We have seen that a non-commutative $l$-point is an algebra$P=S1 \oplus … \oplus Sk$with each $Si$ a simple finite dimensional $l$-algebra with center $Li$ which is a separable extension of $l$. The centers of these non-commutative points (that is the algebras $L1 \oplus … \oplus Lk$) 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$Vi xV W = \{ (vi,w) in Vi x W : fi(vi)=g(w) \}$is again a $G$-set and the collection $\{ Vi xV W \mapsto V \}$ 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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Posted in geometry
Written on Sun, 15 February 2004 at 8:20 pm
Tags: Galois, Grothendieck, non-commutative, noncommutative, topology
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