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 $X_A$ and an arbitrary

variety is then build from \’gluing\’ such things together. Ok, but what

is $X_A$? Commutative tradition whispers $X_A=\mathbf{spec}~A$ 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 $X_A$ something which obviously is

functorial and worry about topologies later. Take $X_A = \mathbf{rep}~A$

the set of all finite dimensional representations of $A$, that is

$\mathbf{rep}~A = \bigsqcup_n \mathbf{rep}_n~A$ where $\mathbf{rep}_n~A

= \{ Chi~:~A \rightarrow M_n(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]:

1/ref=sr_1_8_1/026-3923724-4530018

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