chapter  4
78 Pages

Quiver Technology

Having generalized the classical antiequivalence between commutative algebra and (affine) algebraic geometry to the pair of functors

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trepn .. GL(n)-affine

where ⇑n is a left-inverse for trepn, we will define Cayley-smooth algebras A ∈ [email protected], which are analogous to smooth commutative algebras. The definition is in terms of a lifting property with respect to nilpotent ideals, following Grothendieck’s characterization of regular algebras. We will prove Procesi’s result that a degree n Cayley-Hamilton algebra A is Cayley-smooth if and only if trepn A is a smooth (commutative) affine variety. This result allows us, via the theory of Knop-Luna slices, to describe the

e´tale local structure of Cayley-smooth algebras. We will prove that the local structure of A in a point ξ ∈ trissn A is determined by a combinatorial gadget: a (marked) quiver Q (given by the simple components of the semisimple n-dimensional representation Mξ corresponding to ξ and their (self)extensions)and a dimension vector α (given by the multiplicities of the simple factors in Mxi). In the next chapter we will use this description to classify Cayley-smooth or-

ders (as well as their central singularities) in low dimensions. In this study we will need standard results on the representation theory of quivers: the description of the simple (resp. indecomposable) dimension vectors, the canonical decomposition and the notion of semistable representations.