Download Buchsbaum rings and applications: an interaction between by Jürgen Stückrad PDF

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 earrings. the fundamental underlying notion of a Buchsbaum ring is a continuation of the well known thought of a Cohen-Macaulay ring, its necessity being created by way of open questions of algebraic geometry and algebraic topology. the idea of Buchsbaum earrings all started from a unfavorable solution to an issue of David A. Buchsbaum. the concept that of this idea used to be brought in our joint paper released in 1973.

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Extra resources for Buchsbaum rings and applications: an interaction between algebra, geometry and topology

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The local version of the following very useful lemma can be found in R. Y. 12. Let T be a right exact, covariant additive R-linear functor from the category of graded R-modules to itself which respects shifts of degrees. Then there is for every graded R-module M a natural homomorphism ;,v: M which ~8 ®n T(R) ~ T(M) an isomorphism if M is f%ilitely generated. If T commutes with dired i£mits, ;M is always an isomorphism. Proof: Let M be a graded R-module. For any homogeneous element m E M let h m : R ~ M(deg m) denote the homomorphism defined by hm(r) := rm for all r E R.

In fact H~(X, M) need not even be finitely generated if M is finitely generated. But there are important cases in which I1~(X, M)~ M namely for R = k[Xo, ... , X,,], K a field, X o, ... , Xn indeterminates, n 1 and a) M any free graded R-module, b) M any homogeneous ideal I c::: R which is saturated (that is, I :R mR = I), c) M = I1~(X, :T) where c'r is any quasi-coherent sheaf of modules over X. e. R = kfX o, ... , and H~ define a 1-1 correspondence: {closed subschemes of Xl ........ {saturated homogeneous ideals of R} , where d v is the ideal sheaf of V and Proi (RjI, RjI) +-!

R 2,moaules. (m2' M 2 )} -+ ExtMm, O'(MI' M 2 )} .. 1"(M}), Hq(M ») 2 p+q~ ~q Ml' IW • _" 1 ) ~a(M"MtI ... ~ HfI(a(Mb M i )} .. 12. Let Mb M2 be Noetherian graded R1"resp. R 2,moaules. > M 2 ) '? inf (depth M I , depth Mil. § 3. t. fu 1112 )) --+ a(M 1> M 2 ) --+ HO(a(Ml' M 2 )) --+ H'm(a(Ml> M 2 )) --+ O. If depth Ml 0 or depth M2 0 there is nothing to prove. Hence we assume depth Mi ~ 1 for i 1,2. e. depth a(M}, M 2 ) 1. f1 = 1ordepthM2 1 the conclusion follows. Assume now depth Mi > 2 for i 1,2.

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