other highbrow perspectives as well ... ?? not sure at the moment how to fit them all together ... ??? .... ?? "universally compelling object to become line object" ... ??? "renormalization" ... ??? having trouble seeing how these relate now ... ?? they don't seem particularly visible yet from this lowbrow perspective... ?? because the lowbrowness is in part in the way it seems like you're just doing something very straightforward, namely inducing something from ideal inclusion ... ??maybe stuff is hidden in ideal inclusion ??? ... ??? .... ???? maybe it's something about ... relationship between "symmetric algebra" construction and line bundles .... ???? ... ??wasn't there some idea ... ????.....


??? "apparent inclusion where domain is actually morally bigger" here vaguely reminds me of other situations, maybe of same kind?? ... ?? "localization" ... ???.... ??any interesting relationships here ???.... (??vaguely imagining something about ... localization ... at vs away from ... ?? "completion" ... ??? ... ?? "filtered colimit" .... ???? ...... jets ... ??? ... ??? ... might be phantoms ... ??? .....) ???localization map as (algebraicially ...) blatantly mono and subtly epi ... ??ideal inclusion as blatantly mono ... ??subtly epi in any way ???? .... well, it's clearly not actually epi, right ?? ... being in abelian category ... (!! look at it's cokernel !! ... ??? ...) ??? .... ????but maybe it does induce epi .... ??????.....