By Jürgen Stückrad

Da die algebraische Geometrie wesentlich vom Fundamentalsatz der Algebra ausgeht, den guy nur deshalb in der gewohnten aUgemeinen shape aussprechen kann, weil guy dabei die Vielfachheit der Losungen in Betracht zieht, so mull guy auch bei jedem Resultat der algebra is chen Geometrie beim Zuriickschreiten die gemeinsame QueUe wiederfinden. Das ware aber nicht mehr moglich, wenn guy auf dem Wege das Werkzeug verlore, welches den Fundamentalsatz fruchtbar uud bedeutungsreich macht. Francesco Severi Abh. Math. Sem. Hansischen Univ. 15 (1943), p. a hundred This booklet describes interactions among algebraic geometry, commutative and homo logical algebra, algebraic topology and combinatorics. the most item of analysis are Buchsbaum jewelry. the elemental underlying concept of a Buchsbaum ring is a continuation of the well known suggestion of a Cohen-Macaulay ring, its necessity being created by means of open questions of algebraic geometry and algebraic topology. the speculation of Buchsbaum earrings all started from a adverse resolution to an issue of David A. Buchsbaum. the concept that of this conception was once brought in our joint paper released in 1973.

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Supp iJ-f t;;; V(a). The implication (c) (b) is trivial and (ii) is therefore proven. tI V(a) if and only if 1l~(M) '"'" M and HWlf) = 0 for all i~ 1. We split the exact sequence of (i) into two exact sequences: (a) 0 Ilg(M) ~M 111' --'>- 0 and (b) 0--'>-11/' IlO(M) 1l~(M)--,>-O. M)) for 1 and i~ 2, alll:~ 2. (M)) ~ llWW)): o --'>- /l~(HO(1I1)) 0 and thus (b) gives rise to the following exact sequence H~(M) --'>- HWW) --'>- m(IlO(M)) --'>- O. llf) H~(M/) obtained froll! (a) and this proves (iii).

Af+q) A. d. Remarks. 1. The lemma is true for arbitrary Noetherian rings A. 2. The lemma also has a graded version: It remains true if A is a Noetherian graded ring and M a graded R-module. The homomorphisms occuring here are all of degree zero (compare with the notations of § 2). 3. Let X := Spec A, U X" V(a). 11) ~ r(U, ii), where 1ft is the sheaf on X associated with M. f. R. Hartshorne [4], Chap. 7). § 2. Graded modules and Kiinneth formulas In this paragraph we will develop basic facts and notations concerning graded rings and modules which we will need in the sequel.

Ill (up to order). If R is a graded or a local ring and M a Noetherian (graded) R-module then we denote by fhR(M} the number of elements of a minimal set of generators of M. 3. p =1= m a komogenecnls 'prime ideal of R. ll). This will allow us to show that a projective subscheme is a local complete intersection, or a generic complete intersection, by checking ordinary loealizations instead of degree 0 localizations. 3. Minimal free re8olutions Let M be a graded R-module. Then there is a Bet B of homogeneous elements of M generating M (as anR-module).