Download e-book for kindle: Algebraic and Geometric Topology by James R. Milgram

By James R. Milgram

ISBN-10: 0821814338

ISBN-13: 9780821814338

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CtlJ. = , can be extended t,o a derivation J which we also denote by Since '" H (X) q of t,hf' form = 0 q < 2 for . sineI' u = e + r i=l z 2 (X) 1 r. 2. S. 4) can he extended to a del'ivat,ion J ~:- J:F qr'IX * F q+l X 'file existence of the required derivation J:X X is now clear. k = j{/~ of genel'ators for S III 0 1. ,2.. Since • ,2. :)ince o. ,2. exists an integer ~r­ X £1 ([,) c ~El . we have X~" is a cycle in i > is a minimal set ~ In particular k is a free resolut:ion of the j{-module has finite homological dimension, there j such that s(j) ~ 1 t B(X* m - 33 - 5': k) lJowever, since commutes with the diffel'cntial in X ~~- ® k we have ahsurd.

5.. ), _ s and R- j-{-Sf'qllcnce in for fO s+q = n • J\laximal R-sequences in i! have the same number of elements. ti. K-sequences in The common numr-er of elements in maximal will he called the codimension of m and wi 11 be dcnot f~d codim R • Let Jacobson radical codim j-{ = codim m j-{ - l{ of 1 x be a non-zerodivisor in the l{. Put • - 27 - ~ = K/xR. 4). 4) and the exact sequences (1) an d ( ')) _" . b. m Jacohson radical codim }{ -1 (n-q)-l (q+l) K . (E)=O for 1 i>l. o . ( iii) is an R-sequence, then so is Moreover if any ot,her minimtl 1 set of generators fOl' the idea 1 Pruof.

Ti. K-sequences in The common numr-er of elements in maximal will he called the codimension of m and wi 11 be dcnot f~d codim R • Let Jacobson radical codim j-{ = codim m j-{ - l{ of 1 x be a non-zerodivisor in the l{. Put • - 27 - ~ = K/xR. 4). 4) and the exact sequences (1) an d ( ')) _" . b. m Jacohson radical codim }{ -1 (n-q)-l (q+l) K . (E)=O for 1 i>l. o . ( iii) is an R-sequence, then so is Moreover if any ot,her minimtl 1 set of generators fOl' the idea 1 Pruof. we will first give thf" equivalence of (i), (ii) and (iii).

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Algebraic and Geometric Topology by James R. Milgram


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