Linear Algebraic Groups (Graduate Texts in Mathematics (21))
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Linear Algebraic Groups
"Exceptionally well-written and ideally suited either for independent reading or as a graduate level text for an introduction to everything about linear algebraic groups."―MATHEMATICAL REVIEWS
- Publisher : Springer (May 13, 1975)
- Language : English
- Hardcover : 264 pages
- ISBN-10 : 0387901086
- ISBN-13 : 978-0387901084
- Item Weight : 2.76 pounds
- Dimensions : 6.14 x 0.69 x 9.21 inches
- Best Sellers Rank: #1,167,440 in Books (See Top 100 in Books)
- Customer Reviews:
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This text is relatively self-contained with fairly standard treatment of the subject of linear algebraic groups as varieties over an algebraic closed field (not necessarily characteristic 0). Despite being rooted in algebraic geometry, the subject has a fair mix of non-algebraic geometric arguments. Nonetheless, irreducibility, constructivility, finiteness and completeness are employed often. The scope is about comparable with Borel's, and is a proper subset of TA Springer's.
I aim to talk about the author's exposition. The structure of the book is very rigid. Chapters are organized into sections, and each into subsections. The content is very densely packed; significant points in the exposition are thereby difficult to parse out. At times, concepts were not realized to be significant until they were used often later in the text. Trivial points were belabored, notations were laid down carelessly, arguments were described and not detailed; understandable, though unpleasant.
More than anything else that made this book difficult to read is that the proofs had numerous small to medium gaps. To fill those gaps, readers unfamiliar with the subject are left guessing blindly. Facts were stated as "obvious" or with very little guidance without explanation or reference to earlier results. These occur more or less frequently depending on how familiar one is with the subject. An early student of the subject may find that these gaps make progress unbearably slow. Without perspective, it is easy to miss the forest for the trees.
To summarize the gripe --- borrowing another tree analogy --- reading Humphreys's LAG is like hiking with all significant trail markers removed. Groping is a necessity; being lost a few times, a guarantee.
One can interpret all this as a boon (perhaps with a felicitous application of Stockholm syndrome). The exercises are often simple, but to really read the text carefully, one must necessarily wield deftly a broad range of facts, and be able to conjure them from the slightest of clues. I believe this text is really suited for a month long exam for the student of linear algebraic groups; anyone who can read this book in one comfortable sitting must either know the subject warmly, or is truly equipped with the gift of learning.
In any case, I recommend this not for even a determined self-student. When reading this text, it's best to find a friendly expert, or try Borel's, on which a large portion of this text is based. Borel is lengthier, scheme theoretic, and more classical. It will take about the same amount of time to read, but you will walk away from Borel believing more in yourself. For a mathematician, that may make the difference between a half cup of tea and a pitcher full of coffee.