Read e-book online A Course in Commutative Algebra (Graduate Texts in PDF

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By Gregor Kemper

ISBN-10: 3642035442

ISBN-13: 9783642035449

This textbook deals a radical, smooth advent into commutative algebra. it really is intented more often than not to function a consultant for a process one or semesters, or for self-study. The conscientiously chosen material concentrates at the techniques and effects on the heart of the sphere. The booklet keeps a continuing view at the typical geometric context, permitting the reader to realize a deeper knowing of the fabric. even though it emphasizes idea, 3 chapters are dedicated to computational elements. Many illustrative examples and routines increase the textual content.

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

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

Example text

Zn , then dim(X) = max {dim(Z1 ), . . , dim(Zn ), −1} . 11(b). 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.

By way of contradiction, assume that there exist a1 , . . , an ∈ S such that {a1 + P1 , . . , an + P1 } ⊆ A/P1 is algebraically independent of size n. Then also {a1 , . . , an } ⊆ S is algebraically independent. By the definition of n, all a ∈ S are algebraic over L := Quot (K[a1 , . . , an ]), so Quot(A) is algebraic over L, too. There exists a nonzero element a ∈ P1 . We have a nonzero k i polynomial G = i=0 gi x ∈ L[x] with G(a) = 0. Since a = 0, we may assume g0 = 0. Furthermore, we may assume gi ∈ K[a1 , .

In the special case that R ⊆ S and ϕ is the inclusion, we have ϕ∗ (Q) = R ∩ Q. Notice that the correspondence between ring homomorphisms and morphisms of spectra is not bijective. If ψ: S → T is a further ring homomorphism, then (ψ ◦ ϕ)∗ = ϕ∗ ◦ ψ ∗ . In general, ϕ∗ does not restrict to a map Specmax (S) → Specmax (R); but if ϕ is a homomorphism of affine K-algebras, it does. 1 corresponding to ϕ. 1 (Dominant and injective morphisms). Let X and Y be affine varieties over a field K, and let f : X → Y be a morphism with induced homomorphism ϕ: K[Y ] → K[X].

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

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