By Robert J. Walker

This advent to algebraic geometry examines how the newer summary options relate to conventional analytical and geometrical difficulties. The presentation is saved as effortless as attainable, because the textual content can be utilized both for a starting direction or for self-study.

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E. am = m for all m ∈ M. 28. If M is a finitely generated R-module, any surjective homomorphism ϕ : M → M is an isomorphism. Proof. 8, consider M as an R[x]-module by setting x · m := ϕ(m) for all m ∈ M. 27 with I = (x) to obtain a polynomial f ∈ (x), i. e. a polynomial f = an xn + an−1 xn−1 + · · · + a1 x without constant coefficient, such that f · m = an ϕ n (m) + an−1 ϕ n−1 (m) + · · · + a1 ϕ(m) = m for all m ∈ M. But this means that ϕ(m) = 0 implies m = 0, and so ϕ is injective. 29. For a prime number p ∈ N consider the subring R = { ba : a, b ∈ Z, p | b} of Q, and let M = Q as an R-module.

So essentially we have two choices if we want to continue to carry over our linear algebra results on finitely generated vector spaces to finitely generated modules: 3. Modules 31 • restrict to R-modules that are of the form Rn for some n ∈ N; or • go on with general finitely generated modules, taking care of the fact that generating systems cannot be chosen to be independent, and thus that the coordinates with respect to such systems are no longer unique. In the rest of this chapter, we will follow both strategies to some extent, and see what they lead to.

06 There is another much more subtle way to construct new exact sequences from old ones. This time, instead of gluing two sequences such that the ending module of the first sequence is the starting module of the second, we consider two short exact sequences such that one of them can be mapped to the other by a sequence of homomorphisms. 7 (Snake Lemma). Let ϕ M α 0 M ψ N ϕ 0 P γ β ψ N P be a commutative diagram of R-modules with exact rows. Then there is a long exact sequence ker α −→ ker β −→ ker γ −→ M / im α −→ N / im β −→ P / im γ.