Last time we argued that a noncommutative variety might be an aggregate which locally is of the form $\wis{rep}~A$ 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 $\wis{rep}~A$, let alone on an arbitrary aggregate. Today we will start the construction of a truly non-commutative topology on $\wis{rep}~A$.
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 $\wis{rep}~A$ 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 $\wis{simp}~A$, the set of all fnite dimensional simple representations, as thick subset equipped with the induced Zariski topology on the prime spectrum $\wis{spec}~A$. However, this topology doesn't behave well with respect to the gluings we have in mind so we will extend $\wis{simp}~A$ substantially.

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Posted in geometry
Written on Fri, 28 January 2005 at 4:11 pm
Tags: arxiv, non-commutative, noncommutative, representations, topology
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