By Serre J.-P.
This is often an English translation of the now vintage "Algèbre Locale - Multiplicités" initially released by means of Springer as LNM eleven, in numerous versions because 1965. It offers a brief account of the most theorems of commutative algebra, with emphasis on modules, homological tools and intersection multiplicities ("Tor-formula"). Many alterations to the unique French textual content were made via the writer for this English variation: they make the textual content more uncomplicated to learn, with no altering its meant casual personality.
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It is trivial that a) + b) Conversely, if Tort+,,(A/m, A/m) = 0, then T~f+~(A/rn, A/n) is zero for every maximal ideal n (the annihilator of Tort(M, N) contains the annihilators of M and of N). Thus proj dimA(A/m) 5 n and there exists a projective resolution 0 t L, 4.. i Lo + A/m --t 0. But this implies that Tcx~+~(M,A/III) = 0 for every M; whence a). D: Regular Rings D e f i n i t i o n A regular ring is a no&he&n ring of finite global homological dimension. 1. Properties and characterizations of regular local Let A be a regular local ring, n = glob dim A , m the maximal ideal of A , k = A/m and M a n~nzem finitely generated A-module.
And lifting it to (ei) , with QEM. Now let + Li 5 5 L1% Lo 5 M + 0 be a &ee resolution L. of M Set: N, = Im(L; ---f Li_1) = Ker(L,-l t Li-2). Resolutions 85 Corollary If L. = (Lc) is B minim& free resolution of M, the rank of Li is equal to the dimension of the k-vector space Torf(M, k) Indeed, we have: To&M, k) ” H,(L. ) ” Zi Remark. In particular, the rank of Li is independent of the chosen resolution L.. In fact, it is easy to prove more: any two minimal free resolutions of M are isomorphic (non-canonically in general).
Let us show that d) + a) Let x = (zl,. ) be a minimal system of generators of m Property d) implies that x is an A -sequence (this follows from th. 3 applied to the A-module A). In other words, the Corollary 3. A regular local ring is normal, and Cohen-MacauJay If A is regular, it is Cohen-Macaulay by COT. 2, applied to M = A ; it is normal, because gr,(A) is normal, cf. Chap. II, part A, 54. 78 IV. Homological D: Regular Rings Dimension and Depth Corollary 4 (Auslander-Buchsbaum, is factorial.