COMPUTING COHOMOLOGY GROUPS FROM MAY - RESOLUTIONS

CHRISTIAN NASSAU

Abstract. This paper is a continuation of [N_{1}]. We address the problem of computing the cohomology of the homology algebras A_{i} = H_{∗}(E_{i}, F_{2}), i ∈ {6, 7, 8}, from the May resolutions constructed in that paper. We first give a simplified account of these resolutions and derive a new spectral sequence. After that, the computation is easy for E_{6}, probably managable for E_{7}, but seems to be a formidable problem for E_{8}.

1. A simplified description of the resolutions

We assume the reader is familiar with [N_{1}]. We focus on the case of E_{8}, since the modifications for E_{6} and E_{7} are obvious.

Notation. • A = A_{8} = the algebra we’re resolving.

• Γ = F_{2}[h_{k,l}] = the E_{1}-term of the May spectral sequence.

• The minimal May-resolution can be described as M = A ⊗ Γ with a certain differential d : M → M which is A-linear. We let \overline{M} = Hom_{A}(M, F_{2}) ≅ Γ. The differential on \overline{M} will again be denoted by d.

• Γ̃ = the subring of Γ generated by the permanent h_{k,l}^{n}’s. More explicitly, this is

Γ̃ = F_{2}[h_{1,0}, h_{2,0}, h_{3,0}, h_{5,0}, h_{1,1}, h_{2,1}, h_{3,1}, h_{1,2}, h_{2,2}, h_{1,3}]
⊗ F_{2}[h_{4,0}^{16}, h_{6,0}^{16}, h_{7,0}^{16}, h_{8,0}^{8}, h_{4,1}^{8}]

The reason for introducing Γ̃ is that the differentials d : M → M and d : \overline{M} → \overline{M} are Γ̃-linear.

• B = { h_{k,l}^{2^{j}} | h_{k,l}^{2^{j}} ∉ Γ̃ }. B consists of the 18 elements h_{4,0}^{n}, h_{6,0}^{n}, h_{7,0}^{n} with n ∈ { 1, 2, 4, 8 } and h_{8,0}^{m}, h_{4,1}^{m} with m ∈ { 1, 2, 4 }.

• For K ⊂ B we let h_{K} = ∏_{x∈K} x. Explicitly, we obtain exactly the monomials

h_{4,0}^{n_{1}}h_{6,0}^{n_{2}}h_{7,0}^{n_{3}}h_{8,0}^{n_{4}}h_{4,1}^{n_{5}} with 0 ≤ n_{1}, n_{2}, n_{3} < 16, 0 ≤ n_{4}, n_{5} < 8

this way. The 2^{18} elements h_{K} form a Γ̃-basis of Γ.

We can now explain how this simplifies the description of the resolution of A. This resolution is described by the list of terms in the Appendix of [N_{1}]. These terms have the form

(1) coefficient function · h_{K} · (monomial y from Γ̃)^{−1}, K ⊂ B.

The point is that the coefficient function is redundant, since it can be recovered from K, according to the following

Recipe. For h^{E} ∈ \overline{M} = Γ, d(h^{E}) is obtained as follows. Go through all terms (1) of the description. Call such a term relevant if h_{K} divides h^{E}. If (1) is relevant, form the expression (h^{E}/h_{K}) · y. d(h^{E}) is the sum of all these expressions for the relevant terms (1).

Date: September 1998.

Let B′ = { 1, 2, …, 18 } and choose a bijection ϕ : B′ → B. Let h_{i} = h_{ϕ(i)} ∈ B for i = 1, 2, …, 18 and b_{i} = d(h_{i}). It follows from this recipe that b_{i} ∈ Γ̃; the b_{i} are just the 18 relations listed on page 6 of [N_{1}]. We will see in the next section that the complex \overline{M} is a perturbation of the standard complex which computes

TorΓ̃(Γ̃/(b_{1}), Γ̃/(b_{2}), …, Γ̃/(b_{18}))

2. A new spectral sequence

Let E = EΓ̃(h_{1}, …, h_{18}) be the exterior algebra over Γ̃ generated by the h_{i}. The set of h_{K′} with h_{K′} = h_{i_{1}} ∧ h_{i_{2}} ∧ ⋯ ∧ h_{i_{k}} for K′ = { i_{1}, i_{2}, …, i_{k} } ⊂ B′ is a Γ̃-basis of E. Let C_{i}, i ∈ B′, be the complex

Γ̃ ⟵^{·b_{i}} Γ̃{ h_{i} }.

Obviously H_{0}(C_{i}) = Γ̃/(b_{i}) and H_{k}(C_{i}) = 0 for k > 0, so C_{i} is a Γ̃-resolution of Γ̃/(b_{i}). We can give E a differential d′ by identifying it with C_{1} ⊗ ⋯ ⊗ C_{18}; we then have

H(E, d′) = TorΓ̃(Γ̃/(b_{1}), Γ̃/(b_{2}), …, Γ̃/(b_{18})).

We can identify E with Γ via h_{K′} ↦ h_{ϕ(K′)}. The grading on E is compatible with the filtration on Γ defined by Γ_{n} = Γ̃{ h_{K} | ♯K ≤ n }. We then have Γ̃ = Γ_{0} ⊂ Γ_{1} ⊂ ⋯ ⊂ Γ_{18} = Γ. For the differential d we have d(Γ_{n}) ⊂ Γ_{n−1}.

An inspection of the definitions and the Recipe above now shows that the associated graded complex EΓ with differential

d : Γ_{n}/Γ_{n−1} → Γ_{n−1}/Γ_{n−2}

agrees with (E, d′). So we have

Theorem. There is a spectral sequence of Γ̃-modules.

TorΓ̃(Γ̃/(b_{1}), Γ̃/(b_{2}), …, Γ̃/(b_{18})) ⟹ H(Γ, d) = Ext_{A_{8}}(F_{2}, F_{2})

This spectral sequence is probably not very useful in the case of E_{8}. To see this, one first has to realize that only those terms (1) with ♯K = 1 contribute to the computation of Tor; these are just 46 terms out of the 466 terms listed in the Appendix of [N_{1}]. So the spectral sequence has many nontrivial differentials (more precisely, there seem to be 6 nontrivial differentials and together they account for more than 400 monomials).

The situation is better for E_{6} and E_{7}.

Theorem. The analogous spectral sequences for the computation of Ext_{A_{6}}(F_{2}, F_{2}) and Ext_{A_{7}}(F_{2}, F_{2}) behave as follows.

• For E_{6} the spectral sequence collapses, ie. as Γ̃ modules (up to extensions in the May spectral sequence) there is an isomorphism

Ext_{A_{6}}(F_{2}, F_{2}) = TorΓ̃(Γ̃/(h_{3,0}h_{1,1}), Γ̃/(h_{5,0}h_{1,1})).

Here Γ̃ is generated by h_{1,0}, h_{2,0}, h_{1,1}, h_{3,0}, h_{5,0}, h_{4,0}^{2} and h_{6,0}^{2}.

• For E_{7} the spectral sequence has exactly one nontrivial differential that accounts for the terms h_{6,0}h_{4,0}h_{3,1}^{−1}h_{2,1}^{−2} and h_{7,0}h_{4,0}h_{3,1}^{−2}h_{1,1}^{−1} of May filtration shift 19 and 21.

References

[N_{1}] C. Nassau, Minimal May - resolutions for the homology algebras of E_{6}, E_{7}, and E_{8}, available

from [N_{2}]

[N_{2}] C. Nassau, Internet page at http://www.math.uni-frankfurt.de/~nassau/BE8