A Course in Commutative Algebra (Graduate Texts in by Gregor Kemper

By Gregor Kemper

This textbook bargains an intensive, sleek advent into commutative algebra. it really is intented frequently to function a advisor for a process one or semesters, or for self-study. The rigorously chosen subject material concentrates at the options and effects on the heart of the sector. The booklet continues a relentless view at the common geometric context, allowing the reader to achieve a deeper knowing of the cloth. even though it emphasizes thought, 3 chapters are dedicated to computational points. Many illustrative examples and routines enhance the textual content.

Show description

Read or Download A Course in Commutative Algebra (Graduate Texts in Mathematics, Volume 256) PDF

Best mathematics books

Mathematical Magic Show

This is often the 8th number of Martin Gardner's Mathematical video games columns which were showing per month in clinical American because December 1956.

Amsco's Algebra Two and Trigonometry

Algebra 2 trigonometry textbook will train scholars every little thing there's to grasp made effortless!

Additional resources for A Course in Commutative Algebra (Graduate Texts in Mathematics, Volume 256)

Sample text

Assume dim(X) = 0. Since X is a subset of a Noetherian space, X is Noetherian, too. 11(a), X is a finite union of closed, irreducible subsets Zi . Choose xi ∈ Zi . Then {xi } ⊆ Zi is a chain of closed, irreducible subsets, so Zi = {xi }. It follows that X is finite. Conversely, if X is finite, then the irreducible subsets are precisely the subsets of size 1, so dim(X) = 0. 11: equidimensional algebras whose dimension is only 1 less than the number of generators. These correspond to equidimensional affine varieties in K n of dimension n − 1.

Then dim(A) = trdeg(A). We will prove the theorem together with the following proposition, which often facilitates the computation of the transcendence degree since the set S can be taken to be finite. 10 (Calculating the transcendence degree). Let A be an affine algebra, and let S ⊆ A be a generating set. Then trdeg(A) = sup {|T | | T ⊆ S is finite and algebraically independent} . 10. 6 we have dim(A) ≤ sup {|T | | T ⊆ S is finite and algebraically independent} , and this supremum is clearly less than or equal to trdeg(A).

An ∈ A are algebraically independent over L. By induction, dim(A ) ≥ n − 1, so there exists a chain P0 P1 ··· Pn−1 in Spec(A ). Set Pi := A ∩ Pi ∈ Spec(A). Then Pi−1 ⊆ Pi for i = 1, . . , n−1. These inclusions are strict since clearly L · Pi = Pi for all i. Moreover, L ∩ Pn−1 = {0}, since otherwise Pn−1 would contain an invertible element from L, leading to Pn−1 = A . It follows that a1 + Pn−1 ∈ A/Pn−1 is not algebraic over K. 1(b), A/Pn−1 is not a field, so Pn−1 is not a maximal ideal. Let Pn ⊂ A be a maximal ideal containing Pn−1 .

Download PDF sample

Rated 4.19 of 5 – based on 28 votes