By Alejandro Adem
ISBN-10: 038757025X
ISBN-13: 9780387570259
ISBN-10: 354057025X
ISBN-13: 9783540570257
Adem A., Milgram R.J. Cohomology of finite teams (Springer, 1994)(ISBN 354057025X)
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Additional resources for Cohomology of finite groups
Example text
Passing to group rings we have the corresponding exact sequence Now we describe the kernel R in the map above. 10 If N O---tR---tZ(F(X))---tZ(G)---tO. Tn,gl, ... ,gn)) = ((2tr+1 - 1, ... gn),(2t 1 , .. 2 Let p be an odd prime, then H*(7L/p; IFp) = E(Vl) 0 IFp[b 2 J, making H*(G; IF) into an associative (graded) commutative ring with unit. ) The pairing is natural with respect to restriction, the tensor product of a polynomial algebra on a two dimensional generator and an exterior algebra on a 1-dimensional generator. (resiD* (aU b) = (res~)* (a) U (res~)* (b). Proof. An embedding of our usual resolution for 7L over 7L(7L/p) into the bar construction can be constructed quite simply. Next, let our new module be the kernel of 00 and apply the same construction above to construct a surjective map 0~:C1-+ker(00). Then, including ker(oo) c Co = Us Z(G)mi defines 01 as the composition of inclusion with o~. Now, we can repeat this construction to obtain a resolution. e. direct summands in a free module); all the subsequent constructions will work equally well, as the reader can verify. so these maps fit together to give an action map and consequently an action on cohomology rings r; = id if 9 E H ---+ s = (tl, ... Cohomology of finite groups by Alejandro Adem
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