By Christian Peskine
Peskine does not supply loads of causes (he manages to hide on 30 pages what often takes up part a publication) and the routines are difficult, however the publication is however good written, which makes it beautiful effortless to learn and comprehend. urged for everybody keen to paintings their means via his one-line proofs ("Obvious.")!
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Extra info for An Algebraic Introduction to Complex Projective Geometry: Commutative Algebra
10 (i) If g‘ and g” are injective, so i s g. (ii) If g is injective and g‘ surjective, then g“ i s injective. (iii) If g“is injective and g surjective, then g‘ i s surjective. (iv) If g‘ and g“ are surjective, so i s g. 11 If Ml and Mz are submodules of M , there is a natural exact sequence -+ A/(Z+ J ) --f 0. This proves that Z and J are comaximal if and only if the natural map A/(Z n J ) 4 A/Z @ A / J is an isomorphism. 3. Tensor products and homomorphism modules is an isomorphism. 62. 3 Tensor products and homomorphism modules Although we will not use it immediately, it seems a good time for a first contact with the tensor product, M @ A N , of two A-modules M and N .
From the category of A-modules to itself is right exact. , N) is left exact. These isomorphisms are clear enough. The following is a bit more intricate. ) is left exact. 15 The natural homomorphism HomA ( M @ A N, P ) -+ Consider an ideal Z of A, the exact sequence 0 -+ Z -+ A +. A/Z + 0 and an A-module M . By applying the functor M @ A to the exact sequence, one gets the following easy but important consequence HOmA (M , HomA ( N, p )) is an isomorphism. ) E HomA(N, P ) . Our map is defined and obviously injective.
If A I M 1: HomA(A/M, D ) for all maximal ideals M of A, then for all finitely generated A-modules hl, one has lA(HomA(M, D ) ) 5 ~ A ( M ) . 24 Let A be an artinian ring and D a finitely generated A-module. The following conditions are equivalent: If lA(M)= 1, there is a maximal ideal M such that M cv A / M . Hence (i) the A-module D is dualizing; (ii) the A-module D is faithful and for all maximal ideals M of A one has A I M 21 HomA(A/M, D); (iii) the A-module D satisfies ~ A ( D=) ~ A ( Aand ) for all maximal ideals M of A one has A I M N HomA(A/M, D); lA(HomA(M,D ) ) = lA(HomA(A/M, D)) = ~ A ( A / M= ) 1 If 1A(M) > 1, Let M’ C M be astrict submodule.
An Algebraic Introduction to Complex Projective Geometry: Commutative Algebra by Christian Peskine