Last time we
argued that a noncommutative variety might be an _aggregate_
which locally is of the form $\mathbf{rep}~A$ for some affine (possibly
non-commutative) $C$-algebra $A$. However, we didn't specify what we
meant by 'locally' as we didn't define a topology on
$\mathbf{rep}~A$, let alone on an arbitrary aggregate. Today we will start
the construction of a truly _non-commutative topology_ on
$\mathbf{rep}~A$.
Here is the basic idea : we start with a thick
subset of finite dimensional representations on which we have a natural
(ordinary) topology and then we extend this to a non-commutativce
topology on the whole of $\mathbf{rep}~A$ using extensions. The impatient
can have a look at my old note A noncommutative
topology on rep A but note that we will modify the construction here
in two essential ways.
In that note we took $\mathbf{simp}~A$, the
set of all fnite dimensional simple representations, as thick subset
equipped with the induced Zariski topology on the prime spectrum
$\mathbf{spec}~A$. However, this topology doesn't behave well with
respect to the gluings we have in mind so we will extend $\mathbf{simp}~A$
substantially.
Tag: topology
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
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?