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.

# B for bricks

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