Let us begin with a simple enough question : what are the points of a non-commutative variety? Anyone? Probably you\’d say something like : standard algebra-geometry yoga tells us that we should associate to a non-commutative algebra $A$ on object, say $XA$ and an arbitrary variety is then build from \’gluing\’ such things together. Ok, but what is $XA$? Commutative tradition whispers $XA=\mathbf{spec}~A$ the [prime spectrum][1] of $A$, that is, the set of all twosided prime ideals $P$ (that is, if $aAb \subset P$ then either $a \in P$ or $b \in P$) and \’points\’ of $\mathbf{spec}~A$ would then correspond to _maximal twosided ideals. The good news is that in this set-up, the point-set comes equipped with a natural topology, the [Zariski topology][2]. The bad news is that the prime spectrum is rarely functorial in the noncommutative world. That is, if $\phi~:~A \rightarrow B$ is an algebra morphism then $\phi^{-1}(P)$ for $P \in \mathbf{spec}~B$ is not always a prime ideal of $A$. For example, take $\phi$ the inclusion map $\begin{bmatrix} C[x] & C[x] \ (x) & C[x] \end{bmatrix} \subset \begin{bmatrix} C[x] & C[x] \ C[x] & C[x] \end{bmatrix}$ and $P$ the prime ideal $\begin{bmatrix} (x) & (x) \ (x) & (x) \end{bmatrix}$ then $P Cap \begin{bmatrix} C[x] & C[x] \ (x) & C[x] \end{bmatrix} = P$ but the corresponding quotient is $\begin{bmatrix} C & C \ 0 & C \end{bmatrix}$ which is not a prime algebra so $\phi^{-1}(P)$ is not a prime ideal of the smaller algebra.
Failing this, let us take for $XA$ something which obviously is functorial and worry about topologies later. Take $XA = \mathbf{rep}~A$ the set of all finite dimensional representations of $A$, that is $\mathbf{rep}~A = \bigsqcupn \mathbf{rep}n~A$ where $\mathbf{rep}n~A = { Chi~:~A \rightarrow Mn(C)~}$ with $Chi$ an algebra morphism. Now, for any algebra morphism $\phi~:~A \rightarrow B$ there is an obvious map $\mathbf{rep}~B \rightarrow \mathbf{rep}~A$ sending $Chi \mapsto Chi Circ \phi$. Alernatively, $\mathbf{rep}n~A$ is the set of all $n$-dimensional left $A$-modules $M{Chi} = C^n{Chi}$ with $a.m = Chi(m)m$. As such, $\mathbf{rep}~A$ is not merely a set but a $C$-category, that is, all objects are $C$-vectorspaces and all morphisms $Hom(M,N)$ are $C$-vectorspaces (the left $A$-module morphisms). Moreover, it is an _additive category, that is if $Chi,\psi$ are representations then we also have a direct sum representation $Chi \oplus \psi$ defined by $a \mapsto \begin{bmatrix} Chi(a) & 0 \ 0 & \psi(a) \end{bmatrix}$. Returning at the task at hand let us declare a non-commutative variety $X$ to be (1) an additive $C$-category which \’locally\’ looks like $\mathbf{rep}~A$ for some non-commutative algebra $A$ (even if we do not know at the momemt what we mean by locally as we do not have defined a topology, yet). Let is call objects of teh category $X$ the \’points\’ of our variety and $X$ being additive allows us to speak of indecomposable points (that is, those objects that cannot be written as a direct sum of non-zero objects). By the local description of $X$ an indecomposable point corresponds to an indecomposable representation of a non-commutative algebra and as such has a local endomorphism algebra (that is, all non-invertible endomorphisms form a twosided ideal). But if we have this property for all indecomposable points,our category $X$ will be a Krull-Schmidt category so it is natural to impose also the condition (2) : every point of $X$ can be decomposed uniquely into a finite direct sum of indecomposable points. Further, as the space of left $A$-module morphisms between two finite dimensional modules is clearly finite dimensional we have also the following strong finiteness condition (3) : For all points $x,y \in X$ the space of morphisms $Hom(x,y)$ is a finite dimensional $C$-vectorspace. In their book [Representations of finite-dimensional algebras][3], Peter Gabriel and Andrei V. Roiter call an additive category such that all endomorphism algebras of indecomposable objects are local algebras and such that all morphism spaces are finite dimensional an aggregate. So, we have a first tentative answer to our question the points of a non-commutative variety are the objects of an aggregate Clearly, as $\mathbf{rep}~A$ has stronger properties like being an Abelian category (that is, morphisms allow kernels and cokernels) it might also be natural to replace \’aggregate\’ by \’Abelian Krull-Schmidt category with finite dimensional homs\’ but if Mr. Abelian Category himself finds the generalization to aggregates useful I\’m not going to argue about this. Are all aggregates of the form $\mathbf{rep}~A$ or are there other interesting examples? A motivating commutative example is : the category of all coherent modules $Coh(Y)$ on a projective variety $Y$ form an aggegate giving us a mental picture of what we might expect of a non-commutative variety. Clearly, the above tentative answer cannot be the full story as we haven\’t included the topological condition of being locally of the form $\mathbf{rep}~A$ yet, but we will do that in the next episode B for Bricks. [1]: http://planetmath.org/encyclopedia/PrimeSpectrum.html [2]: http://planetmath.org/encyclopedia/ZariskiTopology.html [3]: http://www.amazon.co.uk/exec/obidos/ASIN/3540629904/qid=1106638540/sr=1- 1/ref=sr18_1/026-3923724-4530018

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1 comment
Posted in geometry
Written on Sun, 09 January 2005 at 12:24 pm
Tags: geometry, non-commutative, noncommutative, representations, topology
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