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 proposal of a Buchsbaum ring is a continuation of the well known notion of a Cohen-Macaulay ring, its necessity being created through open questions of algebraic geometry and algebraic topology. the idea of Buchsbaum earrings began from a detrimental solution to an issue of David A. Buchsbaum. the concept that of this thought was once brought in our joint paper released in 1973.

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This exact sequence is called a free re8olution of M. "Ve note that we work in the category of graded R-modules with degree zero homomorphisms as Illorphisms. This is a complete (and cocomplete) Grothendieck category, (d. Schubert [1], Def. ). e. in the category of k-vector spaces. This category possesses enough projectives (as we have seen before) and therefore enough injectives. This fact we will use in Section 4 of this paragraph. For any graded R-module M we have projective and injective resolutions (even free resolutions).

1 This o. 32 Some foundations of commutative and homological algebra (v) Assume g is injective. For Ass M ~ V(a) let lJ E Ass M" V(a). Then there is a monomorphism A/lJ ...... M giving rise to the following commutative diagram Alp -4 HO(AllJ) 4 (Alp)s t t HO(M) 4 Ms where all homomorphisms are injective (HO( ) is left exact and H~(Alp) 0 since lJ ~V(a». lJ n 8 = O. Assume now :p n S = 0 for all :p E Ass M" V(a). Let a E Ker g and choose 8(ult ) . E Hom(alt ; M) representmg a. e. there is abE S with bs(a") = O.

Let M be a Noetherian graded R-module where R is a graded k-algebra. ) (note that for all n E Z[MJ. is a k-vector space of finite rank). HM is called the Habert junction of M. There is a numerical polynomial hM (E Q[TJ, T an indeterminate) such that HM(n) = hM(n) for all sufficiently large n. The polynomial hM is called the 3* 36 O. Some foundations of commutative and homological algebra Hilbert polynomial of ill. M) (~') + h (M) (a 1 '1' 1) + ... + hdUI1) with d := deg hM (considered as a polynomial in '1'), and where the integers ho(~~f) > 0, hl(JI), ...

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