?? so ... given a dedekind domain ... ???.... consider abelian group of fractional ideals, and commutative monoid over it, where an element over fractional ideal x is an element y of the field of fractions such that ... ??? .... "y gives a module homomomorphism from the unit module to the fractional ideal" ... ?? ... ??in other words, simply y is an element of x .... ??? .....

??wait, did i say that right? ... ???maybe i should have said, y is a _non-fractional_ element of x ... ????..... ...confusion ... ???

?? well, the ideal class group is finite, according to what we've heard ... so
certainly individual elements are of finite order, so ... ?? we should expect that any fractional ideal has morphisms from the unit object if we expect that sufficiently extreme (perhaps in the negative direction) powers of it do ... because the finite order guarantees that it's itself equivalent to one of htose sufficiently extreme powers... ???right ??? ... so ... it shouldn't bug us so much then that all fractional ideals have elements, in contrast to the sort of situation that we're more familiar with, where the fibers outside a certain cone are trivial ... ??? ...

?? so then can we prove that invertible = principal here ?? .... actually i don't quite see it yet ... ??? ....

??so let x be a fractional ideal, with y in x and y' inverse to it (thus in x^[-1]) ... ???.....

???well, so what _is_ the inverse of a fractional ideal x, concretely ??? ... ???fractions which "can be expressed with any denominator taken from x" ??? ??well, perhaps that's an ok way of saying it for the special case of the inverse fractional ideal of a non-fractional ideal ... which perhaps is all we really need here ....

??? "residual ... " ??? .... ???? "residue" ??? ??? ... ???

?? actually let me try saying it more straightforwardly ... for fractional ideal x, fraction y is in x^[-1] iff x1*y integeral for any x1 in x ... ?????

(??which could perhaps be rephrased as "has integeral numerator when expressed with any denominator taken from x" ... ???? .....)

??so then suppose that y in x and that x1*y^[-1] is integeral for any x1 in x ... ??or in other words, x1 = integeral elt * y for any x1 in x ....

??so maybe a more relevant rephrasing here would have been "every x1 in x is an integeral multiple of y^[-1]" ???

??so "y is in x and y^[-1] is in x^[-1]" is ess "y is in x and every x1 in x is an integeral multiple of y" ??? ..... ?? which is pretty much exactly saying that y generates x / y witnesses the principalness of x ... ???

(??? in general, the naive idea of the inverse of a fractional ideal as being formed from the inverses of its elements as wrong ... ???but in the case of principal elements it maybe sort of ends up being true??? ... ??is that sort of what we're saying here ??? ....)

?? hmm, so ... ??? to say that y is in x is to say that the principal ideal <= y, and to say that y^[-1] is in x^[-1] is to say that y <= ... ??? ...



??? ideal class group as "k-group" ... ???forgot about weirdness of this ... tensor product of modules vs direct sum ... requiring invertibility ahead of time vs compelling it afterwards / granting it .... (??ways of thinking about right vs left adjoint ??? .... ????..... ?? "before / after" ... ??? ...) ... ???whether that k-group is _always_ (??for any commutative ring??? ...??? ???....) the group of iso classes of invertible modules, or only in the dedekind case ??? .... ???? ....

??stuff baez said about ... ??? euler characteristic valued in k-group ... ????...