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Download e-book for iPad: A Course in Commutative Algebra (Graduate Texts in by Gregor Kemper

By Gregor Kemper

ISBN-10: 3642035442

ISBN-13: 9783642035449

This textbook deals an intensive, glossy creation into commutative algebra. it's intented typically to function a advisor for a process one or semesters, or for self-study. The conscientiously chosen subject material concentrates at the suggestions and effects on the heart of the sector. The booklet continues a relentless view at the normal geometric context, permitting the reader to achieve a deeper knowing of the fabric. even though it emphasizes idea, 3 chapters are dedicated to computational points. Many illustrative examples and workouts increase the textual content.

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

Example text

Then f g ∈ (S)R ∩ (T )R , so f g ∈ P . Since P is a prime ideal, g ∈ P follows, so P ∈ VSpec(R) (T ). is clear. This follows since prime ideals are always radical ideals, and intersections of radical ideals are again radical ideals. 12. In the light of (c) and (d), we need to show only that VSpec(R) (IR (X)) = X for X ⊆ Spec(R) a closed subset. We have X = VSpec(R) (S) with S ⊆ R, so S ⊆ IR (X). Since the map VSpec(R) is inclusion-reversing, we obtain VSpec(R) (IR (X)) ⊆ VSpec(R) (S) = X ⊆ VSpec(R) (IR (X)) .

We write length(M) := sup {length(C) | C is a chain in M} ∈ N0 ∪ {−1, ∞} (the length −1 occurs if M = ∅). , dim(V ) = length {U ⊆ V | U subspace} . With this in mind, the following definition does not appear too far-fetched. 1 (Krull dimension). (a) Let X be a topological space. Set M to be the set of all closed, irreducible subsets of X. Then the dimension of X (also called the Krull dimension) is defined as dim(X) := length(M). (b) Let R be a ring. Then the dimension of R (also called the Krull dimension) is defined as G.

Again we conclude that f ∈ I . So indeed I = I is a finitely generated ideal. 4. 4, we obtain the following corollary. 12 (Finitely generated algebras). Every finitely generated algebra over a Noetherian ring is Noetherian. In particular, every affine algebra is Noetherian. A special case is the celebrated basis theorem of Hilbert. 13 (Hilbert’s basis theorem). Let K be a field. Then the polynomial ring K[x1 , . . , xn ] is Noetherian. In particular, every ideal in K[x1 , . . , xn ] is finitely generated.

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A Course in Commutative Algebra (Graduate Texts in Mathematics, Volume 256) by Gregor Kemper


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