Etale topology was introduced in algebraic geometry to bypass the coarseness of the Zariski topology for classification problems. Let us give an elementary example: the local classification of smooth varieties in the Zariski topology is a hopeless task, whereas in the e´tale topology there is just one local type of smooth variety in dimension d, namely, affine d-space Ad. A major theme of this book is to generalize this result to noncommutative [email protected] Etale cohomology groups are used to classify central simple algebras over
function fields of varieties. Orders in such central simple algebras (over the central structure sheaf) are an important class of Cayley-Hamilton algebras. Over the years, one has tried to construct a suitable class of smooth orders
that allows an e´tale local description. But, except in the case of curves and surfaces, no such classification is known, say, for orders of finite global dimension. In this book we introduce the class of Cayley-smooth orders, which does allow an e´tale local description in arbitrary dimensions. In this chapter we will lay the foundations for this classification by investigating e´tale slices of representation varieties at semisimple representations. In chapter 5 we will then show that this local structure is determined by a combinatorial gadget: a (marked) quiver setting.