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Chapter 8 Introducing Algebraic Geometry (Commutative Algebra 2)

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Book Id: WPLBN0000673857
Format Type: PDF eBook
File Size: 448.56 KB
Reproduction Date: 2005

Title: Chapter 8 Introducing Algebraic Geometry (Commutative Algebra 2)  
Language: English
Subject: Science., Mathematics, Logic
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Chapter 8 Introducing Algebraic Geometry (Commutative Algebra 2). (n.d.). Chapter 8 Introducing Algebraic Geometry (Commutative Algebra 2). Retrieved from

Mathematics document containing theorems and formulas.

Excerpt: We will develop enough geometry to allow an appreciation of the Hilbert Nullstellensatz, and look at some techniques of commutative algebra that have geometric significance. As in Chapter 7, unless otherwise specified, all rings will be assumed commutative. // Varieties // Definitions and Comments We will be working in k[X1, . . . , Xn], the ring of polynomials in n variables over the field k. (Any application of the Nullstellensatz requires that k be algebraically closed, but we will not make this assumption until it becomes necessary.) The set An = An(k) of all with components in k is called affine n-space. If S is a set of polynomials in k[X1, . . . , Xn], then the zero-set of S, that is, the set V = V (S) of all x E An such that f(x) = 0 for every f E S, is called a variety. (The term ?affine variety? is more precise, but we will use the short form because we will not be discussing projective varieties.) Thus a variety is the solution set of simultaneous polynomial equations.


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