Benson's Test for Completion

The reference for Benson's test is: D. Benson. Dickson invariants, regularity and computation in group cohomology. Illinois J. Math. 48 (2004), 171–197. Published version.

Briefly, Benson's test goes as follows. Suppose we have computed the cohomology ring out to degree N and have found ζ₁,…,ζ_r which are a hsop for the cohomology ring and for the current approximation. Let n_i = max(|ζ_i|,2). Benson's test requires that the p-rank r be at least two.

First one checks whether the hsop is filter-regular. If so, one computes the best possible filter type (d₀,…,d_r). Then one calculates

α = max{i + d_i | 0 ≤ i ≤ r-2 }. The approximation passes Benson's test and is the cohomology ring, provided that N > max{α,0} + n₁ + … + n_r - r. Note that this α is not quite the α of Benson's Theorem 10.1. Rather it is the easily computable upper bound for Benson's α which he himself suggests.

Benson also points out that if the p-rank of the centre is at least two, then the > in the above inequality can be replaced by ≥.

Benson actually requires that n_i := |ζ_i| should be at least two. But I've checked that squaring ζ_i affects neither filter-regularity nor the type.

Computing the filter type

The rules for the type (d₀,…,d_r) are: d₀ ≥ -1; and d_i+1 is either d_i or d_i-1. A hsop for R is filter-regular of this type provided that multiplication by ζ_i+1 is injective on R⁄(ζ₁,…,ζ_i) in degrees > n₁+…+n_i + d_i for 0 ≤ i ≤ r, where ζ_r+1=0.

Amongst the various types of a filter-regular sequence there is one which is smallest in the dominance partial ordering. This best fitting type is determined as follows:

The raw type of the hsop is (b₀,…b_r), where b_i takes the smallest value such that multiplication by ζ_i+1 on the appropriate quotient is injective from degree b_i+1 onwards. If ζ_i+1 is regular one sets b_i = -1. Existence of the raw type is equivalent to the sequence's being filter-regular.
The intermediate type of the hsop is (c₀,…c_r), where c_i is defined by b_i = n₁ + … + n_i + c_i.
The filter type (d₀,…,d_r) is then determined as follows: d₀ = max{c₀,…,c_r}, and d_i+1 = max{d_i-1, c_i+1,…,c_r}.

The regularity conjecture, or SPQR and all that

A filter-regular hsop of type (d₀,…,d_r) is

quasi-regular if d_i ≤ -1 for all i;
strongly quasi-regular if d_i ≤ -i for all i; and
very strongly quasi-regular if d_i ≤ -i-1 for 0 ≤ i ≤ r-1 and d_r ≤ -r.

The regularity conjecture is equivalent to every filter-regular sequence being strongly quasi-regular. As yet there is no counterexample to the stronger statement that every filter-regular sequence is very strongly quasi-regular.

I haven't yet written the code to compute the regularity, should a filter-regular sequence turn up which is not strongly quasi-regular. I presume that Benson's paper holds some clues.

Last modified 7 March 2006 — David J. Green

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