Here pdf-files of older NeverEndingBooks-posts on geometry. For more recent posts go here.

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The categorical cafe has a guest post by Tom Leinster Linear Algebra Done Right on the book with the same title by Sheldon Axler. I haven’t read the book but glanced through his online paper Down with determinants!. Here is ‘his’ proof of the fact that any n by n matrix A has at least one eigenvector. Take a vector , then as the collection of vectors must be linearly dependent, there are complex numbers such that But then as is algebraically closed the polynomial on the left factors into linear factors and therefore as from which it follows that at least one of the linear transformations has a non-trivial kernel, whence A has an eigenvector with eigenvalue . Okay, fine, nice even, but does this simple minded observation warrant the extreme conclusion of his paper (on page 18) ?

As mathematicians, we often read a nice new proof of a known theorem, enjoy the different approach, but continue to derive our internal understanding from the method we originally learned. This paper aims to change drastically the way mathematicians think about and teach crucial aspects of linear algebra.
The simple proof of the existence of eigenvalues given in Theorem 2.1 should be the one imprinted in our minds, written on our blackboards, and published in our textbooks. Generalized eigenvectors should become a central tool for the understanding of linear operators. As we have seen, their use leads to natural definitions of multiplicity and the characteristic polynomial. Every mathematician and every linear algebra student should at least remember that the generalized eigenvectors of an operator always span the domain (Proposition 3.4)—this crucial result leads to easy proofs of upper-triangular form (Theorem 6.2) and the Spectral Theorem (Theorems 7.5 and 8.3).
Determinants appear in many proofs not discussed here. If you scrutinize such proofs, you’ll often discover better alternatives without determinants. Down with Determinants!

I welcome all new proofs of known results as they allow instructors to choose the one best suited to their students (and preferable giving more than one proof showing that there is no such thing as ‘the best way’ to prove a mathematical result). What worries me is Axler’s attitude shared by extremists and dogmatics world-wide : they are so blinded by their own right that they impoverish their own lifes (and if they had their way, also that of others) by not willing to consider other alternatives. A few other comments : 1. I would be far more impressed if he had given a short argument for the one line he skates over in his proof, that of being algebraically closed. Does anyone give a proof of this fact anymore or is this one of the few facts we expect first year students to accept on faith? 2. I dont understand this aversity to the determinant (probably because of its nonlinear character) but at the same time not having any problems with successive powers of matrices. Surely he knows that the determinant is a fixed -polynomial in the traces (which are linear!) of powers of the matrix. 3. The essense of linear algebra is that by choosing a basis cleverly one can express a linear operator in a extremely nice matrix form (a canonical form) so that all computations become much more easy. This crucial idea of considering different bases and their basechange seems to be missing from Axler’s approach. Moreover, I would have thought that everyone would know these days that ‘linear algebra done right’ is a well developed topic called ‘representation theory of quivers’ but I realize this might be viewed as a dogmatic statement. Fortunately someone else is giving the basic linear algebra courses here in Antwerp so students are spared my private obsessions (at least the first few years…). In [his post](http://golem.ph.utexas.edu/category/2007/05/ linearalgebradoneright.html) Leistner askes “What are determinants good for?” I cannot resist mentioning a trivial observation I made last week when thinking once again about THE rationality problem and which may be well known to others. Recall from the previous post that rationality of the quotient variety of matrix-couples under simultaneous conjugation is a very hard problem. On the other hand, the ‘near miss’ problem of the quotient variety of matrix-couples is completely trivial. It is rational for all n. Here is a one-line proof. Consider the quiver then the dimension vector (n-1,n) is a Schur root and the first fundamental theorem of (see for example Hanspeter Krafts excellent book on invariant theory) asserts that the corresponding quotient variety is the one above. The result then follows from Aidan Schofield’s paper Birational classification of moduli spaces of representations of quivers. Btw. in this special case one does not have to use the full force of Aidan’s result. Zinovy Reichstein, who keeps me updated on events in Atlanta, emailed the following elegant short proof Here is an outline of a geometric proof. Let and . Applying the no-name lemma to the -equivariant dominant rational map given by (which makes X into a vector bundle over a dense open -invariant subset of Y), we see that is rational over On the other hand, is an affine space. Thus is rational. The moment I read this I knew how to do this quiver-wise and that it is just another Brauer-Severi type argument so completely inadequate to help settling the genuine matrix-problem. Update on the paper by Esther Beneish : Esther did submit the paper in february.

Half a year ago, it all started with NeverEndingBooks in which I set out a rather modest goal :

Why NeverEndingBooks ? We all complain about exaggerated prices of mathematical books from certain publishers, poor quality of editing and refereeing offered, as well as far too stringent book-contracts. Rather than lamenting about this, NeverEndingBooks gives itself one year to learn (and report) about the many aspects of the book-production cycle and to explore whether an alternative exists. If at the end of this year we will have produced at least one book this experiment will be considered a success. If, however, we find out that it is an impossible task, we will explain where it all went wrong and why it is better to stick to an established PublishingHouse and accept its terms.

I assume we did manage to do it after all as you may check by visiting our storefront : www.lulu.com/neverendingbooks. However, it all turned out to be quite different from what I had in mind half a year ago. So, perhaps it’s time to recap.

Originally, I’d planned to partner-up with the publisher-on-demand LightningSource but in the process they did require a VAT-number. In Belgium, the safest way to get one is to set up a non-profit organization (a VZW as we call it). But then you have to write down your legal statutes, get them published in the Moniteur Belge (at a hefty price) but what really put me off was that you have to set up a “board of directors” consisting of at least three people. I don‚Äôt mind following a folly but if I have to involve others I usually pass, so I abandoned the whole idea. Still, I couldn‚Äôt help talking about the VAT-problem and at a certain time there was an idea to revive a sleeping VZW (=non-profit organization) of the Belgian Mathematical Society, the MaRC (MAthematical Research Centre), the statutes of which allowed to become a publishing house. But, this wouldn‚Äôt involve just two other people but the whole BMS so I decided to forget all about it and have a short vacation in France together with a few (former)PhD-students.

Given plenty of sun, cheese and whine (not necessarily in that order) sooner or later we had to talk about the problem. For Raf it was the first time he heard about it but when we realized I thought one could easily publish books well under 25 dollars he was immediately interested and insisted we should set up a board of directors and continue with the plan. The different roles to play in the board were more or less self-evident : I had to be the trasurer (given the fact that I was the only with a secure, though small, income), Geert had to become chairman (being the only one possessing suits), Raf would be secretary (being the only one who could write better Flemish than English) and Jan or Stijn would do PR (as they are the only ones having enough social skills). So, we went back willing to go through the whole process (at least 3 months) of obtaining a VAT-number.

But then Raf got so interested in the whole idea that he explored other possibilities (I think he was more motivated by the fact that his sister wanted to publish her thesis rather than anything else) and came up with lulu.com. No legal hassle, no VAT-numbers, nothing required (or so it seemed). Still, before risking his sister‚Äôs thesis he wanted to check the service out and as it is a lot easier to take a book lying around rather than write one yourself he took my version 2 and published it at Lulu’s (since then this version is nicknamed Rothko@n).

Although I gave him the permission to do so, it didn‚Äôt feel right that people should pay even a small amount for a nicely bound unedited version 2. So, the last month and a half I‚Äôve been editing and partially rewriting version 2 and the two volumes are now available! Major changes are to the 4 middle chapters. There is now chapter 3 “Etale Technology” which contains all of the etale tricks scattered earlier in two chapters, chapter 4 ‚”Quiver Representations” collects all the quiver material (again, scattered throughout the previous version). Chapter 5 ‚”Semisimple Representations” now includes recent material such as Raf’s characterization of the smooth locus of Cayley-smooth orders and our (together with Geert) classification of the central singularities, and chapter 6 ‚”Nilpotent Representations” now includes the material on Brauer-Severi varities which was in version 1 but somehow didnt make it to version 2 before.

One of the things I like most about returning from a vacation is to have an enormous pile of fresh reading : a week's worth of newspapers, some regular mail and much more email (three quarters junk). Also before getting into bed after the ride I like to browse through the arXiv in search for interesting papers.
This time, the major surprise of my initial survey came from the newspapers. No, not Bush again, that news was headline even in France. On the other hand, I didn't hear a word about Theo Van Gogh being shot and stabbed to death in Amsterdam. I'll come back to this later.
I'd rather mention the two papers that somehow stood out during my scan of this week on the arXiv. The first is Framed quiver moduli, cohomology, and quantum groups by Markus Reineke. By the deframing trick, a framed quiver moduli problem is reduced to an ordinary quiver moduli problem for a dimension vector for which one of the entries is equal to one, hence in particular, an indivisible dimension vector. Such quiver problems are far easier to handle than the divisible ones where everything can at best be reduced to the classical problem of classifying tuples of $n \times n$ matrices up to simultaneous conjugation. Markus deals with the case when the quiver has no oriented cycles. An important examples of a framed moduli quiver problem with oriented cycles is the study of Brauer-Severi varieties of smooth orders. Significant progress on the description of the fibers in this case is achieved by Raf Bocklandt, Stijn Symens and Geert Van de Weyer and will (hopefully) be posted soon.
The second paper is Moduli schemes of rank one Azumaya modules by Norbert Hoffmann and Urich Stuhler which brings back longforgotten memories of my Ph.D. thesis, 21 years ago…

Last time we saw that for $A$ a smooth order with center $R$ the Brauer-Severi variety $XA$ is a smooth variety and we have a projective morphism $XA \rightarrow \mathbf{max}~R$ This situation is very similar to that of a desingularization $~X \rightarrow \mathbf{max}~R$ of the (possibly singular) variety $~\mathbf{max}~R$. The top variety $~X$ is a smooth variety and there is a Zariski open subset of $~\mathbf{max}~R$ where the fibers of this map consist of just one point, or in more bombastic language a $~\mathbb{P}^0$. The only difference in the case of the Brauer-Severi fibration is that we have a Zariski open subset of $~\mathbf{max}~R$ (the Azumaya locus of A) where the fibers of the fibration are isomorphic to $~\mathbb{P}^{n-1}$. In this way one might view the Brauer-Severi fibration of a smooth order as a non-commutative or hyper-desingularization of the central variety.
This might provide a way to attack the old problem of construction desingularizations of quiver-quotients. If $~Q$ is a quiver and $\alpha$ is an indivisible dimension vector (that is, the component dimensions are coprime) then it is well known (a result due to Alastair King) that for a generic stability structure $\theta$ the moduli space $~M^{\theta}(Q,\alpha)$ classifying $\theta$-semistable $\alpha$-dimensional representations will be a smooth variety (as all $\theta$-semistables are actually $\theta$-stable) and the fibration
$~M^{\theta}(Q,\alpha) \rightarrow \mathbf{iss}{\alpha}~Q$ is a desingularization of the quotient-variety $~\mathbf{iss}{\alpha}~Q$ classifying isomorphism classes of $\alpha$-dimensional semi-simple representations. However, if $\alpha$ is not indivisible nobody has the faintest clue as to how to construct a natural desingularization of $~\mathbf{iss}{\alpha}~Q$. Still, we have a perfectly reasonable hyper-desingularization $~X{A(Q,\alpha)} \rightarrow \mathbf{iss}{\alpha}~Q$ where $~A(Q,\alpha)$ is the corresponding quiver order, the generic fibers of which are all projective spaces in case $\alpha$ is the dimension vector of a simple representation of $~Q$. I conjecture (meaning : I hope) that this Brauer-Severi fibration contains already a lot of information on a genuine desingularization of $~\mathbf{iss}{\alpha}~Q$. One obvious test for this seemingly crazy conjecture is to study the flat locus of the Brauer-Severi fibration. If it would contain info about desingularizations one would expect that the fibration can never be flat in a central singularity! In other words, we would like that the flat locus of the fibration is contained in the smooth central locus. This is indeed the case and is a more or less straightforward application of the proof (due to Geert Van de Weyer) of the Popov-conjecture for quiver-quotients (see for example his Ph.D. thesis Nullcones of quiver representations).
However, it is in general not true that the flat-locus and central smooth locus coincide. Sometimes this is because the Brauer-Severi scheme is a blow-up of the Brauer-Severi of a nicer order. The following example was worked out together with Colin Ingalls : Consider the order $~A = \begin{bmatrix} C[x,y] & C[x,y] \ (x,y) & C[x,y] \end{bmatrix}$ which is the quiver order of the quiver setting $~(Q,\alpha)$ then the Brauer-Severi fibration $~XA \rightarrow \mathbf{iss}{\alpha}~Q$ is flat everywhere except over the zero representation where the fiber is $~\mathbb{P}^1 \times \mathbb{P}^2$. On the other hand, for the order $~B = \begin{bmatrix} C[x,y] & C[x,y] \ C[x,y] & C[x,y] \end{bmatrix}$
the Brauer-Severi fibration is flat and $~XB \simeq \mathbb{A}^2 \times \mathbb{P}^1$. It turns out that $~XA$ is a blow-up of $~X_B$ at a point in the fiber over the zero-representation.