We have seen that a *non-commutative $l$-point* is an

algebra$P=S_1 \\oplus … \\oplus S_k$with each $S_i$ a simple

finite dimensional $l$-algebra with center $L_i$ which is a separable

extension of $l$. The centers of these non-commutative points (that is

the algebras $L_1 \\oplus … \\oplus L_k$) are the open sets of a

Grothendieck-topology on

$l$. To define it properly, let $L$ be the separable closure of $l$

and let $G=Gal(L/l)$ be the so called *absolute*

Galois group. Consider the

category with objects the *finite* $G$-sets, that is : finite

sets with an action of $G$, and with morphisms the $G$-equivariant

set-maps, that is: maps respecting the group action. For each object

$V$ we call a finite collection of morphisms $Vi \\mapsto V$ a

*cover* of $V$ if the images of the finite number of $Vi$ is all

of $V$. Let $Cov$ be the set of all covers of finite $G$-sets, then

this is an example of a *Grothendieck*-topology as it satisfies

the following three conditions :

(GT1) : If

$W \\mapsto V$ is an isomorphism of $G$-sets, then $\\{ W \\mapsto

V \\}$ is an element of $Cov$.

(GT2) : If $\\{ Vi \\mapsto

V \\}$ is in $Cov$ and if for every i also $\\{ Wij \\mapsto Vi \\}$

is in $Cov$, then the collection $\\{ Wij \\mapsto V \\}$ is in

$Cov$.

(GT3) : If $\\{ fi : Vi \\mapsto V \\}$ is in $Cov$

and $g : W \\mapsto V$ is a $G$-morphism, then the *fibered
products*$Vi x_V W = \\{ (vi,w) in Vi x W : fi(vi)=g(w) \\}$is

again a $G$-set and the collection $\\{ Vi x_V W \\mapsto V \\}$

is in $Cov$.

Now, finite $G$-sets are just

*commutative separable* $l$-algebras (that is,

*commutative* $l$-points). To see this, decompose a

finite $G$-set into its finitely many orbits $Oj$ and let $Hj$ be the

stabilizer subgroup of an element in $Oj$, then $Hj$ is of finite

index in $G$ and the fixed field $L^Hj$ is a finite dimensional

separable field extension of $l$. So, a finite $G$-set $V$

corresponds uniquely to a separable $l$-algebra $S(V)$. Moreover, a

finite cover $\\{ W \\mapsto V \\}$ is the same thing as saying

that $S(W)$ is a commutative *separable* $S(V)$-algebra. Thus,

the Grothendieck topology of finite $G$-sets and their covers

is anti-equivalent to the category of commutative separable

$l$-algebras and their separable commutative extensions.

This raises the natural question : what happens if we extend the

category to *all* separable $l$-algebras, that is, the category of

non-commutative $l$-points, do we still obtain something like a

Grothendieck topology? Or do we get something like a

*non-commutative Grothendieck topology* as defined by Fred Van

Oystaeyen (essentially replacing the axiom (GT 3) by a left and right

version). And if so, what are the *non-commutative covers*?

Clearly, if $S(V)$ is a commutative separable $l$-algebras, we expect

these non-commutative covers to be the set of all separable

$S(V)$-algebras, but what are they if $S$ is itself non-commutative,

that is, if $S$ is a non-commutative $l$-point?

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