By Jürgen Stückrad

ISBN-10: 0387168443

ISBN-13: 9780387168449

ISBN-10: 3540168443

ISBN-13: 9783540168447

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 research are Buchsbaum jewelry. the fundamental underlying proposal of a Buchsbaum ring is a continuation of the well known notion of a Cohen-Macaulay ring, its necessity being created by means of open questions of algebraic geometry and algebraic topology. the idea of Buchsbaum jewelry began from a unfavorable resolution to an issue of David A. Buchsbaum. the concept that of this idea was once brought in our joint paper released in 1973.

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**Additional info for Buchsbaum rings and applications: an interaction between algebra, geometry and topology **

**Example text**

Denoted by R = 11t(Rl> R 2 ) or R = (f(RI' R2 ), if for every pEN , [R]p = [R1]p ®k [R2 ]p' (ii) Let Mv ~M:2 be graded R 1-, R 2 -modules, resp. Let R be the Segre product of RI and R2 (if it exists). A graded R-module M is called a Seyre product 0/ MI and M2 over k, denoted by 11I = (f,,(M I • M 2 ) or M = (f(MI' M 2 ), if for all p E Z [M]p = [M1)p ®k [M 2 1p, As is known Segre products exist and are uniquely determined (up to isomorphisms) by definition. Also since a sequence of graded modules over a graded k-algebra is exact if and only if the corresponding sequences for each grading are exact (as sequences of k-vector spaces), the Segre product is an exact covariant functor in each variable.

The same assertions made in § 1, 1. of this chapter on the existence of primary decompositions of submodules of Noetherian modules hold word for word in this situation. Therefore we omit an explicit discussion. Note that for any graded submodule N of a Noetherian graded R-module M there are graded primary submodules Ql' .. " Q, of M belonging to homogeneous primes Ph ' .. , 1', of R where {Ph"" Pr) = Ass M /N such that N Ql n ". n Qr. s, 8'gstem8 of parameters and Hilbert functions Let k be any field.

Is a Cohen-Macaulay module (over R Il ) if and only if M(ll) is a Cohen-Macaulay module (over R(tJ)}. 2. 1 has a more general version: Assume T c R is a multiplicatively closed set consisting of homogeneous elements 0. e. S = [T-IR]o)' '* § 2. Graded modules and Kiinneth formulas 35 If t E 'P n [Rh and if X is an indeterminate then we have T-IR S[tJt~S[Xlx = s[x, ~l If M is a graded R-module and if we put N;= [T-IMJo, then T- 11Y N[tJ/ ~N[XJx N lx, ~ J X1 . Set SUPPn M;= (I' c:: R I I' prime, M'p =1= 0).

### Buchsbaum rings and applications: an interaction between algebra, geometry and topology by Jürgen Stückrad

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