By Robert J. Walker

This advent to algebraic geometry examines how the more moderen summary techniques relate to conventional analytical and geometrical difficulties. The presentation is stored as straight forward as attainable, because the textual content can be utilized both for a starting direction or for self-study.

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**Example text**

Unique factorization 47 Proof. Note that if p is a prime divisor, then Ap − {0} = {f ∈ K ∗ | νp (f ) ≥ 0}. 2). 17. The cokernel of the divisor map div : K ∗ → Div(A) is called the class group of A and will be denoted Cl(A). The canonical projection will be denoted c : Div(A) → Cl(A). Let U (A) denote the group of multiplicatively invertible elements in A. 16 that we have a canonical exact sequence 1 −−→ U (A) −−→ K ∗ −−→ Div(A) −−→ Cl(A) −−→ 0. 16. 18. Let p be a prime divisor. Then p is a principal ideal if and only if c(p) = 0.

Then Ass(N ) ⊆ Ass(M ) ⊆ Ass(N ) ∪ Ass(M/N ). Proof. The first inclusion is trivial. As for the second, suppose given p ∈ Ass(M ) such that p ∈ / Ass(N ). Choose a submodule P of M such that P A/p. We have Ass(P ∩ N ) ⊆ Ass(P ) ∩ Ass(N ) from which it follows that P ∩ N = 0, and therefore M/N contains a submodule isomorphic to A/p. 10. Let M = 0 be a finitely generated A-module and p ∈ Ass(M ). Then there exists a submodule N of M with Ass(N ) = {p}, Ass(M/N ) = Ass(M ) − {p}. 2. Associated prime ideals 21 Proof.

Ai−1 )M , i = 1, . . , r. We have dim((M/(a1 , . . , ar )M )p ) = 0, thus p ∈ Ass(M/(a1 , . . 29 dim(A/p) = dim(M ) − r. Since dim((M/(a1 , . . , ar )M )p ) = 0 we get depth((M/(a1 , . . , ar )M )p ) = 0 and whence depth(Mp ) = dim(Mp ). 6. 31. Suppose there exists a finitely generated Cohen–Macaulay module M with Supp(M ) = Spec(A). Then for any two prime ideals p ⊆ q we have dim(A/p) = dim(Aq /pq ) + dim(A/q). Proof. 30 that we can find a Cohen–Macaulay module N with p ∈ Ass(N ). 30 yields dim(A/q) + dimAp (Np ) = dim(N ).