Category Of Algebraic Varieties Research Articles

Representations of complex tori and GL(2, C)

Groups and their representations have been studied for a long time. One can extend the notion of a group by asking the group axioms to hold in other categories. A group in the category of smooth manifolds is a Lie group, and a group in the category of algebraic varieties is an algebraic group. In this paper, we discuss the representation theory of algebraic groups, in particular complex tori and GL(2,C).

Factorial algebraic group actions and categorical quotients

Given an action of an affine algebraic group with only trivial characters on a factorial variety, we ask for categorical quotients. We characterize existence in the category of algebraic varieties. Moreover, allowing constructible sets as quotients, we obtain a more general existence result, which, for example, settles the case of a finitely generated algebra of invariants. As an application, we provide a combinatorial GIT-type construction of categorical quotients for actions of not necessarily reductive groups on, e.g. complete varieties with finitely generated Cox ring via lifting to the characteristic space.

A categorical quotient in the category of dense constructible subsets

A. A'Campo-Neuen and J. Hausen gave an example of an algebraic torus action on an open subset of the affine four space that admits no quotient in the category of algebraic varieties. We show that this example admits a quotient in the category of dense constructible subsets and thereby answer a question of A. Bialynicki-Birula.

Fibrant Presheaves of Spectra and Guillén–Navarro Extension

In this remark we prove that the Guillén–Navarro extension of a presheaf of spectra in the category of algebraic varieties over a field of characteristic zero, when exists, coincides up to weak equivalence with the fibrant replacement of the presheaf in the injective model category structure with the cd-topology of abstract blow-ups.

Spherical unirational homotopy groups

Spherical unirational homotopy objects as an example of a homotopy theory internal to the category of algebraic varieties over a field are introduced. Calculations of homotopy groups of spheres are made when the l-sphere is involved. The fundamental groups of the general and special linear groups are determined. A polynomial representation of 2 in the usual second homotopy group of the complex sphere is found. Let k be a field whose characteristic is not 2 and VAR the category of algebraic varieties defined over k. For this and other notions from algebraic geometry, see [4]. Then VAR* will denote the category of based algebraic varieties, whose objects are of the form (V, *) with * E I/ (the basepoint of a variety will always be denoted * unless there is possible confusion) and whose morphisms f : (V, *) + ( W, *) are the morphisms in VAR such that f(*) = *. We call the subset S”(k) of k”+l defined by 2 X1 +*.*+x,2+, = 1 a k-n-sphere. Pick a base point * for S”(k). We will show in Section 3 that there are isomorphisms in VAR sending any choice of basepoint for S”(k) to any other choice of basepoint. Thus the homotopy objects that we define below will be (up to an equivalence of functors) independent of choice of basepoint for S”(k). Let (X, *) be a based algebraic variety. Consider the collection [(S”(k), *), (X, *)] of maps (S”(k), *) -+(X, *) in VAR*. Then we associate to (X, *) the nth spherical unirational homotopy object (group operations are defined only in certain cases).

On the simple object associated to a diagram in a closed model category

In this paper we develop a descent technique for generalized (co)-homology theories defined in the category of algebraic varieties. By such a theory we mean a functor Sch→C, where C is a closed model category in the sense of Quillen satisfying certain axioms (cf. §4). We have chosen to work in such a general context so as to include two situations for which the results of SGA 4 of Deligne and Saint-Donat are not applicable: descent of multiplicative structures (i.e. of differential graded algebras) and descent for generalized sheaf cohomology (such as the algebraic K-theory of coherent sheaves over a noetherian scheme).