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Algebraic Geometry

By James R. Milgram

ISBN-10: 0821814338

ISBN-13: 9780821814338

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Here are some instances of exact sequences: (i) 0 - A L B is exact if f : A --3 B is monic; (ii) A B -+ 0 is exact if f : A - B is onto; (iii) 0 - A L B - 0 is exact if f is an isomorphism; (iv) 0 -* K L M - N ---+ 0 is exact if f is monic, g is onto, and (by Noether I) g induces an isomorphism Ml f (K) ^_' N. Since the sequence (iv) is the shortest exact sequence which starts and ends with 0 and says something nontrivial, it is called a short exact sequence, and is fundamental in module theory.

Finite When A< R, this is a submodule of M, as seen by direct verification. When A = R, the submodule RS is called the submodule generated by S. We shall review this important concept as we come across the special cases that we need. One major approach to module theory is to study a module M in terms of its submodules. The following facts are basic in this perspective. 3. Suppose M is a module and M1, M2 < M. Then 1. MI+M2

9. Suppose C is a commutative ring and al, , an E C. The Vandermonde matrix of the ai is the matrix A whose i, j entry is ai-1. We claim the determinant d of A is precisely fi>j (ai - aj). Indeed, it is . . enough to consider the generic situation, where all the ai are indeterminates over Z. Specializing some ai " aj (i j) would send d to 0, so ai - aj divides d for each i j; hence [i>j (ai - aj) divides d, and comparison of degrees shows d = a fi>j (ai - aj) for some a E Q. But the monomial a1a2a3 ...

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