Algebraic Groups by Jürgen Müller

By Jürgen Müller

Show description

Read or Download Algebraic Groups PDF

Similar algebraic geometry books

Equidistribution in Number Theory, An Introduction

Written for graduate scholars and researchers alike, this set of lectures presents a established creation to the idea that of equidistribution in quantity thought. this idea is of starting to be value in lots of components, together with cryptography, zeros of L-functions, Heegner issues, major quantity concept, the speculation of quadratic types, and the mathematics facets of quantum chaos.

Undergraduate algebraic geometry

Algebraic geometry is, basically, the learn of the answer of equations and occupies a primary place in natural arithmetic. With the minimal of necessities, Dr. Reid introduces the reader to the elemental ideas of algebraic geometry, together with: aircraft conics, cubics and the gang legislation, affine and projective forms, and nonsingularity and size.

Constructible Sets in Real Geometry

This e-book offers a scientific and unified file at the minimum description of constructible units. It starts off at a truly uncomplicated point (almost undergraduate) and leads as much as cutting-edge effects, a lot of that are released in ebook shape for the first actual time. The ebook includes quite a few examples, sixty three figures and every bankruptcy ends with a piece containing ancient notes.

Algebraic geometry for scientists and engineers

This booklet, in line with lectures offered in classes on algebraic geometry taught via the writer at Purdue college, is meant for engineers and scientists (especially machine scientists), in addition to graduate scholars and complicated undergraduates in arithmetic. as well as supplying a concrete or algorithmic method of algebraic geometry, the writer additionally makes an attempt to encourage and clarify its hyperlink to extra sleek algebraic geometry in response to summary algebra.

Additional info for Algebraic Groups

Sample text

C) Show that dim(RU ) = sup{di − di−1 ; i ∈ N} ∈ N ∪ {∞}. Proof. 6]. 15) Exercise: Dimension and height. Give an example of a finitely generated K-algebra, where K is a field, which is not a domain, possessing an ideal I R such that dim(I) + ht(I) = dim(R). 16) Exercise: Catenary rings. A finite dimensional Noetherian ring R is called catenary, if for any prime ideals P ⊆ Q R all maximal chains P = P0 ⊂ · · · ⊂ Pr = Q of prime ideals have length r = ht(Q) − ht(P ). Let K be a field, and let R be a finitely generated K-algebra which is a domain.

We may assume that G ≤ GLn closed, for some n ∈ N. Since G is abelian, Gs , Gu ≤ G are subgroups, and µ is a bijective homomorphism of algebraic groups. The set Gu ⊆ G is closed, and we show that Gs ⊆ G also is closed: For any family Λ := {λA ∈ K; A ∈ Gs } let WΛ := A∈Gs Eλ (A) ≤ Kn . r Hence we have Kn ∼ = i=1 WΛr , for some r ∈ N and certain families Λr , where the WΛr are G-invariant. 4). Hence Gs = G ∩ Tn ⊆ G is closed. The morphism Bn → Tn : [aij ] → diag[a11 , . . , ann ] restricts to the morphism G → Gs : g → gs , hence µ−1 : G → Gs × Gu : g → [gs , gs−1 g] is a morphism.

R−1 , λr + 1, λr+1 , . . , λs−1 , λs − 1, λs+1 , . . 26): II Algebraic groups 32 If λ max µ, let r := min{i ∈ {1, . . , n}; λi = µi } and r < s := min{k ∈ k k {r + 1, . . , n}; i=1 λi = i=1 µi } ≤ n. Hence we have λr < µr , and µr ≤ µr−1 = λr−1 if r > 1, as well as λs > µs ≥ µs+1 ≥ λs+1 . This yields λ ν := [λ1 , . . , λr−1 , λr + 1, λr+1 , . . , λs−1 , λs − 1, λs+1 , . . , λn ] µ, hence ν = µ. It remains to show λr = λs whenever s > r + 1: Assume to the contrary that λr > λs , and let r < t := 1 + min{i ∈ {r, .

Download PDF sample

Rated 4.08 of 5 – based on 40 votes