Affine varieties and finite morphisms

An affine algebraic group

Let [Formula: see text] be a [Formula: see text]-equivariant algebraic principal [Formula: see text]-bundle over a normal complex affine variety [Formula: see text] equipped with an action of [Formula: see text], where [Formula: see text] and [Formula: see text] are complex linear algebraic groups. Suppose [Formula: see text] is contractible as a topological [Formula: see text]-space with a dense orbit, and [Formula: see text] is a [Formula: see text]-fixed point. We show that if [Formula: see text] is reductive, then [Formula: see text] admits a [Formula: see text]-equivariant isomorphism with the product principal [Formula: see text]-bundle [Formula: see text], where [Formula: see text] is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal [Formula: see text]-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal [Formula: see text]-bundles over any complex toric variety, generalizing the main result of [A classification of equivariant principal bundles over nonsingular toric varieties, Internat. J. Math. 27(14) (2016)].

This chapter investigates the objects and mappings of algebraic geometry from a geometric point of view, making use especially of the algebraic tools of Chapter VII and of Sections 7–10 of Chapter VIII. In Sections 1–12, k denotes a fixed algebraically closed field. Sections 1–6 establish the definitions and elementary properties of varieties, maps between varieties, and dimension, all over k. Sections 1–3 concern varieties and dimension. Affine algebraic sets, affine varieties, and the Zariski topology on affine space are introduced in Section 1, and projective algebraic sets and projective varieties are introduced in Section 3. Section 2 defines the geometric dimension of an affine algebraic set, relating the notion to Krull dimension and transcendence degree. The actual context of Section 2 is a Noetherian topological space, the Zariski topology on affine space being an example. In such a space every closed subset is the finite union of irreducible closed subsets, and the union can be written in a certainway thatmakes the decomposition unique. Every nonempty closed set has a meaningful geometric dimension. In affine space the irreducible closed sets are the varieties, and each variety acquires a geometric dimension. The discussion in Section 2 applies in the context of projective space as well, and thus each projective variety acquires a geometric dimension. Moreover, any nonempty open subset of a Noetherian space is Noetherian. A nonempty open subset of an affine variety is called quasi-affine, and a nonempty open subset of a projective variety is called quasiprojective. Each quasi-affine variety or quasiprojective variety has a dimension equal to that of its closure, which is a variety. Sections 4–6 take up maps between varieties. Section 4 introduces spaces of scalar-valued functions on quasiprojective varieties–rational functions, functions regular at a point, and functions regular on an open set. The section goes on to relate these notions for the different kinds of varieties. Section 5 introduces morphisms, which are a restricted kind of function between varieties. The tools of Sections 4–5 together show that for many purposes all the different kinds of varieties can be treated as quasiprojective varieties. Section 6 introduces rational maps between varieties; these are not everywhere-defined functions, but each can be restricted to an open dense subset on which it is a morphism. Rational maps with dense image correspond to field mappings of the fields of rational functions, with the order of the mappings reversed. Section 7 concerns singularities at points of varieties, still over the field k. Zariski's Theorem was stated in Chapter VII for affine varieties and partly proved at that time. In the current context it has a meaning for any point of any quasiprojective variety. The section proves the full theorem, which characterizes singular points in a way that shows they remain singular under isomorphisms of varieties. Section 8 concerns classification questions over k for irreducible curves, i.e., quasiprojective varieties of dimension 1. From Section 6 it is known that two irreducible curves are equivalent under rational maps if and only if their fields of rational functions are isomorphic. The main theorem of Section 8 is that each such equivalence class of irreducible curves contains an everywhere nonsingular projective curve, and this curve is unique up to isomorphism of varieties. The points of this curve are parametrized by those discrete valuations of the underlying function field that are defined over k. Sections 9–12 relate the general theory of Sections 1–6 to the topic of solutions of simultaneous solutions of polynomial equations, as treated at length in Chapter VIII. Section 9 treats monomial ideals in k[X1, . . . , Xn], identifying their zero loci concretely and computing their dimension. The section goes on to introduce the affineHilbert function of this ideal, which measures the proportion of polynomials of degree≤ s not in the ideal. In the way that this function is defined, it is a polynomial for large s called the affine Hilbert polynomial of the ideal. Its degree equals the dimension of the zero locus of the ideal. Section 10 extends this theory from monomial ideals to all ideals, again concretely computing the dimension of the zero loci, obtaining an affine Hilbert polynomial, and showing that its degree equals the dimension of the zero locus of the ideal. Section 11 adapts the theory to homogeneous ideals and projective algebraic sets by making use of the cone in affine space over the set in projective space. Section 12 applies the theory of Section 11 to address the question how the dimension of a projective algebraic set is cut down when the set is intersected with a projective hypersurface. A consequence of the theory is the result that a homogeneous system of polynomial equations over an algebraically closed field with more unknowns than equations has a nonzero solution. Section 13 is a brief introduction to the theory of schemes, which extends the theory of varieties by replacing the underlying algebraically closed field by an arbitrary commutative ring with identity.

We study the group of automorphisms Aut(G) of a given group G from the groups of automorphisms (supposed known) of some of its subgroups; basic results are applied to topological groups and widely developed inside the Borel–Chevalley theory of linear groups; affine algebraic complete groups are studied in detail.

First, we prove that any inner automorphism in the stabilizer of a graded-simple unital associative algebra whose grading group is abelian is the conjugation by a homogeneous element. Now consider a grading by an abelian group on an associative algebra such that the algebra is graded-simple and satisfies the DCC on graded left ideals. We give necessary and sufficient conditions for the grading to be fine. Then we assume that one of these necessary conditions to be fine is satisfied, and we compute the automorphism groups of the grading; the results are expressed in terms of the automorphism groups of a graded-division algebra. Finally, we compute the automorphism groups of graded-division algebras in the case in which the ground field is the field of real numbers, and the underlying algebra (disregarding the grading) is simple and of finite dimension.

A table algebra was defined in [1] in order to consider in a uniform way the common properties of conjugacy classes and irreducible characters. Non-commutative table algebras were introduced in [5]. They generalize properties of such well-known objects as coherent and Hecke algebras. Here we extend the main definition of a non-commutative table algebra by letting the ground field be an integral domain. We call these algebrasgeneralized table algebras (GT-algebras, in brief). It is worth mentioning that this class of algebras includes generic Hecke-Iwahori algebras of finite Coxeter groups. We develop the general theory for this type of algebras which includes their representation theory and theory of closed subsets. We also study the properties of primitive integral table algebras.

In a 2004 paper, Totaro asked whether a G-torsor X that has a zero-cycle of degree d > 0 will necessarily have a closed etale point of degree dividing d, where G is a connected algebraic group. This question is closely related to several conjectures regarding exceptional algebraic groups. Totaro gave a positive answer to his question in the following cases: G simple, split, and of type G_2, type F_4, or simply connected of type E_6. We extend the list of cases where the answer is "yes" to all groups of type G_2 and some nonsplit groups of type F_4 and E_6. No assumption on the characteristic of the base field is made. The key tool is a lemma regarding linkage of Pfister forms.

We show that for every $n\ge 7$ there is a smooth rational affine variety with exotic model. Moreover, we show that for every $n\ge 6$ there are Zariski open subsets $U_1, U_2$ of $\mathbb {C}^n$, such that they are holomorphically but not algebraically equivalent.

The action of an affine algebraic group G on an algebraic variety V can be differentiated to a representation of the Lie algebra L(G) of G by derivations on the sheaf of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a homomorphism ρ : L → DerK(K[U]) for an affine algebraic variety U, one may wonder whether it comes from an algebraic group action on U or on a variety V containing U as an open subset. In this paper, we prove two results on this integration problem. First, if L acts faithfully and locally finitely on K[U], then it can be embedded in L(G), for some affine algebraic group G acting on U, in such a way that the representation of L(G) corresponding to that action restricts to ρ on L. In the second theorem, we assume from the start that L = L(G) for some connected affine algebraic group G and show that some technical but necessary conditions on ρ allow us to integrate ρ to an action of G on an algebraic variety V containing U as an open dense subset. In the interesting cases where L is nilpotent or semisimple, there is a natural choice for G, and our technical conditions take a more appealing form.

Reductive subgroups of reductive groups in nonzero characteristic

A connected component of an affine algebraic group is called periodic if all its elements have finite order. We give a characterization of periodic components in terms of automorphisms with finitely many fixed points. Also discussed is which connected groups have finite extensions with periodic components. The results are applied to the study of the normalizer of a maximal torus in a simple algebraic group.Bibliography: 10 titles.

Let us fic an algebraically closed field K. We work in the category of affine varieties over K. The purpose of this section is to give a brief exposition on the basic facts of algebraic groups.

The equivariant cohomology ring of a G-space X defines a homogeneous affine variety. Quillen [Q] and W. Y. Hsiang [Hs] have determined the relation between such varieties and the family of isotropy subgroups as well as their fixed point sets when dim X < oo. In modular representation theory, J. Carlson [Cj] introduced cohomological support varieties and rank varieties (the latter depending on the group algebra) and explored their relationship. We define rank and support varieties for G-spaces and G-chain complexes and apply them to cohomological problems in transformation groups. As a corollary, a useful criterion for ZG-projectivity of the reduced total homology of certain G-spaces is obtained, which improves the projectivity criteria of Rim [R], Chouinard [Ch], and Dade [D].

In this article, we find all irreducible three-dimensional affine algebraic varieties that admit a quasi-transitive algebraic group of biregular automorphisms (that is, there is an orbit under the group action whose complement has dimension at most zero). The ground field is algebraically closed and has characteristic zero.Bibliography: 29 items.

On the structure of commutative pointed Hopf algebras