By Christian Peskine
Peskine does not provide loads of factors (he manages to hide on 30 pages what frequently takes up part a e-book) and the routines are difficult, however the e-book is however good written, which makes it beautiful effortless to learn and comprehend. steered for everybody prepared 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
To be the ideal generated by the ( m- r)-minors of M and show that the ideals ZT do not depend on the presentation f but only on N . The ideal ZT is called the rth Fitting ideal of N and often denoted by FT ( N ) . 4. Let A be a Noetherian ring and N a finitely generated A-module. (N’). 5. Let A be a Noetherian domain such that each non-zero prime ideal is maximal. Show that for every finitely generated A-module M , the torsion submodule T ( M ) of M has finite length. 7. Let A be a Noetherian ring and a E JR(A) an element such that aA is a non-minimal prime ideal.
Dualizing module on an artinian ring 6. 24. = 0, hence T = (0) and we are done. (v) + (i). To begin with, note that if the evaluation homomorphism eD,M = 0, then f ( x ) = 0 for all f E HomA(M,D) and all x E M . This shows HomA(M, D ) = (0). Let M be a maximal ideal. Since HomA(A/M, D ) f 0, then eD,A/M : A / M HomA(HomA(A/M, D ) ,D ) (ii) + (iii). Using (*) twice, we find + is different from zero. Note next that (iii) + (iv). By (*), lA(HomA(M,D ) ) 5 ~ A ( Mfor ) all finitely generated A-modules M.
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. Now if g E HomA(M, HOmA(N, P ) ) , note that M xN +. 17 M @A A/Z 2 M/ZM. 18 U (i) An A-module P is flat if for each exact sequence of A-modules 111' + M + M", the complex M' @ A P -+ M @ A P -+ M" @ A P is an exact Sequence.
An Algebraic Introduction to Complex Projective Geometry: Commutative Algebra by Christian Peskine