started trying to respond to ben-zvi, but i might be getting bogged down... i'll try thinking outloud here...

well, i'm pretty confused now, though perhaps mainly just because you're reminding me of questions about d-modules that have had me confused for a long time. in hopes of getting unconfused i'll try thinking outloud here about some of my confusions.

given a cocommutative hopf algebra g acting on a commutative algebra r, we can construct a new (generally noncommutative) algebra from this. (perhaps this is called a "semi-direct product" in some terminology?) a module of the new algebra is essentially "a way of extending the action of g on r to an action of g on the pair (r,m) for some r-module m". if g acts on the triple (r,m1,m2) then it also acts on the pair (r,m1 tensor m2), so the modules of the new algebra form a tensor category.

specializing to the case where r is the coordinate algebra of an affine scheme x and g is the lie algebra of derivations of r, this gives the tensor category of d-modules over x. (is that correct?)

the general construction here is morally similar to taking the "orbit stack" of an action of an algebraic group acting on an affine scheme...

??hmm, but ben-zvi said something about "quotienting by the formal neighborhood of the diagonal" (or something like that...), which maybe has a pretty different flavor from "taking the orbit stack of the action by the formal group generated by the vector fields" (or something like that...) .... the flavor difference having to do with functoriality / variance ...???or something.... ???.... ??so what about how this relates to issues about pullback and pushforward of d-modules??? ... and so forth ... ????....

(???sa history of ideas about "noncommutative geometry" os... ???asf os... ????)

??so what about how this (???...) might tie in with... ???all sorts of stuff about "dg spaces" and so forth... and their relationship to stacks and so forth.... ???? or something??? .... ???hmm, _did_ we think about this before?? maybe even think about resolving certain functoriality paradoxes this way ???? or something???...

what about discrete analogs here???...

??what about morita equivalence here??? .... something about matrix algebras ... ???or something?? .... ??some sort of level slip (or something??) here???

??hmm, so maybe i _have_ been making a silly mistake here for decades... about what d-modules are... ??something about the orbit stack of the action of [the formal translation group of a vector space v] on v ... vs the orbit stack of the action of [the formal diffeomorphism group of v] on v ... ??or something like that?? ... i wonder whether we might have straightened this out before and then forgotten about it... perhaps not...

??of course part of the point is that "orbit stack of the action of [the formal translation group of a vector space v] on v" has much more functoriality than manifestly suggested by that phrase ... ???...

??so how might this affect our attempts to learn other stuff involving d-modules?? for example the alleged relationship to perverse sheaves and so forth... ???.... hmmm.... "beilinson-bernstein localization" ... ??ideas about d-modules and "ayntax/semantics completeness theorem for differential equations" and so forth... ??... ??something about "dirac delta function" and so forth... ??... "distributional solutions" ... ...

??so is it true that d-modules over the underlying affine scheme of an affine algebraic group can be thought of as .... ???modules of the "quantum double" (or something...) of something... ???or something???? ... not sure i said that anywhere close to correct yet... what i'm trying to get at here is that the idea that d-modules over a vector space v can be thought of as quasicoherent sheaves over the orbit stack of the formal translation group of v seems like it shouldn't depend on the abeliannes of v (thinking of a vector space as a sort of abelian affine algebraic group.... or something...) ... ???try to work out more details here...

??ok, so not "quantum double" here, i guess, because that's about the conjugation action... we actually want the translation action here, i think....

??hmm, so what about something about... ???contrast to quantum double here?? ... something about... quantum double as hopf algebra over base field (or something) vs this other (??) orbit stack as not hopf algebra over commutative algebra...??? and so forth ... ?? ... braidedness of tensor product.... ??_is_ there another tensor product around here, symmetric instead of braided???....

??anyway, is it actually obvious at a "concrete" level that this orbit stack is independent of the algebraic group structure... ??? ...or something like that .. ??... ??hmm, perhaps directly related to that similar question ...(??not to be confused with slightly dissimilar question... ??about "duflo isomorphism" or something??)... ??about poincare-birkhoff-witt theorem??...

???so what about the idea of "d-modules over a dg space"?? ... and so forth??? ...??what _about_ relationship to stuff like "perverse sheaves" and so forth???....

??so now that we may have straightened out some confusions about d-modules here, can we get any idea of what ben-zvi is talking about about some sort of "non-affine behavior" here????? ....

this is what ben-zvi said:

"The main technical problem with D-modules from our point of view is that pushforward is essentially never conservative (except for finite maps) - that’s the sense in which D-modules behave non-affinely even on affine varieties..and which is why on stacks with affine diagonal Tannakian constructions with D-modules – eg description of sheaves on a fiber product as a categorical tensor product of categories of sheaves – fail dramatically.. though they DO hold for schemes."

??so what in the world are they talking about?? ... i don't get it yet... ??very vaguely though it does remind me of something about... how geometric pullbacks and equalizers (or something like that... hope that i didn't get the arrows backwards here...) seem screwed up for cocommutative coalgebras... ???maybe dual to geometric colimits being screwed up for commutative algebras?? ??? or something ... ??...

??while searching for "non-affine" in the "alg geom for category theorists" thread i accidentally came across earlier parts of the dicussion from 2009... possibly interesting evolution (and/or lack thereof) in viewpoints of some participants...

??so what about tensor product for perverse sheaves compared to for d-modules??
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