The second edition, thoroughly revised and expanded, extends the theory over arbitrary fields, which are not necessarily algebraically closed. It thus represents a higher aim. As in the first edition, the book includes a self-contained treatment of the prerequisites from algebraic geometry and commutative algebra, as well as basic results on reductive groups. As a result, the first part of the book can well serve as a text for an introductory graduate course on linear algebraic groups.
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It is an very good book. Moreover it is printed clearly. July 9, - Published on Amazon. There is a lot of good and a lot of bad in this book.
Some sections are organized beautifully, others are an absolute trainwreck. The first chapter is pretty horrible.
The book has been checked out of my university library, and I checked Amazon, and it says that the book is out of print. When one wants to work over a base ring R commutative , there is the group scheme concept: that is, a group object in the category of schemes over R. Orthogonal group schemes with simple degeneration Over a base scheme, I will discuss a class of quadratic forms that have the simplest type of nontrivial degeneration along a divisor. Algebraic sets and algebraic groups 2. Algebraic representations and Borel subgroups 4. The course will survey some of this research.
It's fine if you already know algebraic geometry, but if you don't, you're better off learning from Milne or somewhere else. The biggest problem with the book is how unfriendly it is. Springer leaves way too much to the reader to figure out on their own for the sake of brevity. If he claims something, and it isn't immediately clear to you, there are three possibilities: 1 it is actually see to see 2 you will have to flip through the previous couple of chapters scouring for a result that can help you, which Springer could have just mentioned or 3 there is actually a lot of work to be done in establishing his claim.
Seriously, when he says something without an explanation, and you endeavor to explain it yourself, you never know whether you'll have two sentences or half a page. It's like he has not really thought about the proofs of the theorems he is writing.
It's like if Serge Lang had an evil twin brother. Just as frustrating, is how averse Springer is to making the chapters of the book relatively independent.
I get it, the material of linear algebraic groups is hard and everything is connected. But it's one thing if you're reading a theorem from Chapter 5 and Springer is citing Lemma 5 from Chapter 2. The following is a tentative list of topics to be covered: Understanding algebraic groups as representable group functors.
Understanding the connection between algebraic groups and commutative Hopf algebras. Linearity of affine algebraic groups. Proving a version of the first isomorphism theorem for affine algebraic groups. Defining the Lie algebra of an algebraic group and the adjoint representation. Studying structure of a torus and a solvable group. Lie-Kolchin Theorem.
Studying Borel subgroups. Along the way, we have to cover some of the needed background from Algebraic Geometry: Algebraic sets and their connection with k-algebras of finite type. Nullstellensatz theorem. Variety: sheaf of regular functions; ringed space; separation axiom.
Chevalley Theorem: constructible sets are mapped to constructible sets. Tanget spaces of varieties.
Module of differentials. Simple points: generically simple; separability criterion. A version of Zariski's main theorem. Complete varieties.
I will be using the following books and lecturenotes: Springer, Linear algebraic groups 2nd edition , Birkhauser. Humphreys, Linear algebraic groups, Springer-Verlag. Borel, Linear algebraic groups 2nd edition , Springer-Verlag. Waterhouse, Introduction to affine group schemes, Springer-Verlag.
First we started by a naive approach towards linear algebraic groups; then we viewed it as a representable functor from k-algebras to groups; then we found out the necessary and sufficient conditions for an algebra to represent a group functor: commutative Hopf algebra. We defined the regular representation of a representable group functor ; proved that any representable group functor is linear.
Algebraic groups from algebraic sets point of view; connected component of the identity; closed finite-index subgroups; Zariski closure of an abstract subgroup; kernel from two points of view: functorial and k-points; Chevalley's theorem : image of a morphism contains a non-empty open subset of its closure; image of an algebraic group is an algebraic group;. Background on AG : Sheaf of regular functions on an algebraic sets; the stalk of the sheaf of regular functions at a point; the isomorphism between the coordinate ring of an algerbaic set and its ring of regular functions; defining prevariety ; separation axiom; quasi-affine varieties; a criterion for being a variety; projective and quasi-projective varities;.
Actions of affine algebraic groups; any orbit is quasi-affine ; existence of closed orbits; the induced action on the ring of regular functions; locally finiteness of the ring of regular functions as a G-mod; linearity of affine algebraic groups;.