Fix a base ring $R$, and consider pairs $(X,f)$ where $X$ is a scheme
of finite type over $R$ and $f:X\to R$ is an $R$-valued algebraic (not
constructible!) function on $X$.
I want to consider a Grothendieck $R$-algebra of such pairs, where
if $X = Y \coprod Z$, then $[(X,f)] = [(Y,f|_Y)] + [(Z,f|_Z)]$,
but also $[(X,f+g)] = [(X,f)] + [(X,g)]$ and $[(X,rf)] = r[(X,f)]$.
Surely this is a standard extension of the usual notion of the
Grothendieck ring of varieties (which only has $f=1$, and the first
sort of relation)? If so, where can I read about it?
Maybe I'm misreading the motivic integration survey literature (by
K. Smith, and E. Looijenga), but it seems like they're insisting
on constructible functions, not algebraic.
Ordinarily when a construction like this isn't in the literature, I assume it's
because it has too many relations and is $0$, but if $R = {\mathbb Z}$
it seems to me that this ring has many functionals, like
$[(X,f)] \mapsto \sum_{x \in X_p} (f(x) \bmod p) \in {\mathbb Z}/p.$ (I don't
see an analogue of $[X] \mapsto$ the Euler characteristic $\chi(X_{\mathbb C})$.)
EDIT: One problem I see is that $({\mathbb A}^1, f(x)=x)$ is
isomorphic under translation to $({\mathbb A}^1, f(x)=x+1)$. So
$[({\mathbb A^1}, 1)] = [({\mathbb A}^1, (x+1)-x)]
= [({\mathbb A}^1, x+1)] – [({\mathbb A}^1, x)] = 0$. Of course
this fits with point-counting $\bmod p$.
Best Answer
I have just come across this question, and I have no idea if the OP still has any interest in it, but theories built out of pairs $(X, f)$ go under the name of "exponential motives", and there has been quite a lot of recent interest in developing and applying them. Two recent papers that come to mind, emphasising different aspects, are Motivic classes of Nakajima quiver varieties, https://arxiv.org/pdf/1603.03200.pdf, and Exponential motives http://javier.fresan.perso.math.cnrs.fr/expmot.pdf; these papers in turn have several further references.