For a

qurve (aka formally smooth algebra) A a *block* is a (possibly infinite

dimensional over the basefield) left A-module X such that its

endomorphism algebra $D = End_A(X)$ is a division algebra and X

(considered as a right D-module) is finite dimensional over D. If a

block X is finite dimensional over the basefield, we call it a *brick*

(aka a *Schur representation*). We want to endow the set of all blocks

with a topology and look at the induced topology on the subset of

bricks. It is an old result due to Claus Ringel

that there is a natural one-to-one correspondence between blocks of A

and algebra epimorphisms (in the categorical sense meaning that identify

equality of morphisms to another algebra) $A \rightarrow M_n(D) =

End_D(X_D)$. This result is important as it allows us to define a

partial order on teh set of all A-blocks via the notion of

*specialization*. If X and Y are two A-blocks with corresponding

epimorphisms $A \rightarrow M_n(D),~A \rightarrow M_m(E)$ we say that Y

is a specialization of X and we denote $X \leq Y$ provided there is an

epimorphism $A \rightarrow B$ making the diagram below commute

$\xymatrix{& M_n(D) \\\ A \ar[ru] \ar[r] \ar[rd] & B \ar[u]^i

\ar[d]^p \\\ & M_m(E)} $

where i is an inclusion and p is a

onto. This partial ordering was studied by Paul Cohn, George Bergman and

Aidan Schofield who use

the partial order to define the _closed subsets_ of blocks to be

those closed under specialization.

There are two important

constructions of A-blocks for a qurve A. One is Aidan’s construction of

a universal localization wrt. a *Sylvester rank function* (and which

should be of use in noncommutative rationality problems), the other

comes from invariant theory and is related to Markus Reineke’s monoid in

the special case when A is the path algebra of a quiver. Let X be a

GL(n)-closed irreducible subvariety of an irreducible component of

n-dimensional A-representations such that X contains a brick (and hence

a Zariski open subset of bricks), then taking PGL(n)-equivariant maps

from X to $M_n(\mathbb{C})$ determines a block (by inverting all central

elements). Now, take a *sensible* topology on the set of all A-bricks.

I would go for defining as the open wrt. a block X, the set of all

A-bricks which become simples after extending by the epimorphism

determined by a block Y such that $Y \leq X$. (note that this seems to

be different from the topology coming from the partial ordering…).

Still, wrt. this topology one can then again define a *noncommutative

topology* on the Abelian category $\mathbf{rep}~A$ of all finite

dimensional A-representations

but this time using filtrations with successive quotients being bricks

rather than simples.

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