Naively A1-Connected Components of Varieties

Abstract

newline Vii newlineAbstract newlineA 1 -homotopy theory is a homotopy theory for schemes in which the affine line newlineA 1 plays the role of the unit interval. The main objects of study are simplicial newlineSheaves on the nisnevich site of smooth schemes of finite type over a field. For newlineThese objects, one constructs analogues of various devices from the classical ho- newlineMotopy theory of topological spaces. One such device is the sheaf of a 1 -connected newlineComponents of a simplicial sheaves. newlineFor a general simplicial sheaf x , the sheaf and#960; 0 a (x ) of a 1 -connected components newline1 newlineOf x is generally hard to compute. However, one can attempt to study it by means newlineOf the sheaf of naively a 1 -connected components, denoted by s(x ). The sheaf newlineS(x ) may be viewed as a crude approximation to and#960; 0 a (x ), but it is easier to define newline1 newlineAnd compute, at least when x is a sheaf of sets. The functor s is the main object newlineOf study in this thesis. newlineWhen x is a sheaf of sets, the direct limit of the sheaves s n (x ), which we newlineDenote by l(x ) can be proved to be a 1 -invariant. In fact, this is the universal newlineA 1 -homotopic quotient of x . When and#960; 0 a (x ) is a 1 -invariant, it can be proved to newline1 newlineBe isomorphic to l(x ). A recent example of ayoub has shown that and#960; 0 a (x ) is not newline1 newlineAlways a 1 -invariant. However, we show that there is a natural bijection between newlineField valued points of the sheaves l(x ) and and#960; 0 a (x ) for any sheaf of sets x . newline1 newlineThe sheaf l(x ) is obtained by iterating s on a the sheaf x infinitely many newlineTimes. Our second main result is to show that the infinitely many iterations are newlineIndeed necessary. We achieve this by constructing a family of sheaves {x n } n , newlineIndexed by the positive integers, such that s I (x n ) and#824; = s I+1 (x n ) for any I lt n. newlineThe third main result of this thesis is regarding retract rational varieties over newlineAn infinite field k. A result of kahn and sujatha shows that for a retract rational newlineVariety x, the sheaf and#960; 0 a (x) is the point sheaf. We strengthen this result by newline1 newlineShowing that s(x) is the point sheaf.