More
than a month ago I started a long term project trying to make the best
of our little home network. The first couple of weeks I managed to get
iTunes, iPhoto and iMovie-files flowing from any computer to the living
room (the TV-set for photo and mpeg-files and squeezebox for audio files). The last couple of weeks I
have been making my hands dirty with some hard-ware upgrades. The key
problem being that some of our Macs have a too small hard disk for
present day needs. For example, PD2 could no longer play The Sims (and
their never ending extensions) on her 6 Gb iMac, so one day she simply
decided to get rid off most things in het Applications-folder, a
desperate cry for attention. Together with Jan I took our two 6 Gb
slot-loading iMacs apart and replaced them by a 120 Gb resp. 80 Gb hard
disk, giving the Sims ample virtual space to expand (I hope). Beginning
of this week I finished the slightly more daunting task of upgrading an
original 4 Gb front-loading iMac to a performing 120 Gb potential
Server. But I knew that the worst part was still to come : my old
(colored) iBook was making so much noise that I didn’t use it anymore
for anything demanding some kind of concentration (like writing papers).
So I wanted to replace the old 6 Gb noisy disk with a silent Hitachi
2.5 HDD 20GB 5400RPM ATA100 8Mb Cache-hard disk. However I did read
the instructions and was a bit put off by this.
Luckily, I had to wait because I didn’t have the appropriate material.
Whereas any super-market sells Torx- 10 and 15 screwdrivers, I needed an
8 or 9. Eventually I found one in a good shop (they even have torx 6 and
7, it seems you need those to take your mobile apart), so no more
excuses. Tuesday afternoon I had a first try but already between stage 2
and 3 of the instructions I cut an essential connection (for the
trackpad)… and quickly assembled everything again (I could still use
my iBook with a USB-mouse…). This morning, when the rest of my family
left at 8 o’clock, I had another go (btw. never try to do this unless
you can afford to loose your iBook). The whole process is pretty scary :
you have to take out your keyboard, modem, CD-player, display and a few
minor ingredients before you get at the hard-disk. At the time you do,
the whole table is filled with parts and several cups containing some
screws which you will hopefully remember to put back in the correct
place. Still, in less than two hours I managed to replace the harddisk
and put everything back together (I lost some tome because at the end
one needs to remove some bolts and I didnt have a good tool available so
I had to improvise). I didn’t expect anything when I powered up the
iBook but somehow it reacted ok, I could start up from a System9 CD and
initialize the harddisk and even put System9 on it, but there was
something strange : all message-windows appeared at the lower right hand
side of the display… When I did restart from the HD, it became
apparent that I lost about 30% of my screen, including the part where
one normally sees the HD, CDs etc., so I had a small problem. But, after
my success of conquering my hard-ware phobia, I was not alarmed, I
cycled to the university and had a chat with Jan about it. He suggested
zapping the PRAM which I did in the afternoon, without any effect
on my partial display. So, perhaps it was a hard-ware thing after all
and I had to take my iBook apart again until I got at the connections
for the display (which is step 6 out of 8 of the instructions). Then,
the problem became clear : in reassembling the display-connectors I had
used a bit too much force so that some of the spikes were bend. But,
after repairing this and closing up the iBook again, the problem was
solved. So I have now a 20 Gb iBook with a nice quiet harddisk and I
“only” lost my trackpad and sound in the process… a good deal I
would say.
neverendingbooks Posts
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.
Non-commutative geometry seems pretty trivial compared
to commutative geometry : there are just two types of manifolds,
points and curves. However, nobody knows how to start classifying
these non-commutative curves. I do have a conjecture that any
non-commutative curve can (up to non-commutative birationality) be
constructed from hereditary orders over commutative curves
by universal methods but I’ll try to explain that another
time.
On the other hand, non-commutative points
have been classified (at least in principle) for at least 50
years over an arbitrary basefield $l$. non-commutative
$l$-points $P$ is an $l$-algebra such that its double
$d(P) = P \\otimes P^o$ ( where $P^o$ is the opposite algebra,
that is with the reverse multiplication) has an element$c=\\sum_i
a_i \\otimes b_i with \\sum_i a_ib_i = 1 (in $P$)$ and such that for
all p in $P$ we have that $(1 \\otimes a).c = (a \\otimes 1).c$ For
people of my generation, c is called a separability idempotent
and $P$ itself is called a separable $l$-algebra.
Examples of $l$-points include direct sums of full matrixrings
(of varying sizes) over $l$ or group-algebras $lG$ for $G$ a
finite group of n elements where n is invertible in $l$. Hence, in
particular, the group-algebra $lG$ of a p-group $G$ over a field $l$
of characteristic p is a non-commutative singular point and
modular representation theory (a theory build almost single
handed by
Richard Brauer) can be viewed as
the methods needed to resolve this singularity. Brauer’s name is
still mentioned a lot in modular representation theory, but another
of his inventions, the Brauer group, seems to be hardly known
among youngsters.
Still, it is a crucial tool
in classifying all non-commutative $l$-points. The algebraic
structure of an $l$-point $P$ is as follows : $$P = S_1 + S_2 + …
+ S_k$$ where each S_i is a simple algebra (that is, it
contains no proper twosided ideals), finite dimensional over
its center $l_i$ which is in its turn a finite dimensional
separable field extension of $l$. So we need to know all
simple algebras $S$, finite dimensional over their center $L$ which
is any finite dimensional separable field extension of $l$. The
algebraic structure of such an $S$ is of the form$$S = M(a,D)$$ that
is, full axa matrices with entries in $D$ where $D$ is a
skew-field (or some say, a division algebra) with
center $L$. The $L$-dimension of such a $D$ is always a square,
say b^2, so that the $L$-dimension of $S$ itself is also a square
a^2b^2. There are usually plenty such division algebras, the simplest
examples being quaternion algebras. Let p and q be two
non-zero elements of $L$ such that the conic $C : X^2-pY^2-bZ^2 =
0$ has no $L$-points in the projective $L$-plane, then the
algebra$D=(p,q)_2 = L.1 + L.i + L.j + L.ij where i^2=p, j^2=q and
ji=-ij$ is a four-dimensional skew-field over $L$. Brauer’s idea to
classify all simple $L$-algebras was to associate a group to them,
the Brauer group, $Br(L)$. Its elements are equivalence
classes of simple algebras where two simple algebras $S$ and
$S’$ are equivalent if and only if$M(m,S) = M(n,S’)$ for some sizes
m and n. Multiplication on these classes in induced by
the tensor-product (over $L$) as $S_1 \\otimes S_2$ is again a simple
$L$-algebra if $S_1$ and $S_2$ are. The Brauer group $Br(L)$ is an
Abelian torsion group and if we know its structure we know all
$L$-simple algebras so if we know $Br(L)$ for all finite dimensional
separable extensions $L$ of $l$ we have a full classification of
all non-commutative $l$-points.
Here are some examples
of Brauer groups : if $L$ is algebraically closed (or separable
closed), then $Br(L)=0$ so in particular, if $l$ is algebraically
closed, then the only non-commutative points are sums of matrix rings.
If $R$ is the field of real numbers, then $Br(R) = Z/2Z$ generated by
the Hamilton quaternion algebra (-1,-1)_2. If $L$ is a complete
valued number field, then $Br(L)=Q/Z$ which allows to describe also
the Brauer group of a number field in terms of its places. Brauer groups
of function fields of (commutative) varieties over an algebraically
closed basefield is usually huge but there is one noteworthy
exception $Tsen’s theorem$ which states that $Br(L)=0$ if $L$ is the
function field of a curve C over an algebraically closed field. In 1982
Merkurjev and Suslin proved a marvelous result about generators of
$Br(L)$ whenever $L$ is large enough to contain all primitive roots
of unity. They showed, in present day lingo, that $Br(L)$
is generated by non-commutative points of the quantum-planes
over $L$ at roots of unity. That is, it is generated by cyclic
algebras of the form$(p,q)_n = L
\\< X,Y>/(X^n=p,Y^n=q,YX=zXY)$where z is an n-th primitive root of
unity. Next time we will recall some basic results on the relation
between the Brauer group and Galois cohomology.