Do Weil cohomology theories for schemes over arbitrary rings exist, and do the standard theorems (Lefschetz fixed point, Tr. Formula etc.) still hold?

Question

A Weil cohomology theory is a functor that assigns to a smooth projective variety $X$ of dimension $d$ over a field $k$ a graded ring of cohomology groups with values in a field $K$ of characteristic $0$, so that

1. $H^{i}(X)$ is finite-dimensional for all $i$.
2. $H^{i}(X)=0$ if $i<0$ or $i>2d$.
3. $H^{2d}(X)\cong K$.
4. There exists a perfect pairing $H^{i}(X)\times H^{2d-i}(X)\longrightarrow H^{2d}(X)=K$.
5. The map $H^{*}(X)\otimes H^{*}(Y)\longrightarrow H^{*}(X\otimes Y)$ is an isomorphism.
6. Existence of the cycle map $\gamma_{r}:Z^{r}(X)\longrightarrow H^{2r}(X)$ where $Z^{r}(X)$ is the group of algebraic cycles of codimension $r$.
7. If $W\subseteq X$ is a smooth hyperplane section of the ambient projective space then the maps $H^{i}(X)\longrightarrow H^{i}(W)$ is an injection for $i\leq d-1$ and an isomorphism for $i\leq d-2$.
8. If $W\subseteq X$ is a smooth hyperplane section of the ambient projective space then the map $L:H^{d-i}(X)\longrightarrow H^{d+i}(X)$ defined by $x\longrightarrow\gamma_{X}(W)\cdot x$ is an isomorphism for $i\in\left\{1,\dotsc,n\right\}$.

I have been pondering for quite some time now the question whether or not it is possible to generalize this to a setting where $X$ is defined not over $k$ but over any ring $R$.

Furthermore I am asking myself what would happen if the groups $H^{i}(X)$ were not defined over some field $K$ but over an arbitrary ring $S$, as long as one replaces "finite-dimensional" by "finitely generated over $S$" in condition 1, or even if one allows the $H^{i}(X)$ to take values in an arbitrary Abelian category as long as the Grothendieck group $K^{0}$ of that category is isomorphic to $\mathbb{Z}$ (so that the notion of a "trace" makes sense).

Would the usual theorems such as Lefschetz's fixed point theorem, the trace formula etc., still hold?

I do believe that if one requires the modules in condition 1. to be finitely generated and projective the usual theorems would still hold, but this is no more than a gut feeling.

I also believe that if one only lets go of the "characteristic $0$" condition for $K$ and allows $K$ to be an arbitrary field, then the main theorems would still hold — but the resulting $L$ functions would not be "useful" anymore in the sense that one could not possibly conceive them as generating functions counting the sets of points that $X$ has at the geometric points.