Download An Algebraic Introduction to Complex Projective Geometry: by Christian Peskine PDF

By Christian Peskine

Peskine does not provide loads of motives (he manages to hide on 30 pages what frequently takes up part a publication) and the routines are difficult, however the e-book is however good written, which makes it lovely effortless to learn and comprehend. instructed for everybody prepared to paintings their approach via his one-line proofs ("Obvious.")!

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Extra info for An Algebraic Introduction to Complex Projective Geometry: Commutative Algebra

Example text

0 56 5. 35 (Krull’s theorem) Let A be a Noetherian ring and M afinitely generated A-module. 5. Exercises 57 6. Let R be a principal ideal ring and M a finitely generated R-module. If M’ is a submodule of M , show ~ R ( T ( M5 ) ) ~R(T(M’))~R(T(M/M’)). + Show that if equality holds, then M / T ( M ) N M’/T(M’) @ ( M / M ’ ) / T ( M / M ’ ) Proof Put E = n,Z”M. We have ZnM n E = E for all n 2 0. Hence by the Artin-Rees lemma ZE = E . Applying Nakayama’s lemma, we have proved E = (0). 5 Exercises 1.

S,, we have s f ( e , / l ) = z,/l, with z, E N. If we define g E HomA(nA,N) by g(e,) = z,, we have f = g / s E Homs-1A(nS-'A,S-'N). With this in mind, 0 an easy chase round the diagram proves (iii) in our proposition. 28 Let f : M --+ N be an A-modules homomorphism. The following conditions are equivalent: (i) f is injective (resp. surjective, bijective); (ii) f o r all prime ideals P E Supp(M) (resp. P E Supp(N), P E Supp(M) U SUPP(NI1, fP MP NP Support of a module + We recall that to a ring A, we have associated a topological space Spec(A).

We define thus a "covariant functor from the category of A-modules to the category of S-lA-modules". We also prove that this functor is "exact". (ii) If N' is another submodule of M , then S-'(N If q5 : M -+ and S-'(N n N') = S-lN n S-lN'. 14 (i) + N ' ) = S-lN + S-lN' F N is a homomorphism of A-modules, the map S-lq5 : S-'M + S-lN, (iv) S-'~~(Z/S) = $(x)/s If N is a submodule S-'N n M = = S-'(F n Ad). of M , then {x E M , there exists s E S such that sx E N } . is well defined and is a homomorphism of S-'A-modules.

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