By Christian Peskine
Peskine does not supply loads of causes (he manages to hide on 30 pages what frequently takes up part a e-book) and the workouts are difficult, however the publication is however good written, which makes it beautiful effortless to learn and comprehend. advised for everybody prepared to paintings their method via his one-line proofs ("Obvious.")!
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Extra resources for An Algebraic Introduction to Complex Projective Geometry: Commutative Algebra
This shows that the natural surjective homomorphism A/Z an isomorphism. 4. Dualizing module on an artinian ring 6. A first contact with homological algebra (v) for all injective homomorphisms N M of finitely generated A-modules, the natural homomorphism HomA(M, D ) + HomA(N, D) is surjective and for all maximal ideals M of A, one has A I M 11’ HomA(A/M, D ) . Pro0f (i) + (ii). Assume D is a dualizing A-module. 21 Let A be an artinian ring. A finitely generated A-module D is dualizing if the natural evaluation application shows that D is faithful.
Cl Proof Note first that if ((0) : M ) $ P , there exists s E (A \ P ) n ((0) : M ) . This shows M = ker[M + Mp], hence Mp = (0), and P 6 Supp(M ) . ,x,) be a finite system of generators of M . We have ((0) : M ) = n,(O : x,). If ((0) : M ) c P , there exists z such that (0 : x,) c P . This shows sx, # 0 for all s 6 P , in other words x, 6 ker[M + Mp] and x , / l # 0 E Mp. 27 Let M be an A-module. The following conditions are equivalent: (i) M 87 = 0; 0 (ii) Supp(M) = 0; (iii) Suppm(M) = 0. 30 If M as a finately generated A-module, then Supp(M) as a closed set of Spec(A) for the Zarzska topology.
3. Support of a module 7. Fractions Proof (i) Consider z E ker( f ) . We have (Ax)M = (0) for all M E Suppm(M). 27. (ii) We denote by M‘ the submodule of JJME~UPPm(M)MM formed by all ( z M / s M ) ~ ~ s such ~ ~ that ~ ~ X( M~S M) ‘ = X M ~ S M for all M, M ’ ~ S u p p ~ ( A 4 ) . It is clear that f ( M ) c M’. Note that (0) : M = (0) : M’. 29, hence Supp(M’) = Supp(M). Consider an element x = ( Z M / S M ) M E M’. We have S M X M , / S M ! = Z M / ~E MM, for all M‘ E Suppm(M). kfh This shows S M X E f(M), hence (f(M)M = for all M E Suppm(M).