Springer - Linear Algebraic Groups



Birkhäuser Progress in Mathematics 9 (First Edition)

Modern Birkhäuser Classics (Second Edition)

"The aim of [the first edition] was to present the theory of linear algebraic groups over an algebraically closed field, including the basic results on reductive groups. A distinguishing feature was a self-contained treatment of the prerequisites from algebraic geometry and commutative algebra.  The present book has a wider scope.  Its aim is to treat the theory of linear algebraic groups over arbitrary fields, which are not necessarily algebraically closed."

= Chapter 1 : Some Algebraic Geometry =

Section 1.9 : Some Results on Morphisms

 * Lemma 1.9.1: http://math.stackexchange.com/questions/71623/why-does-surjectivity-of-the-induced-map-show-that-a-morphism-of-affine-varietie


 * Lemma 1.9.3: First, note that the conditions on A and B come from the paragraph preceding the statement of the lemma: B is a reduced ring which is of finite type over its subring A. Now, as for the relatively baffling statement of the lemma: We want to perform polynomial long division in the polynomial ring A[T], but this is only possible in very restrictive settings, namely Euclidean domains, which A[T] is not. Consequently, we instead map into K[T], which is a Euclidean domain, and perform the long division there. The condition that $$\phi(\mathcal{J}(I)) \neq \{0\}$$ is precisely what is required to make this work.


 * Lemma 1.9.4: This statement is also somewhat baffling. It's much easier to see what's going on if we translate into scheme-theoretic language: let i denote the inclusion of A into B, and consider the induced map $$i* : {\rm Spec} B \rightarrow {\rm Spec} A$$.

= Chapter 2 : Linear Algebraic Groups, First Properties =

= Chapter 3 : Commutative Algebraic Groups =

= Chapter 4 : Derivations, Differentials, Lie Algebra =

= Chapter 5 =

= Chapter 6 =

= Chapter 7 =

= Chapter 8 =

= Chapter 9 =

= Chapter 10 =

= Chapter 11 =

= Chapter 12 =

= Chapter 13 =

= Chapter 14 =

= Chapter 15 =

= Chapter 16 =

= Chapter 17 =