Commutative Algebra

1. Front Matter

   Pages i-x

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2. Valuation Theory

   • Oscar Zariski, Pierre Samuel

   Pages 1-128

3. Polynomial and Power Series Rings

   • Oscar Zariski, Pierre Samuel

   Pages 129-247

4. Local Algebra

   • Oscar Zariski, Pierre Samuel

   Pages 248-320

5. Back Matter

   Pages 321-416

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This second volume of our treatise on commutative algebra deals largely with three basic topics, which go beyond the more or less classical material of volume I and are on the whole of a more advanced nature and a more recent vintage. These topics are: (a) valuation theory; (b) theory of polynomial and power series rings (including generalizations to graded rings and modules); (c) local algebra. Because most of these topics have either their source or their best motivation in algebraic geom­ etry, the algebro-geometric connections and applications of the purely algebraic material are constantly stressed and abundantly scattered through­ out the exposition. Thus, this volume can be used in part as an introduc­ tion to some basic concepts and the arithmetic foundations of algebraic geometry. The reader who is not immediately concerned with geometric applications may omit the algebro-geometric material in a first reading (see" Instructions to the reader," page vii), but it is only fair to say that many a reader will find it more instructive to find out immediately what is the geometric motivation behind the purely algebraic material of this volume. The first 8 sections of Chapter VI (including § 5bis) deal directly with properties of places, rather than with those of the valuation associated with a place. These, therefore, are properties of valuations in which the value group of the valuation is not involved.

• Grad
• algebra
• algebraic geometry
• commutative algebra
• polynomial

• Department of Mathematics, Harvard University, Cambridge, USA

  Oscar Zariski

• Mathématique / Bâtiment 425, Université de Paris-Sud, Orsay, France

  Pierre Samuel