Tweedledum is a first-generation iMac (233 MHz
slot-loading, 192Mb RAM, No Airport) whereas Tweedledee is
2nd-generation (350 MHz front-loading, 192Mb RAM, Airport card). A
couple of weeks ago I replaced their original hard-discs (4 Gb resp. 6
Gb) by fat 120 Gb discs and from this weekend they serve as our
backup-facility. Tweedledee is connected via Airport to our network and
is a fully functional 10.3 computer, everyone has a login on it and is
encouraged to dump important files onto it as a secondary copy.
Tweedledum. on the other hand, is invisible to the network but forms a
one-wire network with Tweedledee (they are connected by a crossed
ethernet cable which results in having a self-assigned IP address in the
169.254 range and hence they can see each other; moreover using the
Sharing-pane in the System Preferences I allowed
Tweedledee to share its internet connection to other computers,
connected to it via Ethernet, so Tweedledum can go online to get
system-updates when necessary).
A house-computer rule is that
all important files (which means those you don’t like to loose in a
crash) are kept in the Documents folder of your
Home-folder on your own computer. At regular intervals I make
sure that these folders are synchronized with backup-copies on both
Tweedledee & Tweedledum, so at any given time there are at least 3
computers containing the essential files (usually more as everyone has a
login at each of the 4 ‘work’-computers and can drop extra copies
around, but must clean-up when asked).
To synchronise I use
the shareware program ExecutiveSync. It is no longer possible to
obtain this from its original homepage as they seem to have been taken
over and invite you to buy You Sinc instead which costs more than
twice what ExecutiveSync costs (19.95$). Fortunately, for now you can
still download it from the Apple site. I have
ExecutiveSync running on Tweedledee (you are only allowed to run it on
one computer, you can install it on every computer but then the
synchronizing process is sometimes not possible which is why I came to
the following work-around). In ExecutiveSync you make several
Projects which involve choosing a Local folder and a
Remote folder somewhere on your network which you want to keep in
Sync. In my Home folder on Tweedledee I made several (originally
empty) folders such as docsGitte. Then my ExecutiveSync-project
syncGitte takes docsGitte as the local folder and the
/Users/gitte/Documents-folder on iBookGitte as the remote
folder. The first time you synchronise takes a lot of time (especially
over the wireless network, it may be better to do the first sync via
ethernet) but afterwards it works pleasantly.
Once I
synchronised all the local Documents-folders with the corresponding
folders in my home-folder on Tweedledee, I have another
ExecutiveSync-project BACKUP which takes as the Local-folder my
Home-folder and as the remote folder a folder BACKUP I did create
on Tweedledum. Fortunately, here the synchronising is done over Ethernet
or it would take forever.
neverendingbooks Posts
Today I
did prepare my lectures for tomorrow for the NOG master-class on
non-commutative geometry. I\’m still doubting whether it is worth TeXing
my handwritten notes. Anyway, here is what I will cover tomorrow :
– Examples of l-algebras (btw. l is an
arbitrary field) : matrix-algebras, group-algebras lG of finite
groups, polynomial algebras, free and tensor-algebras, path algebras
lQ of a finite quiver, coordinaterings O(C) of affine smooth
curves C etc.
– Refresher on homological algebra : free and
projective modules, exact sequences and complexes, Hom and Ext groups
and how to calculate them from projective resolutions, interpretation of
Ext^1 via (non-split) short exact sequences and stuff like that.
– Hochschild cohomology and noncommutative differential forms.
Bimodules and their Hochschild cohomology, standard complex and
connection with differential forms, universal bimodule of derivations
etc.
– Non-commutative manifolds. Interpretation of low degree
Hochschild cohomology, characterization of non-commutative points as
separable l-algebras and examples. Formally smooth algebras
(non-commutative curves) characterised by the lifting property for
square-free extensions and a proof that formally smooth algebras are
hereditary.
Next week I will cover the representation
varieties of formally smooth algebras and the semigroup on their
connected (or irreducible) components.
This
week I reread with pleasure all 918 pages of Cryptonomicon by Neal Stephenson and found out
that last time I had been extremely choosy in subplots. There are 4
major plots : one contemporary (a couple of geeks trying to set up a data-haven) and three
WW2 stories : the Waterhouse-plot about cracking Enigma and other
crypto-systems featuring a.o. Alan Turing, the Shaftoe-plot about the
crazy division 2702 trying to cover-up that Enigma has been broken and
the Goto Dengo-plot about hiding the Japanese Gold reserve in the
jungle. Five years ago I was mostly interested in the first two subplots
and later on in the book I jumped chapters quite a bit, it seems.
During the first read I assumed that the Van Eck
phreaking-bit was just another paranoid misconception of the two present
day main players Randy&Avi, but this week I wasn\’t so sure anymore so
the first thing I did when we came home was Googling on Van Eck phreaking which really does
exist!
Van Eck phreaking is a form of eavesdropping in which
special equipment is used to pick up telecommunication signals or data
within a computer device by monitoring and picking up the
electromagnetic fields.
The U.S. government has been involved
with EM interpretation for many years under a top-secret program
code-named “TEMPEST”.
It seems that in 1985 the
Dutch scientist Wim Van Eck wrote a paper \’Electromagnetic Radiation from Video Display Units:
An Eavesdropping Risk?\’ He concluded: “If no preventive measures
are taken, eavesdropping on a video-display unit is possible at several
hundred meters distance, using only a normal black-and-white TV
receiver, a directional antenna, and an antenna amplifier.” He proved
it by taking a BBC crew around London in a van, showing them what was on
the computer screens at various companies.
To me it seems that
putting your computer inside a Faraday cage is a simpler counter-measure than
the cumbersome method of Randy in the novel (but I have to admit, he
was in prison at the time…). There is a more detailed manual on
phreaking available, but the best text I found (sofar) on Van Eck
phreaking is Soft Tempest: Hidden Data Transmission Using
Electromagnetic Emanations by Markus G. Kuhn and Ross J.
Anderson.
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?
The game of Fox and Geese is usually played on a cross-like
board. I learned about it from the second volume of the first edition of
Winning Ways
for your Mathematical Plays which is now reprinted as number 3 of
the series. In the first edition, Elwyn Berlekamp,
John Conway and
Richard Guy claimed that the value of their
starting position (they play it on an 8×8 chess board with the Geese on
places a1,c1,e1 and g1 and the Fox at place e8) has exact value
1 + 1/on
where on is the class of all ordinal numbers so
1/on is by far the smallest infinitesimal number you can think
of. In this second edition which I bought a week ago, they write about
this :
We remained steadfast in that belief until we heard
objections from John Tromp. We then also received correspondence
from Jonathan Weldon, who seemed to prove to somewhat higher standards
of rigor that
“The value of Fox-and-Geese is 2 +
1/on”
Oops! But of course they try to talk themselves out
of it
Who was right? As often happens when good folks
disagree, the answer is “both!” because it turns out that the parties
are thinking of different things. The Winning Ways argument
supposed an indefinitely long board, while Welton more reasonably
considered the standard 8×8 checkerboard.
Anyway, let us be
happy that the matter is settled now and even more because they add an
enormous amount of new material on the game to this second edition (in
chapter 20; btw. if after yesterday you are still interested in the game of sprouts you might be interested in
chapter 17 of the same volume). Most of the calculations were done with
the combinatorial game suite program of Aaron
Siegel.
The
game of SNORT was invented by Simon Norton. The
rules of its SNORTgo-version are : black and white take turns in
putting a stone on a go-board such that no two stones of different
colour occupy neighbouring spots. In contrast to COLgo it is a
hot game meaning that many of its positions are non-numbers. The picture
on the left is a SNORTgo endgame of exact value
{{3|2}|-1}
and is therefore a fuzzy game meaning that the first player to
move has a winning strategy. Can you find the best move for white? and
for black?
If not, you can always consult the slides of the talk I gave this afternoon for the
WIS-seminar. Afterwards there was a game playing
afternoon but I felt too tired and went home early.
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).
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.
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.