## Noncommutative Manifolds

By now we have developed enough machinery to study the representation varieties trepn A and trissn A of a Cayley-smooth algebra A ∈ [email protected] In particular, we now understand the varieties

repn A = trepn

∫ n

A and issn A = trissn ∫ n

A

for the level n approximation ∫ n A of a Quillen-smooth algebra A, for all

n. In this chapter we begin to study noncommutative manifolds, that is, families (Xn)n of commutative varieties that are locally controlled by Quillensmooth algebras. Observe that for every C-algebra A, the direct sum of representations induces sum maps

repn A× repm A - repn+m A and issn A× issm A - issn+m A

The characteristic feature of a family (Xn)n of varieties defining a noncommutative variety is that they are connected by sum-maps

Xn ×Xm - Xn+m and that these morphisms are locally of the form issn A × issm A - issn+m A for a Quillen-smooth algebra A. An important class of examples of such noncommutative manifolds is given by moduli spaces of quiver representations. In order to prove that they are indeed of the above type, we have to recall results on semi-invariants of quiver representations and universal localization.