By Richard M. Kane

ISBN-10: 0821823426

ISBN-13: 9780821823422

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**Extra resources for Implications in Morava K-Theory**

**Sample text**

Let X. generate A.. We know H*K is a polynomial algebra. (See §2). We now make special choices for the polynomial generators. For each X. choose y. where f*y. = X.. 1 Choose the remaining polynomial generators o*__ let y! ,nq Let If height X. } of H*K from ker f*. }. i Observe that A = H*K//r. e ideal (r). 1. E 2 = H*X//A 8 Tor (Z/2;Z/2). PROOF: We mimic the argument of Smith in [21]. A = rn//r. 2) Let r Q = H*K. Thus over which F n Is a free r G 0 term as G 2 = Tor (H*X;Z/2) 3 Tor (Z/2;Z/2). 3) TorA(H*X;Z/2 A = H*X ®A Z/2.

For, suppose we have found y such that x + y survived to B . Since hr{x + y} = 0 in D r it follows from 9-3 that {x + y} = y1 + t + d 1 where y e P ,t e T and d is decomposable. T Thus {x + y} = t + d where y e PA. Since d t = 0 we have dr(x + y) = drd. — even However, x + y e P B r forces dr(x + y) e P o d d B r c Q ° ^ r while drd = 0 in QBr. Thus, dr(x + y) = drd = 0. 2) {x + y} = 0 in B w . Suppose {x + y} ^ 0 in B . We will show {x + y} = y where y e P . For 00 then we can alter y to obtain { x + y } = 0 .

We want to show that Jy survives to B . Since x e C we can choose some z e B r r r where h z = x. Suppose y has survived to B . Consider z e B . Then h y7 = h z in C . s* s s So Jy-z € Ker h . 2 J s y - z = t + d where t c T c and d is decomposable. However How d o z = d^t = 0. d J y = d d. s s d y e P°d% c QoddB Thus IMPLICATIONS IN MORAVA K-THEORY while dsd = 0 in ^9 f dd B . So dgy = 0. D. 6. n+s+1 n+1 * H*X if x = 0 in H*X/I then there 2n+s+l+2n+l_2 exists y e P A such that x + y € Tor(n). Given x e P PROOF: We want to find y e PA such that x + y is a permanent cycle in B and {x + y} = 0 in B .

### Implications in Morava K-Theory by Richard M. Kane

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