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 variations due to the fact that 1965. It supplies a quick account of the most theorems of commutative algebra, with emphasis on modules, homological equipment and intersection multiplicities ("Tor-formula"). Many variations to the unique French textual content were made through the writer for this English variation: they make the textual content more straightforward to learn, with no altering its meant casual personality.
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Extra info for Local Algebra - Multiplicities
I claim that p = ~1. +~ in p which belongs to none of the pi ; the set zl,. ,zr+l would thus be a subset of a system of parameters of E / contrary to the maximality of ~1,. , z7 Thus p = p1 , which shows that the zi satisfy the stated condition, and proves at the same time that dim(A/p) = n-r Moreover, the z; form an A,, -sequence of EP , which is at the same time a system of parameters, since p is a minimal element of Supp(E/(sl, , z,)E) This proves that EP is a Cohen-Macaulay module of dimension T , qed.
Let A be a noetherian local ring, with maximal ideal m = m(A) ; let E be a nonzero finitely generated A -module. Let A and B be two noetherian local rings and let Proposition 11. r$ : A + B be a homomorphism which makes B into a finitely generated A-module. If E is a finitely generated B-module, then E is a CohenMacauJay A module if and only if it is a Cohen-Macaulay B-module. This follows from the following more general proposition: Let A and B be two noetherian local rings, and Jet Proposition 12.
14, and observing that dim(E/(zl,. , zk)E) = n - k since the zi form a system of parameters of E The converse is trivial. g. use prop. 3 of Chap. III). 2. Several characterizations of Cohen-Macaulay modules If E is a Cohen-Macaulay module, and if a is an ideal Corollary of A generated by a subset of k elements of a system of pammeters of A, the module E/aE is a Cohen-Macaulay module of dimension equal to dim(E) - k. This has been proved along the way. Proposition 13. Let E be a Cohen-Macaulay A -module of dimension n For every p l Ass(E) , we have dim A/p = n , and p is a minimal element of Supp(E) Indeed, we have dim(E) > dim(A/p) 2 depth(E) (cf.