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By Tammo tom Dieck

This publication is a jewel– it explains very important, important and deep issues in Algebraic Topology that you just won`t locate somewhere else, conscientiously and in detail."""" Prof. Günter M. Ziegler, TU Berlin

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Lf is a finite-dimensional vector space over the residue field k::::: Rim. A set {m l , ••• , mil of elements of M is called a minimal set of generators for M if the residues mH ••• , mt in M form a vector space basis of M. The graded version of Nakayama's Lemma guarantees that a minimal set of generators is in faet a set of generators for M as an R-module. 2. ill be a Noetherian graded R-module. 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.

By a Noetherian graded R-mooule we mean a graded R-module satisfying the ascending chain condition for gradcd submodules. Since R is Noetherian itself, a graded R-module M is Noetherian if and only if M is finetely generated by homogeneous elements. 1. Associated primes Let R be a graded ring and let M be a graded R-module. A homogeneous prime ideal pnme of M if one of the following equivalent conditions is fulfilled: (i) There is an homogeneous element x E 1'J. ):I. (ii) M contains a submodule isomorphic to (R/:p) (n) for some n E Z.

Notice that fi'm,,( ) and fii(R, ) are the right derived functors of the left exact functors = 0 :M (mR) for any graded R-module M. If M is Noetherian then fi'mR(M) is Artinian for all i. Now we have fi~n,,( ) resp. fiO(R, ). 3. Let X:= Proj Rand aS8Ume that M is a graded R-nwdule. Then tor all i here are natural i8omorphisms H~(X, Ai) ~ fi'(R, M). Proof: We prove Hi(X, M) c::::. Hi(R, M) (then taking the direct sum over all shifts of degrees the result will follow). f. 8). This isomorphism is a natural one.

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Algebraic Topology and Tranformation Groups by Tammo tom Dieck


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