Reciprocal Polynomials and p-Group Cohomology |
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Authors: | Chris Woodcock |
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Institution: | 1. Institute of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, CT2 7NF, UK
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Abstract: | Let p be a prime and let G be a finite p-group. In a recent paper (Woodcock, J Pure Appl Algebra 210:193–199, 2007) we introduced a commutative graded ?-algebra R G . This classifies, for each commutative ring R with identity element, the G-invariant commutative R-algebra multiplications on the group algebra RG] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity element. We show here that, up to inseparable isogeny, the “graded-commutative” mod p cohomology ring $H^\ast(G, \mathbb{F}_p)Let p be a prime and let G be a finite p-group. In a recent paper (Woodcock, J Pure Appl Algebra 210:193–199, 2007) we introduced a commutative graded ℤ-algebra R
G
. This classifies, for each commutative ring R with identity element, the G-invariant commutative R-algebra multiplications on the group algebra RG] which are cocycles (in fact coboundaries) with respect to the standard “direct sum” multiplication and have the same identity
element. We show here that, up to inseparable isogeny, the “graded-commutative” mod p cohomology ring
H*(G, \mathbbFp)H^\ast(G, \mathbb{F}_p) of G has the same spectrum as the ring of invariants of R
G
mod p
(RG ?\mathbbZ \mathbbFp)G(R_G \otimes_{\mathbb{Z}} \mathbb{F}_p)^G where the action of G is induced by conjugation. |
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