Download Cohomology Rings of Finite Groups: With an Appendix: by Jon F. Carlson, L. Townsley, Luís Valero-Elizondo, Mucheng PDF

By Jon F. Carlson, L. Townsley, Luís Valero-Elizondo, Mucheng Zhang

Group cohomology has a wealthy heritage that is going again a century or extra. Its origins are rooted in investigations of workforce concept and num­ ber thought, and it grew into an crucial element of algebraic topology. within the final thirty years, staff cohomology has constructed a strong con­ nection with finite workforce representations. not like the early purposes that have been essentially fascinated by cohomology in low levels, the in­ teractions with illustration concept contain cohomology jewelry and the geometry of spectra over those earrings. it really is this connection to represen­ tation thought that we take as our basic motivation for this e-book. The publication includes separate items. Chronologically, the 1st half was once the pc calculations of the mod-2 cohomology jewelry of the teams whose orders divide sixty four. the guidelines and the courses for the calculations have been constructed over the past 10 years. a number of new positive aspects have been additional over the process that point. We had initially deliberate to incorporate just a short advent to the calculations. even if, we have been persuaded to supply a extra immense textual content that will comprise in larger aspect the innovations which are the topic of the calculations and are the resource of a few of the motivating conjectures for the com­ putations. we now have collected jointly some of the effects and ideas which are the focal point of the calculations from through the mathematical literature.

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Extra info for Cohomology Rings of Finite Groups: With an Appendix: Calculations of Cohomology Rings of Groups of Order Dividing 64

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A consequence of the theorem is that a projective resolution for any module can be obtained by tensoring the module with a projective resolution of the trivial module. More generally we have the following. 5. Suppose that L and M are kG-modules and that (P*,c) is a projective resolution of L. Then (P* 0 M,c 0 ldM) is a projective resolution of L 0 M. Proof. 9 is exact except in degree zero. Moreover, Ho (P* 0 M) = Ho (P*) 0 M = L 0 M by the Kiinneth Tensor Formula. Finally, note that the kernel of c 0 IdM coincides with the image of 0 ldM.

So the image of 1 0 o: is contained in the kernel of 1 0 ,B. Thus we need only show the reverse inequality. So suppose that W is the cokernel of 1 0 o:. Then we have a natural map(): W---+ X 0N given by O(x0m+lmage(10o:)) = x0,B(m). We 20 COHOMOLOGY RINGS OF FINITE GROUPS also have a map cr: XxN---+ W given by cr(x, n) = (xc>9m+Image(lc>9a)) where m E M is any element such that (3(m) = n. Note that cr(x (><) n) does not depend on the choice of m E M and that cr is middle linear. Hence, there exists a homomorphism 1/;: X(><) N---+ W with 1/;(x (><) n) = cr(x, n).

Cohomology Rings of Finite Groups © Springer Science+Business Media Dordrecht 2003 48 COHOMOLOGY RINGS OF FINITE GROUPS discuss minimal resolutions and bar resolutions in some detail. In a later section we give some applications of low dimensional cohomology. Although we do not discuss it in detail, it is worth mentioning another method for constructing resolution. This is by perturbations on a resolutions for the associated graded algebra of a filtered algebra. Such a filtration in group cohomology might come from the lower central series if the group were nilpotent.

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