By Serre J.-P.
This can be an English translation of the now vintage "Algèbre Locale - Multiplicités" initially released through Springer as LNM eleven, in numerous variants given that 1965. It supplies a brief account of the most theorems of commutative algebra, with emphasis on modules, homological equipment and intersection multiplicities ("Tor-formula"). Many adjustments to the unique French textual content were made through the writer for this English version: they make the textual content more uncomplicated to learn, with out altering its meant casual personality.
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In mLo ), which means that the boundary operator d on L. is zero. The map f is injective, and identifies K. with a direct factor of L. (as A-modules). We need to show that the fi : KS + Li are left-invertible. But, we have the following lemma (whose proof is easy): Lemma Let L and L’ be two free A -modules, and let g : L + L’ be a homomorphism. For g to be let%invertible (rep. right-invertible), it is necessary and sufficient that 3 : z + Z’ is injective (resp. surjective). i + zi are injective.
DimM - 1, cf. Chap. III, car. 5 to th. 1. By induction on i , we see that dim(M/(al,... and 61 ,ai)M) = dimM -i for i=l,... ,p, which implies (i), cf. Chap. III, prop. 6. Let NOR’ p be an element of Ass(M). We have an exact sequa~~ 0 + Hom(A/p, M) 2 Hom(A/p, M) + Hom(A/p,M/alM). Since p belongs to Ass(M) , Hom(A/p, M) is nonzero. By Nakayama’s lemma, the same is true for Hom(A/p, M)/al Hom(A/p, M) , hence also for Hom(A/p, M/alM) This means that there is a nonzero element of M/alM which is annihilated by p + alA.
For these to be minimal, it is necessary and sufficient that their kernels N,+, (rap. Nl ) are conta+d i” mLi (resp. in mLo ), which means that the boundary operator d on L. is zero. The map f is injective, and identifies K. with a direct factor of L. (as A-modules). We need to show that the fi : KS + Li are left-invertible. But, we have the following lemma (whose proof is easy): Lemma Let L and L’ be two free A -modules, and let g : L + L’ be a homomorphism. For g to be let%invertible (rep. right-invertible), it is necessary and sufficient that 3 : z + Z’ is injective (resp.