Download e-book for kindle: A Course in Commutative Algebra by Gregor Kemper

By Gregor Kemper

ISBN-10: 3642035442

ISBN-13: 9783642035449

ISBN-10: 3642035450

ISBN-13: 9783642035456

This textbook bargains a radical, smooth creation into commutative algebra. it truly is intented generally to function a advisor for a process one or semesters, or for self-study. The rigorously chosen subject material concentrates at the techniques and effects on the middle of the sphere. The booklet keeps a continuing view at the usual geometric context, permitting the reader to realize a deeper realizing of the cloth. even though it emphasizes concept, 3 chapters are dedicated to computational features. Many illustrative examples and routines enhance the text.

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Xn ]. To show that X is irreducible, let X = X1 ∪X2 with Xi closed in X, so Xi = X ∩VK n (Ii ) with Ii ⊆ K[x1 , . . , xn ] ideals. 1(a) for the equality. ,xn ] (X) = I. Since I is a prime ideal, there exists i with Ii ⊆ I, so X ⊆ VK n (I) ⊆ VK n (Ii ). This implies Xi = X. Therefore X is irreducible. (b) The proof of this part is obtained from the proof of part (a) by changing K[x1 , . . ” The following theorem allows us to view irreducible spaces as the “atoms” of a Noetherian space. 11 (Decomposition into irreducibles).

G) All polynomials f ∈ K[x1 , . . , xn ], viewed as functions K n → K, are continuous with respect to the Zariski topology. In fact, the Zariski topology is the coarsest topology such that all polynomials are continuous (assuming that {0} ⊂ K 1 is closed). , the function C → C, x → x (complex conjugation). (h) The Zariski-open subsets of K n are unions of solution sets of polynomial inequalities. 1 Affine Varieties 35 open sets U1 and U2 with Pi ∈ Ui . If K is an infinite field, then K n with the Zariski topology is never Hausdorff.

We will call X equidimensional if all Zi have the same dimension. Likewise, a Noetherian ring R is called equidimensional if Spec(R) is equidimensional. 1 directly for determining the dimension of a variety. At this point we are not even able to determine the dimension of K n (or of the polynomial ring K[x1 , . . , xn ]), although we easily get n as a lower bound. Another disadvantage is that at this point it is far from clear that the Krull dimension of an affine variety coincides with what we intuitively understand by dimension.

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