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a noncommutative Grothendieck topology

Published February 15, 2004 by lievenlb

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?

Galois and the Brauer group

Published February 9, 2004 by lievenlb

Last time we have seen that in order to classify all
non-commutative $l$ -points one needs to control the finite
dimensional simple algebras having as their center a finite
dimensional field-extension of $l$ . We have seen that the equivalence
classes of simple algebras with the same center $L$ form an Abelian
group, the
Brauer group. The calculation of Brauer groups
is best done using
Galois-cohomology. As an aside :
Evariste Galois was one of the more tragic figures in the history of
mathematics, he died at the age of 20 as a result of a duel. There is
a whole site the Evariste Galois archive dedicated to his
work.

But let us return to a simple algebra $T$ over the
field $L$ which we have seen to be of the form $M(k,S)$ , full
matrices over a division algebra $S$ . We know that the dimension of
$S$ over $L$ is a square, say $n^2$ , and it can be shown that all
maximal commutative subfields of $S$ have dimension n over $L$ .
In this way one can view a simple algebra as a bag containing all
sorts of degree n extensions of its center. All these maximal
subfields are also splitting fields for $S$ , meaning that
if you tensor $S$ with one of them, say $M$ , one obtains full nxn
matrices $M(n,M)$ . Among this collection there is at least one
separable field but for a long time it was an open question
whether the collection of all maximal commutative subfields also
contains a Galois-extension of $L$ . If this is the case, then
one could describe the division algebra $S$ as a crossed
product. It was known for some time that there is always a simple
algebra $S’$ equivalent to $S$ which is a crossed product (usually
corresponding to a different number n’), that is, all elements of
the Brauer group can be represented by crossed products. It came as a
surprise when S.A. Amitsur in 1972 came up with examples of
non-crossed product division algebras, that is, division algebras $D$
such that none of its maximal commutative subfields is a Galois
extension of the center. His examples were generic
division algebras $D(n)$ . To define $D(n)$ take two generic
nxn matrices, that is, nxn matrices A and B such that all its
entries are algebraically independent over $L$ and consider the
$L$ -subalgebra generated by A and B in the full nxn matrixring over the
field $F$ generated by all entries of A and B. Somewhat surprisingly,
one can show that this subalgebra is a domain and inverting all its
central elements (which, again, is somewhat of a surprise that
there are lots of them apart from elements of $L$ , the so called
central polynomials) one obtains the division algebra $D(n)$ with
center $F(n)$ which has trancendence degree n^2 1 over $L$ . By the
way, it is still unknown (apart from some low n cases) whether $F(n)$
is purely trancendental over $L$ . Now, utilising the generic
nature of $D(n)$ , Amitsur was able to prove that when $L=Q$ , the
field of rational numbers, $D(n)$ cannot be a crossed product unless
$n=2^s p_1…p_k$ with the p_i prime numbers and s at most 2. So, for
example $D(8)$ is not a crossed product.

One can then
ask whether any division algebra $S$ , of dimension n^2 over $L$ , is a
crossed whenever n is squarefree. Even teh simplest case, when n is a
prime number is not known unless p=2 or 3. This shows how little we do
know about finite dimensional division algebras : nobody knows
whether a division algebra of dimension 25 contains a maximal
cyclic subfield (the main problem in deciding this type of
problems is that we know so few methods to construct division
algebras; either they are constructed quite explicitly as a crossed
product or otherwise they are constructed by some generic construction
but then it is very hard to make explicit calculations with
them).

connected component coalgebra

Published February 5, 2004 by lievenlb

Never thought that I would ever consider Galois descent of semigroup
coalgebras but in preparing for my talks for the master-class it
came about naturally. Let A be a formally smooth algebra
(sometimes called a quasi-free algebra, I prefer the terminology
noncommutative curve) over an arbitrary base-field k. What, if
anything, can be said about the connected components of the affine
k-schemes rep(n,A) of n-dimensional representations
of A? If k is algebraically closed, then one can put a
commutative semigroup structure on the connected components induced by
the sum map

rep(n,A) x rep(m,A) -> rep(n + m,A)   :  (M,N)
-> M + N

as introduced and studied by Kent
Morrison a long while ago. So what would be a natural substitute for
this if k is arbitrary? Well, define pi(n) to be the
maximal unramified sub k-algebra of k(rep(n,A)),
the coordinate ring of rep(n,A), then corresponding to the
sum-map above is a map

pi(n + m) -> pi(n) \\otimes
pi(m)

and these maps define on the graded
space

Pi(A) = pi(0) + pi(1) + pi(2) + ...

the
structure of a graded commutative k-coalgebra with
comultiplication

pi(n) -> sum(a + b=n) pi(a) \\otimes
pi(b)

The relevance of Pi(A) is that if we consider it
over the algebraic closure K of k we get the semigroup
coalgebra

K G  with  g -> sum(h.h\' = g) h \\otimes
h\'

where G is Morrison\’s connected component
semigroup. That is, Pi(A) is a k-form of this semigroup
coalgebra. Perhaps it is a good project for one of the students to work
this out in detail (and correct possible mistakes I made) and give some
concrete examples for formally smooth algebras A. If you know of
a reference on this, please let me know.

Tag: Galois

a noncommutative Grothendieck topology

Galois and the Brauer group

connected component coalgebra