By Tamás Szamuely

Show description

Read Online or Download Lectures on Linear Algebraic Groups PDF

Similar differential geometry books

Download PDF by Norman Steenrod: The topology of fibre bundles

Fibre bundles, now a vital part of differential geometry, also are of serious significance in smooth physics - corresponding to in gauge idea. This ebook, a succinct advent to the topic via renown mathematician Norman Steenrod, used to be the 1st to provide the topic systematically. It starts with a common creation to bundles, together with such subject matters as differentiable manifolds and overlaying areas.

Differential geometry and complex analysis: a volume - download pdf or read online

Chavel I. , Farkas H. M. (eds. ) Differential geometry and complicated research (Springer, 1985)(ISBN 354013543X)(236s)

Theorems on regularity and singularity of energy minimizing by Leon Simon PDF

The purpose of those lecture notes is to offer an basically self-contained advent to the fundamental regularity idea for power minimizing maps, together with contemporary advancements about the constitution of the singular set and asymptotics on method of the singular set. really good wisdom in partial differential equations or the geometric calculus of diversifications is no longer required.

Additional info for Lectures on Linear Algebraic Groups

Example text

3. Let G be an affine algebraic group, H ⊂ G a closed subgroup. Then the quotient of G by H exists. Moreover, it is a homogeneous space for G such that H is the stabilizer of a point. Proof. 4. 2. It is enough to check this property for a connected affine open subset U ⊂ X. Pick f ∈ O(ρ−1 (U )) constant on the left cosets of H, and (ρ,f ) consider the composite map ρ−1 (U ) −→ U × A1 → U , where the last map is the natural projection. Note that ρ−1 (U ) is a finite disjoint union of connected open sets, each one dense in a component of G.

So h ∈ U U −1 ⊂ Im (ϕn )Im (ϕn ). This shows H = [G, G]. 1. 7. Let ϕ : X → Y be an injective morphism of irreducible quasi-projective varieties with Zariski dense image. If the induced field extension k(X)|k(Y ) is separable, then it is an isomorphism. 1, we see that the proposition is a consequence of the following lemma. 8. Assume that ϕ : X → Y is a morphism of affine varieties and ϕ∗ induces an isomorphism AX ∼ = AY [f ] with f separable over k(Y ). Then there is an open subset V ⊂ Y such that each point of V has exactly [k(X) : ϕ∗ k(Y )] preimages in X.

2 we find a representation of G on some finite-dimensional V with GP stabilizing a one-dimensional subspace in V , hence fixing a point Q in the induced action of G on the projective space P(V ). Let Y be the orbit of Q in P(V ) and Z that of (P, Q) in X × P(V ) (equipped with the product action). The natural projections Z → X and Z → Y are bijective G-morphisms, so it is enough to find a fixed point in Y (which must then be the whole of Y ). 5 below, but this was not used in the proof of the proposition).

Download PDF sample

Lectures on Linear Algebraic Groups by Tamás Szamuely

by Paul

Rated 4.59 of 5 – based on 35 votes