The Numerical Solution of Algebraic Equations by R. Wait

By R. Wait

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Benson 1994/95] D. J. Benson, “Cohomology of modules in the principal block of a finite group”, New York J. Math. 1 (1994/95), 196–205. [Benson 2001] D. Benson, “Modules with injective cohomology, and local duality for a finite group”, New York J. Math. 7 (2001), 201–215. [Benson 2004] D. Benson, “Dickson invariants, regularity and computation in group cohomology”, Illinois J. Math. 48:1 (2004), 171–197. [Benson and Carlson 1992] D. J. Benson and J. F. Carlson, “Products in negative cohomology”, J.

2), H ∗ (G, k) = k[x1 , . . , xr ] is a polynomial ring on r generators of degree one. In s,t ∗ this case, Hm H (G, k) vanishes except when s = r, and13 r,∗ ∗ −1 Hm H (G, k) = k[x−1 1 , . . , xr ], where the right hand side is graded in such a way that the identity element is in r,−r ∗ Hm H (G, k). There are no differentials, and it is easy to see how the spectral sequence converges to the dual of the cohomology ring. On the other hand, if G ∼ = (Z/p)r with p odd, then H ∗ (G, k) = Λ(x1 , . .

Let K denote the collection of all elementary abelian p-subgroups K of G with the property that the Sylow p-subgroups of the centralizer CG (K) are not conjugate to a subgroup of any of the groups in H . 1). Let J be the intersection of the kernels of restriction to subgroups in K , which is again an ideal in H ∗ (G, k) (in case K is empty, this intersection is taken to be the√ideal √ of all elements of positive degree). Then J and J have the same radical , J = J . 1 is the main ingredient in the proof of the following theorem of Carlson [1995] relating the associated primes with detection on centralizers.

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