# Is there a good notion of an étale topos associated to a noncommutative ring?

The title of my question basically sums up what I wish to know. I'm looking into trying to generalize étale cohomology to noncommutative rings (without having to go through some sort of De Rham or motivic cohomology theory) by trying to find what a good candidate for an étale topos of a noncommutative ring might be. I'm most interested in any references to what people have tried in the past, as nLab states that people have tried to look at such a topos in the past, but gives no references for me to chase down and learn from. Here is where I have gotten so far:

While we may want to look at algebras over a noncommutative ring $R$ with unit as objects in the coslice category $R/\mathbf{Ring}$ (as the category of unital algebras over a commutative ring $A$ may be identified with the coslice categories $A/\mathbf{Ring}$ or $A/\mathbf{Cring}$ depending on one's taste for commutativity), and while we certainly have a natural analogue of finite generation of an algebra $R \xrightarrow{f} S$ over $R$, what I wish to know is primarily if there has been any work in looking at lifting properties and extensions. In particular, for commutative algebras $A \xrightarrow{f} B$ over a commutative ring with unit $A$, $B$ is étale over $A$ if and only if $B$ is finitely generated over $A$ and for any algebra $A \xrightarrow{g} C$ with $N \trianglelefteq C$ nilpotent, there exists a unique $h:B \to C/N$ making the diagram $$\begin{array} s & A & \xrightarrow{f} & B \\ & g\downarrow & & \downarrow\exists!h \\ & C & \xrightarrow{\pi_N} & C/N \end{array}$$ commute. This gives many important geometric properties, as étale maps are stable under base change, they allow a descent theory, and localizations give an étale map $A \xrightarrow{\lambda_S} S^{-1}A$ for finitely generated monoids $S \subseteq A$. What I'm worried about primarily is in the noncommutative case what kind of (let's say two-sided) ideal we would want $N \trianglelefteq C$ to be; while nilpotent ideals could remain useful to look at, I have been led to believe that it may be more appropriate to look at ideals $N$ contained inside the Jacobson radical $J(C)$. The reason I suspect this is more appropriate is because when $S \subseteq R$ is a (two-sided) Ore set (so that $S^{-1}R$ has all its elements take the form $\lambda_S(r)\lambda_S(s)$ for $r \in R, s \in S$, and universal $S$-inverting morphism $\lambda_S$, or at least be uniquely isomorphic to a representation of this form, and two-sided so we don't have to worry about any stupid things happening with sidedness) we should be able to lift any element of $C/N$ in the diagram $$\begin{array} s & R & \xrightarrow{\lambda_S} & S^{-1}R \\ & f\downarrow & & \\ & C & \xrightarrow{\pi_N} & C/N \end{array}$$ and induce a morphism $h:S^{-1}R \to C/N$ rendering the whole thing commutative. The problem I am having is that I am not sure if we would need $N$ to be contained instead in some sort of radical that allows the lifting of nilpotents of $C$ that is distinct from the Jacobson radical, like the upper or lower nilradical, or if someone has already done this in full detail and I have just could not find it. As I said earlier, I am especially interested in this because it should give a notion of étale cohomology for a noncommutative ring, and I would like to see if this gives an aspect of noncommutative algebraic geometry that works in a quasi-sane fashion. Please let me know as well if you think this question is more relevant on MO instead of here. Thanks in advance!

• Probably you'll have more answers on MO than here.
– paf
Aug 12 '16 at 0:31