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=\sumi ai \otimes bi with \sumi aibi = 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 : $$P = S1 + S2 + … + Sk$$ where each Si is a simple algebra (that is, it contains no proper twosided ideals), finite dimensional over its center $li$ 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 $S1 \otimes S2$ is again a simple $L$-algebra if $S1$ and $S2$ 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.

• I Love Social Bookmarking
  
  • Subscribe
  • Digg
  • del.icio.us
  • Ma.gnolia
  • StumbleUpon
  • Technorati

2 comments
Posted in geometry
Written on Wed, 04 February 2004 at 4:55 pm
Tags: Brauer, Galois, geometry, groups, modular, non-commutative, rationality, simples
If you liked this post, then consider subscribing to our full RSS feed.