By Ernst Kunz (auth.)

Originally released in 1985, this vintage textbook is an English translation of *Einführung in die kommutative Algebra und algebraische Geometrie*. As a part of the trendy Birkhäuser Classics sequence, the writer is proud to make *Introduction to Commutative Algebra and Algebraic Geometry* to be had to a much wider audience.

Aimed at scholars who've taken a uncomplicated direction in algebra, the target of the textual content is to offer very important effects in regards to the illustration of algebraic types as intersections of the least attainable variety of hypersurfaces and—a heavily comparable problem—with the main inexpensive iteration of beliefs in Noetherian earrings. alongside the best way, one encounters many simple thoughts of commutative algebra and algebraic geometry and proves many proof that can then function a uncomplicated inventory for a deeper research of those subjects.

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Xn]/M is a finite field extension of K. b) For every extension field L of K the ideal M has at most finitely many zeros in Ln. c) If K is algebraically closed, then there are elements 6, ... , en E K such that M = (Xt- 6, ... ,Xn- en)· 18 CHAPTER I. 2. b) If L f K is a field extension, then by a) there are at most finitely many K-homomorphisms K[Xt. ,Xn]/M--+ L, hence M has at most finitely many zeros in Ln. c) If K is algebraically closed, then M has a zero (et. 1. The polynomials not belonging toM then do not have (6, ...

Passing to R/ I, we see that it suffices to prove the second statement. It is clear that npeSpec(R) ll contains an the nilpotent elements of R. If X E npESpec(R) p, considers:= {xn In EN}. 4 there is a prime ideal p with p n S = 0, contradicting the hypothesis xE p. 3. U(I) P = p = Rad(/). 2. 6. Let R be a ring, X its spectrum, J -spectrum, or maximal spectrum. A closed subset A C X is irreducible if and only if J (A) is a prime ideal. Proof. a) Let A be irreducible and I· g E J(A) with I, g E R.

The inverse mapping is given by the formation of the zero set. For any homogeneous ideal I¥:- K[Xo, ... U(I)) = Rad(/). This bijection is inclusion-reversing and the empty variety is assigned the ideal (Xo, ... , Xn)· The irreducible K-varieties correspond bijectively to the homogeneous prime ideals ¥:- K[Xo, ... 10. A system of equations Fi = 0 (i = 1, ... , m) with nonconstant homogeneous polynomials Fi E K[Xo, ... , Xn] has a nontrivial solution in £"+ 1 , where L is an algebraically closed extension field of K, if and only if Rad(Fl, ...