| Algebraic Geometry (Graduate Texts in Mathematics) | ||||||||||||||
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no more than pure algebra, schemes and cohomology • • • • • (bewertet mit 5 von 5 Punkten) Alle meine Rezensionen ansehen Rezension bezieht sich auf: Algebraic Geometry (Graduate Texts in Mathematics) (Gebundene Ausgabe) After a sketchy Chapter I about classical Varieties concentrating on equivalences of categories, Chapter II and III give an *outstanding* account on Schemes and Cohomology. So good in fact, that it definitely will be hard for you to find something competitive. The style is purely algebraic and very clearly structured with lots of examples and instructive exercises. You don't need no Differential Geometry or Complex Analysis, just some Commutative Algebra and elementary Topology. The proofs are short, elegant and good to remember. Maybe you can compare the writing style with Lang's Algebra book (which seems to be good at hand, when reading Hartshorne) - but much better! The following Chapters IV and V contain an impressive treatment of the classical problems and results concerning Curves and Surfaces using Cohomology. E.g. Riemann-Roch (using Serre duality), Hurwitz Theorem, Elliptic Curves and Classifications of Curves and Surfaces. Appendix A, B and C are briefly about Intersection Theory (Chern Classes, Riemann-Roch), an overview of links to Complex Analysis and an introductory section on the Weil Conjectures and l-adic Cohomology. After all, Hartshorne's Book is all about the *basic* techniques of modern Algebraic Geometry and the presented Theorems are rather illustrations of it's fruitfullness. Topics like l-adic Cohomology, group varieties and arithmetic schemes are only briefly mentioned or celebrated in exercises. As a Reference and a preparation for current applications and more advanced books in abstract algebraic geometry but, this book is the best choice, i think. Also consider, that this book is written with a view towards massive applications in current research concerning today's unsolved problems in Arithmetic or Algebraic Geometry and Number Theory. So if one only needs a reference on classical results (and techniques) of Algebraic Geometry, e.g. for applications in Computer Algebra or Analysis, this may be too heavy. Hartshorne's book is well accompanied by Lang's "Algebra" and Eisenbud's "The Geometry of Schemes". Eine Rezension von Ein Kunde vom 11. Januar 2004 | ||||||||||||||
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