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 Edit
Section 1.9 : Some Results on Morphisms Edit
- 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 $.