By Gregor Kemper
This textbook bargains an intensive, glossy creation into commutative algebra. it's intented more often than not 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 middle of the sphere. The booklet keeps a continuing view at the average geometric context, allowing the reader to realize a deeper realizing of the cloth. even though it emphasizes thought, 3 chapters are dedicated to computational points. Many illustrative examples and routines improve the text.
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Extra info for A Course in Commutative Algebra
A closed subset of Spec(R) is irreducible if and only if the corresponding ideal in R is a prime ideal. So chains of closed, irreducible subsets of Spec(R) correspond to chains of prime ideals in R, but with all inclusions reversed. 3) to the points. 2). (2) A ring homomorphism ϕ: R → S induces a morphism ϕ∗ : Spec(S) → Spec(R), Q → ϕ−1 (Q) of spectra. 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.
Since there are inﬁnite strictly descending chains of subsets of X, there are also inﬁnite strictly ascending chains of ideals in R. So R is not Noetherian. (4) The rings Z and K[x] considered above are not Artinian. (5) Every ﬁeld and every ﬁnite ring or module is Artinian. (6) The ring K X , as deﬁned in (3), is Artinian if and only if X is a ﬁnite set. (7) Let R := K[x] be a polynomial ring over a ﬁeld. Then S := R/(x2 ) is Artinian, and S is also Artinian as an R-module. 2(3) is a rather pathological example of a nonNoetherian ring.
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 ). 38 3 The Zariski Topology We will often say that ϕ∗ is induced from ϕ. For I ⊆ R a subset, we have (ϕ∗ )−1 VSpec(R) (I) = VSpec(S) (ϕ(I)), so ϕ∗ is continuous. Maps between spectra of rings that are induced from ring homomorphisms are called morphisms. Going from ϕ to ϕ∗ is compatible with, and a generalization of, the process of obtaining a morphism X → Y of aﬃne varieties from a homomorphism K[Y ] → K[X] described on page 35.
A Course in Commutative Algebra by Gregor Kemper