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Get A Course in Commutative Algebra PDF

By Gregor Kemper

ISBN-10: 3642035442

ISBN-13: 9783642035449

ISBN-10: 3642035450

ISBN-13: 9783642035456

This textbook deals a radical, glossy creation into commutative algebra. it really is intented normally to function a consultant 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 keeps a relentless view at the usual geometric context, allowing the reader to achieve a deeper figuring out of the fabric. even though it emphasizes conception, 3 chapters are dedicated to computational features. Many illustrative examples and workouts improve the text.

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1 True Geometry: Affine Varieties In this section, K is assumed to be an algebraically closed field. We have the following correspondences between algebraic and geometric objects: (1) Hilbert’s Nullstellensatz gives rise to a bijective correspondence affine varieties in K n ←→ radical ideals in K[x1 , . . , xn ]. 1) In fact, assigning to any set X ⊆ K n of points the vanishing ideal I(X) yields a map from the power set of K n to the set of radical ideals in K[x1 , . . , xn ], and assigning to a set S ⊆ K[x1 , .

To prove the reverse inclusion, let P ∈ V(I ∩ J). Assume P ∈ / V(I), so there exists G. 1007/978-3-642-03545-6 4, c Springer-Verlag Berlin Heidelberg 2011 33 34 3 The Zariski Topology f ∈ I with f (P ) = 0. We need to show that P ∈ V(J), so let g ∈ J. Then f g ∈ I ∩ J, so f (P )g(P ) = 0. But this implies g(P ) = 0. Part (b) is clear. 1 tells us that finite unions and arbitrary intersections of affine varieties in K n are again affine varieties. Since K n and ∅ are also affine varieties, this suggests that we can define a topology using the affine varieties as closed sets.

It may also be interesting to note that a ring R is a Jacobson ring if and only if for every closed subset Y ⊆ Spec(R) we have that Specmax (R) ∩ Y is dense in Y . ) In fact, this is nothing but a translation of the Jacobson property. To every ring R we have assigned a topological space Spec(R). We will make this assignment into a contravariant functor as follows. Let R and S be rings and let ϕ: R → S be a homomorphism. For every P ∈ Spec(S), the preimage ϕ−1 (P ) is obviously a prime ideal of R, so we obtain a map ϕ∗ : Spec(S) → Spec(R), P → ϕ−1 (P ).

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A Course in Commutative Algebra by Gregor Kemper

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