Download A Course in Commutative Algebra by Gregor Kemper PDF

By Gregor Kemper

This textbook bargains a radical, sleek advent into commutative algebra. it's intented frequently to function a advisor for a process one or semesters, or for self-study. The conscientiously chosen subject material concentrates at the options and effects on the middle of the sphere. The publication continues a continuing view at the typical geometric context, allowing the reader to achieve a deeper knowing of the cloth. even though it emphasizes conception, 3 chapters are dedicated to computational facets. Many illustrative examples and workouts improve the text.

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Example text

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. The name basis theorem comes from the fact that generating sets of ideals are sometimes called bases.

6, and since ∅ = VSpec(R) ({1}) and Spec(R) = VSpec(R) (∅), this indeed defines a topology. A subset of Spec(R) is equipped with the subspace topology induced from the Zariski topology on Spec(R). The following proposition contains all the important general facts about the maps VSpec(R) and IR defined above. 19. 6 (Properties of VSpec(R) and IR ). Let R be a ring. (a) Let S, T ⊆ R be subsets. Then VSpec(R) (S) ∪ VSpec(R) (T ) = VSpec(R) (S)R ∩ (T )R . 2 Spectra 37 (b) Let M be a nonempty set of subsets of R.

Xn ] with Rd the space of all homogeneous polynomials of degree d (including the zero polynomial). Let R be graded and set Rd , I= d∈N>0 which obviously is an ideal. Sometimes I is called the irrelevant ideal. Prove the equivalence of the following statements. (a) R is Noetherian. (b) R0 is Noetherian and I is finitely generated. (c) R0 is Noetherian and R is finitely generated as an R0 -algebra. 12, a finitely generated algebra over a Noetherian rings is Noetherian. However, Noetherian algebras are not always finitely generated.

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