Computing May Resolutions

C. Nassau

First, maybe final, draft — June 13, 1996

Abstract

This is a description of a program I wrote between Jan - Aug 1995 to compute “May resolutions” of ℤ_{2} over finite dimensional sub Hopf algebras A′ of the mod 2 Steenrod algebra A. Theoretically, these May resolutions allow the determination of differentials in the May spectral sequence converging to the cohomology of the given sub Hopf algebra A′, along with the determination of multiplicative extensions. Practically, however, they are of little or no use, and the program succeeded only in very simple cases.

1 Theory

Let us first recall the basic theory behind the May spectral sequence. Let

A_{∗} = ℤ_{2}[ξ_{1}, ξ_{2}, ξ_{3}, …]

be the mod 2 dual Steenrod algebra, and let Sq(k_{1}, k_{2}, …, k_{m}) ∈ A denote the dual of ξ_{1}^{k_{1}}ξ_{2}^{k_{2}} ⋯ ξ_{m}^{k_{m}}. We give A the variant of the May filtration described in [Rav], Proof of Theorem 3.2.3. That is we let

|| Sq(k_{1}, k_{2}, …, k_{m}) || = ∑_{i=1}^{m}(2i − 1)α(k_{i})

where α(l) denotes the number of ones in the binary expansion of l, and then

F^{s}A = span_{ℤ_{2}}{ x = Sq(k_{1}, …, k_{m}) | || x || ≥ s }

defines a decreasing multiplicative convergent filtration of A such that the associated bigraded algebra E^{∗,∗}A is the exterior algebra

E([P_{t}^{s}] | t ≥ 1, s ≥ 0)

where as usual P_{t}^{s} ∈ A is dual to ξ_{t}^{2^{s}} and where [x] for a homogenous x ∈ A denotes the class of x in E^{s,∗} if x ∈ F^{s}A \ F^{s+1}A.

Standard homological algebra gives now an E_{1} spectral sequence

E_{1}^{∗,∗} = Ext_{E^{∗,∗}A}(ℤ_{2}, ℤ_{2}) ⇒ Ext_{A}(ℤ_{2}, ℤ_{2})

and it’s this we wish to compute.

The existence of this spectral sequence is merely a statement about the existence of a suitably filtered resolution of ℤ_{2} over A. The standard proof consists of showing that the bar resolution C_{A}^{∗,∗}(ℤ_{2}, ℤ_{2}) inherits a filtration from the filtration of A, and that the associated trigraded complex E^{∗}C_{A}^{∗,∗}(ℤ_{2}, ℤ_{2}) coincides with the bar resolution C_{E^{∗,∗}A}^{∗}(ℤ_{2}, ℤ_{2}) of ℤ_{2} over E^{∗,∗}A.

Since the bar complex is much too big for direct computations we will instead try to find a filtered A free resolution of ℤ_{2}

\underline{M}_{∗}^{∗} : 0 ⟵ ℤ_{2} ⟵ M_{0} ⟵ M_{1} ⟵ M_{2} ⟵ ⋯

such that the associated trigraded complex E^{∗}\underline{M}_{∗}^{∗} is the minimal resolution of ℤ_{2} over E^{∗}A^{∗}. We call such an \underline{M} a May resolution (though “minimal May resolution” would be more appropriate).

Since, of course, we know that

Ext_{E^{∗}A^{∗}}^{∗}(ℤ_{2}, ℤ_{2}) = P (h_{i,j} | i ≥ 1, j ≥ 0),

with h_{i,j} corresponding to P_{i}^{j}, we can be more explicit here. The minimal resolution of ℤ_{2} over E^{∗}A^{∗} is the Koszul resolution

\underline{KM}_{∗}^{∗} : 0 ⟵ ℤ_{2} ⟵ KM_{0} ⟵ KM_{1} ⟵ KM_{2} ⟵ · · · .

(1)

This has the graded set M_{∗} of monomials in the h_{i,j} as generators and the differential is given on h_{i_{1},j_{1}}^{k_{1}}h_{i_{2},j_{2}}^{k_{2}} ⋯ h_{i_{n},j_{n}}^{k_{n}} with k_{1}, k_{2}, …, k_{n} > 0 by formal multiplication with

[P_{i_{1}}^{j_{1}}] h_{i_{1},j_{1}}^{−1} + [P_{i_{2}}^{j_{2}}] h_{i_{2},j_{2}}^{−1} + ⋯ + [P_{i_{n}}^{j_{n}}] h_{i_{n},j_{n}}^{−1}.

So we are looking for a free resolution \underline{M} of ℤ_{2} over A, indexed on the same set of generators M_{∗}, such that the associated trigraded complex is \underline{KM}, when \underline{M} is endowed with the filtration degree

|| ah_{i_{1},j_{1}}^{k_{1}} ⋯ h_{i_{n},j_{n}}^{k_{n}} || = || a || + (2i_{1} − 1)k_{1} + ⋯ + (2i_{n} − 1)k_{n}, a ∈ A.

Equivalently, we require that

d\underline{M}(x) = d\underline{KM}(x) + terms of higher May filtration.  (2)

This will be satisfied iff

d\underline{M}(m) = d\underline{KM}(m) + y_{m}, || y_{m} || > || m ||, ∀m ∈ M_{∗}.

So for each m ∈ M_{k} we have to find a y_{m} ∈ M_{k−1} such that

1. || y_{m} || > || m || and

2. d_{k−1}d_{k}(m) = 0

where d : M_{k} → M_{k−1} is the A linear map with d_{k}(m) = d_{\underline{KM},k}(m) + y_{m}.

The first condition asures that (2) is fulfilled, and the second that \underline{M} really is a complex.

To show the existence of such y_{m} we use the following standard Lemma.

Lemma 1. Assume that y_{m} have been chosen for all m ∈ M_{k} with k ≤ n for a fixed n, such that 1. and 2. above are satisfied. Then for every s the sequence

0 ⟵ F^{s}ℤ_{2} ⟵^{d_{0}} F^{s}M_{0} ⟵^{d_{1}} F^{s}M_{1} ⟵^{d_{2}} ⋯ ⟵^{d_{n}} F^{s}M_{n}

is exact.

Proof. Without the restriction k ≤ n this would be an easy induction on s using the exact sequences

0 ⟶ F^{s+1}\underline{M} ⟶ F^{s}\underline{M} ⟶ E^{s}\underline{M} ⟶ 0

and the exactness of E^{∗}\underline{M}. (Restrict to a finite dimensional sub algebra A′ ⊂ A to ground the induction.) The diagram chases involved in that proof carry over immediately to prove it under the restriction k ≤ n, too.

So in particular, once the y_{m} satisfying 1. and 2. have been chosen, the sequence

0 ⟵ ℤ_{2} ⟵^{d_{0}} M_{0} ⟵^{d_{1}} M_{1} ⟵^{d_{2}} ⋯ ⟵^{d_{n}} M_{n}

will be exact.

Using this Lemma we get

Lemma 2. There are y_{m} as above such that 1. and 2. are fulfilled.

Proof. Assume that such y_{m} have been chosen for all m ∈ M_{k} with k ≤ n for a given n and let x ∈ M_{n+1}. Since the Koszul resolution is a complex we have || z || > || x || for z := d_{n}(d_{\underline{KM},n+1}(x)). Since d_{n−1}d_{n} = 0 we have d_{n−1}(z) = 0. The Lemma above gives the existence of an y with || y || ≥ || z || > || x || and d_{n}(y) = z. Since d_{n}(d_{\underline{KM},n+1}(x) + y) = z + z = 0, letting y_{x} := y will do.

The most convenient way to find a correction term y_{m} for a given m ∈ M_{n+1} makes use of the null homotopy of the Koszul complex \underline{KM}. Note that the terms of lowest May filtration in d_{n}(d_{\underline{KM},n+1}(m)) represent a cycle in the associated graded complex since we know that a correction term y_{m} of that May filtration exists. Hence we can apply a null homotopy of \underline{KM} to these terms, giving a y^{(1)} such that d_{n}(d_{\underline{KM},n+1}(m) + y^{(1)}) has strictly higher May filtration than d_{n}(d_{\underline{KM},n+1}(m)). Continuing this way we obtain a (necessarily finite) sequence y^{(1)}, y^{(2)}, y^{(3)}, … such that y_{m} = y^{(1)} + y^{(2)} + ⋯ is a correction term of the form we were looking for.

This completes our survey of the theory involved. In the next section we will do some sample calculations and discuss the algorithm that will eventually be used.

2 The algorithm

From now on we will restrict ourselves to finite dimensional sub Hopf algebras A′ of A. Recall that such an algebra is associated to a profile function π : ℕ → ℕ_{0} with finite support. Given π the corresponding subalgebra A′ ⊂ A is generated by the P_{t}^{s} with s < π(t). The theory of the previous section carries over verbatim if we impose the restriction s < π(t) on all the P_{t}^{s} and h_{t,s}.

A naive way to implement the method outlined above would be to compute the correction terms y_{m}, m ∈ M_{k}, to the Koszul differential d\underline{KM}(m) by induction on k, using a null homotopy of the Koszul resolution as outlined above, thereby keeping a list of all the correction terms for m ∈ M_{k−1} while working on M_{k}. The enormous size of M_{k} even for small k, however, prevents this from being of any practical use.

So we try to be a bit smarter and compute symbolically instead. We illustrate this method in the simplest nontrivial case, i.e. we try to resolve ℤ_{2} over A(1), the subalgebra of A generated by Sq(1) and Sq(2). So now only h_{1,0}, h_{2,0} and h_{1,1} are present.

The “universal monomial” m ∈ M_{∗} has the form

m_{univ} = h_{1,0}^{a}h_{2,0}^{b}h_{1,1}^{c} with a, b, c > 0.

We’ll focus on m_{univ} which means that we ignore the edge cases a = 0 or b = 0 or c = 0 for the moment (and indeed for the rest of this article).

The Koszul differential d\underline{KM} =: d^{(1)} is given on m_{univ} by formal multiplication with

KD = Sq(1)h_{1,0}^{−1} + Sq(0, 1)h_{2,0}^{−1} + Sq(2)h_{1,1}^{−1}.

We compute d^{(1)}d^{(1)}: this is given on m_{univ} by formal multiplication with

KD ∘ KD = Sq(1)Sq(1)h_{1,0}^{−2} + Sq(1)Sq(0, 1)h_{1,0}^{−1}h_{2,0}^{−1}
+ Sq(1)Sq(2)h_{1,0}^{−1}h_{1,1}^{−1} + Sq(0, 1)Sq(1)h_{2,0}^{−1}h_{1,0}^{−1}
+ Sq(0, 1)Sq(0, 1)h_{2,0}^{−2} + Sq(0, 1)Sq(2)h_{2,0}^{−1}h_{1,1}^{−1}
+ Sq(2)Sq(1)h_{1,1}^{−1}h_{1,0}^{−1} + Sq(2)Sq(0, 1)h_{1,1}^{−1}h_{2,0}^{−1}
+ Sq(2)Sq(2)h_{1,1}^{−2}
= Sq(1)Sq(2)h_{1,0}^{−1}h_{1,1}^{−1} + Sq(2)Sq(1)h_{1,1}^{−1}h_{1,0}^{−1}
+ Sq(2)Sq(2)h_{1,1}^{−2}
= Sq(0, 1)h_{1,0}^{−1}h_{1,1}^{−1} + Sq(1, 1)h_{1,1}^{−2}.

Sq(0, 1)h_{1,0}^{−1}h_{1,1}^{−1} inflicts a shift of 1 in the May filtration and Sq(1, 1)h_{1,1}^{−2} a shift of 2, so we should try to correct the former first.

Since h_{2,0} ↦ Sq(0, 1), it seems plausible that we should try to replace KD by KD + h_{1,0}^{−1}h_{1,1}^{−1}h_{2,0}, since the Koszul differential of h_{1,0}^{−1}h_{1,1}^{−1}h_{2,0} contains Sq(0, 1) h_{1,0}^{−1} h_{1,1}^{−1} as a summand. But straightforward computation shows that this would not work. The reason is that adding this correction term to d^{(1)}(m_{univ}) for even exponents of h_{2,0} only, solves the problem for odd exponents too. Adding it for odd exponents would then reintroduce the Sq(0, 1)h_{1,0}^{−1}h_{1,1}^{−1} that we wanted to kill.

To see this let us add

γ = ϕ(b)h_{1,0}^{−1}h_{1,1}^{−1}h_{2,0}

to KD, where b as above is the exponent of h_{2,0} in m_{univ} and ϕ : ℤ → ℤ_{2} a function to be determined. Let’s call the new differential d^{(2)}. d^{(2)}d^{(2)} is given on m_{univ} by formal multiplication with

KD ∘ KD + KD ∘ γ + γ ∘ KD + γ ∘ γ.

We computed KD ∘ KD earlier and can ignore the “quadratic” term γ ∘ γ since it will give a shift of at least 2 in the May filtration and we are concerned with May shifts of 1. For the remaining terms we get

KD ∘ γ = ϕ(b)Sq(1)h_{1,0}^{−2}h_{1,1}^{−1}h_{2,0}
+ ϕ(b)Sq(0, 1)h_{1,0}^{−1}h_{1,1}^{−1}
+ ϕ(b)Sq(2)h_{1,1}^{−2}h_{1,0}^{−1}h_{2,0}

and

γ ∘ KD = ϕ(b)Sq(1)h_{1,0}^{−2}h_{1,1}^{−1}h_{2,0}
+ ϕ(b − 1)Sq(0, 1)h_{1,0}^{−1}h_{1,1}^{−1}
+ ϕ(b)Sq(2)h_{1,1}^{−2}h_{1,0}^{−1}h_{2,0}.

So we see that ϕ(b) should satisfy ϕ(b) + ϕ(b − 1) = 1 for all b if we want to kill Sq(0, 1)h_{1,0}^{−1}h_{1,1}^{−1}. We take ϕ(2n) = 1, ϕ(2n + 1) = 0 and write this as

ϕ(b) = (b : \overline{10}).

In this notation (f : \overline{11001010}), for example, would denote the periodic ℤ_{2}-valued function of f ∈ ℤ which is one for f congruent 0, 1, 4 or 6 modulo 8 and zero otherwise.

Having fixed ϕ(b) = (b : \overline{10}) we compute γ ∘ γ:

γ ∘ γ = γ ∘ (b : \overline{10})h_{1,0}^{−1}h_{1,1}^{−1}h_{2,0}
= (b + 1 : \overline{10})(b : \overline{10})h_{1,0}^{−2}h_{1,1}^{−2}h_{2,0}^{2}
= (b : \overline{01})(b : \overline{10})h_{1,0}^{−2}h_{1,1}^{−2}h_{2,0}^{2}
= 0.

Taken all the partial results together we get

d^{(2)}d^{(2)}(m_{univ}) = Sq(1, 1)h_{1,1}^{−2}.

To kill this term we might want to rely on the Koszul differential h_{2,0} ↦ Sq(0, 1) again since Sq(1, 1) = Sq(1)Sq(0, 1). So we introduce a new correction term

δ = ψ(b)Sq(1)h_{1,1}^{−2}h_{2,0}.

A similar computation as above suggests that we should require ψ(b) to satisfy ψ(b) + ψ(b − 1) = 1 again, so we take ψ(b) = (b : \overline{10}) and let d^{(3)} be given on m_{univ} by formal multiplication with

KD + γ + δ.

A straightforward computation then gives that d^{(3)}d^{(3)} is given by formal multiplication with

(b : \overline{10})Sq(0, 1)h_{1,1}^{3}h_{2,0}.

So we have been succesfull in killing the offending term of May filtration shift 2, but got a new one of May shift 3.

To kill this one we again rely on the Koszul differential h_{2,0} ↦ Sq(0, 1). So we introduce the new term

κ(b)h_{1,1}^{−3}h_{2,0}^{2}

with κ : ℤ → ℤ_{2}. Our experience above suggests that we should require κ to fulfill

κ(b) + κ(b − 1) = (b : \overline{10}).

So we take κ(b) = (b : \overline{1100}).

(In general, if we encounter a term of the form ω(a, b, c, …) x h_{i_{1},j_{1}}^{k_{1}} h_{i_{2},j_{2}}^{k_{2}} ⋯, x ∈ A that we wish to kill by relying on h_{i,j} ↦ P_{i}^{j} where the exponent of h_{i,j} is, say, d, then we will choose a function \overline{ω}(a, b, c, …) with

\overline{ω}(a, b, c, d, e, …) − \overline{ω}(a, b, c, d − 1, e, …) = ω(a, b, c, d, …)

to do so. This is justified by experience, not by a rigorous proof.)

The new d^{(4)} is given on m_{univ} by formal multiplication with

Sq(1)h_{1,0}^{−1}
+ Sq(0, 1)h_{2,0}^{−1}
+ Sq(2)h_{1,1}^{−1}
+ (b : \overline{10})h_{1,0}^{−1}h_{1,0}^{−1}h_{2,0}
+ (b : \overline{10})Sq(1)h_{1,1}^{−2}h_{2,0}
+ (b : \overline{1100})h_{1,1}^{−3}h_{2,0}^{2}.

A straightforward but tedious computation now shows that d^{(4)}d^{(4)} = 0 so we finally got our May resolution, at last.

It has taken me about an hour to write up the above computation, even though I cheated in the last two computations, since I didn’t do them myself, but relied on the computations of the computer instead. The computer finds (a slightly different) May resolution in no time at all as the following sample output shows:

     Mon Jun 03 18:19:43 1996
     Under consideration:

h_{1,1}^{a} h_{1,0}^{b} h_{2,0}^{c} h_{0,0}^{?}

     Lowest Filtration = 1.  List follows
     1 terms
     Integrating h_{2,0}. New terms:

(c : \overline{10}) h_{1,1}^{−1} h_{1,0}^{−1} h_{2,0}

     1 terms.
     Lowest filtration = 2
     Lowest Filtration = 2.  List follows
     1 terms
     Integrating h_{1,0}. New terms:

(b : \overline{10}) Sq(0,1) h_{1,1}^{−2} h_{1,0}

     1 terms.
     Lowest filtration = 3
     Lowest Filtration = 3.  List follows
     1 terms
     Integrating h_{2,0}. New terms:

(c : \overline{1100}) h_{1,1}^{−3} h_{2,0}^{2}

     1 terms.
     Lowest filtration = 1111111
     Differential:

Sq(2) h_{1,1}^{−1} [0] Sq(1) h_{1,0}^{−1} [0] Sq(0,1) h_{2,0}^{−1} [0] (c : \overline{10}) h_{1,1}^{−1} h_{1,0}^{−1} h_{2,0} [1] (b : \overline{10}) Sq(0,1) h_{1,1}^{−2} h_{1,0} [2] (c : \overline{1100}) h_{1,1}^{−3} h_{2,0}^{2} [3]

     Under consideration:

h_{1,1}^{a} h_{1,0}^{b} h_{2,0}^{c} h_{0,0}^{?}

     Mon Jun 03 18:19:43 1996

A few comments are in order. Firstly, note that the exponents are labelled differently, so our b is the machine’s c. Note again that the machine relied on the Koszul differential h_{1,0} ↦ Sq(1) to kill Sq(1, 1)h_{1,1}^{−2}, not on h_{2,0} ↦ Sq(0, 1) which is what we used. This explains why the machine found (b : \overline{10})Sq(0, 1)h_{1,1}^{−2}h_{1,0} instead of our (c : \overline{10})Sq(1)h_{1,1}^{−2}h_{2,0}.

The numbers in square brackets behind each term give the shift in the May filtration. The phrase “integrating h_{i,j}” is what we described as “relying on the Koszul differential h_{i,j} ↦ P_{i}^{j}”. Finally, the “List follows” that litter this computation are reminiscent of previous versions of the program which included a listing of all terms of lowest May filtration at each stage of the computation, and the h_{0,0}^{?} is just an error in the routine that lists the relevant h_{i,j}.

Now that we have a May resolution — but indexed on monomials h_{1,0}^{a}h_{2,0}^{b}h_{1,1}^{c} for a, b, c ∈ ℤ, not just a, b, c ∈ ℕ_{0} — we can go on and try to compute Ext_{A(1)}(ℤ_{2}, ℤ_{2}).

(In general it might be necessary at this point to adjust the exponents — by adding a constant — so that the edge cases are properly taken care of. Fortunately, we don’t need to do this in this case.)

Let \overline{M} = Hom_{A(1)}(\underline{M}, ℤ_{2}) with differential δ induced by d^{(4)}. \overline{M} is the ℤ_{2} vector space on the set of monomials in r_{1,0}, r_{2,0} and r_{1,1} dual to h_{1,0}, h_{2,0} and h_{1,1} (which regretably conflicts with standard notation). From the explicit description of d^{(4)} we get

δ(r_{1,0}^{a}r_{2,0}^{b}r_{1,1}^{c}) = (b : \overline{01})r_{1,0}^{a+1}r_{2,0}^{b−1}r_{1,1}^{c+1} + (b : \overline{0011})r_{1,0}^{a}r_{2,0}^{b−2}r_{1,1}^{c+3}.

The first term gives the well known May d_{1} differential d_{1}(r_{2,0}) = r_{1,0}r_{1,1}, and the second the equally well known d_{2}(r_{2,0}^{2}) = r_{1,1}^{3} (compare [Rav], 3.1.21 and 3.1.23). From these we can conclude that H^{∗}(\overline{M}, δ) is as it should be.

In general, however, this last step of computing Ext_{A′}(ℤ_{2}, ℤ_{2}) from a given May resolution \underline{M} will be too difficult to be done by hand calculations (at least as far as my hands are concerned). So there really should be a second program that computes the May E_{∞} term H^{∗}(\overline{M}, δ) from the May resolution \underline{M}, along with hidden multiplicative extensions. I have not set up such a thing, nor even thought about the details of how this could work, since only the A′ with profile function π satisfying π(t) ≤ 3 for all t could be handled succesfully by the program.

Before giving a few amusing but useless samples of May resolutions in the next section, let me finally remark that there is a technical difficulty to the algorithm sketched above, at least the way I implemented it. The program, as it stands, computes dd(m_{univ}), collects the terms of lowest May filtration therein, and then tries to kill one term after the other, computing dd(m_{univ}) again after each modification of d. The order in which offending terms are attacked is partly prescribed by an ordering of the h_{i,j}, but there still remains quite a choice. In previous versions it happened that the program ran into loops, producing the same offending term after a few iterations that it tried to kill earlier. After that happened I introduced a new parameter, “descent”, that is used to sort the terms of lowest May filtration. For m = h_{i_{1},j_{1}}^{k_{1}}h_{i_{2},j_{2}}^{k_{2}} ⋯ h_{i_{n},j_{n}}^{k_{n}} its decent is the sum of all negative exponents k_{i}, 1 ≤ i ≤ n. The idea is that in honest May resolutions with no monomials with negative exponents, contributions to the differential with larger descent will be found before those with smaller descent. After taking descent into account these loops never happened again, though I’m not certain that they can’t. “The program runs, so why bother.”

3 Samples

The following is the input file that configures the program to look for a May resolution of A(2), the subalgebra of A generated by Sq(1), Sq(2) and Sq(4).

     INTRO
     d : h_{1,1},
     e : h_{2,1},
     g : h_{1,0},
     h : h_{2,0},
     i : h_{3,0},
     b : h_{1,2}
     END;

TEXOPEN;
INFO;
DIFF Sq(2) h_{1,1}^{-1} ;
DIFF Sq(0,2) h_{2,1}^{-1} ;
DIFF Sq(4) h_{1,2}^{-1} ;
DIFF Sq(1) h_{1,0}^{-1} ;
DIFF Sq(0,1) h_{2,0}^{-1} ;
DIFF Sq(0,0,1) h_{3,0}^{-1} ;
BE NICE;
INFO;
TEXCLOSE;

The INTRO ... END; part declares the h_{i,j} present in A(2) and establishes names for the exponents. INFO causes these exponents to be shown in the output file. The various DIFF ... ; give the initial Koszul differentials. TEXOPEN and TEXCLOSE produce the necessary \begin{document} and \end{document} in the outputfile out.tex. And BE NICE means “Do all the work on your own and don’t bother me for advice!”. (Before BE NICE was available, most of the computation was done interactively.)

The following is (part of) the output:

Mon Jun 03 18:01:49 1996
Under consideration:

h_{1,1}^{d} h_{2,1}^{e} h_{1,0}^{g} h_{2,0}^{h} h_{3,0}^{i} h_{1,2}^{b} h_{0,0}^{?}

Lowest Filtration = 1.  List follows
4 terms
Integrating h_{2,1}. New terms:

(e : \overline{10}) h_{1,1}^{−1} h_{2,1} h_{1,2}^{−1}

1 terms.
Lowest filtration = 1
Lowest Filtration = 1.  List follows
3 terms
Integrating h_{2,0}. New terms:

(h : \overline{10}) h_{1,1}^{−1} h_{1,0}^{−1} h_{2,0}

1 terms.
Lowest filtration = 1
Lowest Filtration = 1.  List follows
2 terms
Integrating h_{3,0}. New terms:

(i : \overline{10}) h_{2,0}^{−1} h_{3,0} h_{1,2}^{−1}

⋮

Here I spare you 9 pages of similar nonsense

⋮

1 terms.
Lowest filtration = 12

Lowest Filtration = 12.  List follows
1 terms
Integrating h_{2,0}. New terms:

((h : \overline{1110})(i : \overline{0011}) + (h : \overline{0010})(i : \overline{1100})) Sq(0,0,1) h_{2,1}^{−3} h_{2,0}^{3} h_{3,0}^{2} h_{1,2}^{−3}

1 terms.
Lowest filtration = 12
Lowest Filtration = 12.  List follows
1 terms
Integrating h_{2,0}. New terms:

(h : \overline{10})(i : \overline{0011}) Sq(0,0,1) h_{2,1}^{−3} h_{2,0}^{3} h_{3,0}^{2} h_{1,2}^{−3}

1 terms.
Lowest filtration = 12
Lowest Filtration = 12.  List follows
1 terms
Integrating h_{2,0}. New terms:

(h : \overline{10}) Sq(0,0,1) h_{2,1}^{−3} h_{2,0}^{3} h_{3,0}^{2} h_{1,2}^{−3}

1 terms.
Lowest filtration = 1111111
Differential:

Sq(2) h_{1,1}^{−1} [0] Sq(0,2) h_{2,1}^{−1} [0] Sq(4) h_{1,2}^{−1} [0] Sq(1) h_{1,0}^{−1} [0] Sq(0,1) h_{2,0}^{−1} [0] Sq(0,0,1) h_{3,0}^{−1} [0] (e : \overline{10}) h_{1,1}^{−1} h_{2,1} h_{1,2}^{−1} [1] (h : \overline{10}) h_{1,1}^{−1} h_{1,0}^{−1} h_{2,0} [1] (i : \overline{10}) h_{2,0}^{−1} h_{3,0} h_{1,2}^{−1} [1] (i : \overline{10}) h_{2,1}^{−1} h_{1,0}^{−1} h_{3,0} [1] (d : \overline{10}) Sq(0,2) h_{1,1} h_{1,2}^{−2} [2] (d : \overline{10}) Sq(0,1) h_{1,1} h_{1,0}^{−1} h_{1,2}^{−1} [2] (g : \overline{10}) Sq(0,1) h_{1,1}^{−2} h_{1,0} [2] (g : \overline{10}) Sq(0,0,1) h_{1,1}^{−1} h_{2,1}^{−1} h_{1,0} [2] (d : \overline{10}) Sq(1,0,1) h_{1,1} h_{2,1}^{−1} h_{1,2}^{−1} [3] (e : \overline{10})(h : \overline{10}) h_{2,1} h_{1,0}^{−1} h_{2,0} h_{1,2}^{−2} [3] (e : \overline{1100}) h_{2,1}^{2} h_{1,2}^{−3} [3] (h : \overline{10})(i : \overline{10}) h_{1,1}^{−2} h_{2,1}^{−1} h_{2,0} h_{3,0} [3] (h : \overline{1100}) h_{1,0}^{−2} h_{2,0}^{2} h_{1,2}^{−1} [3] (h : \overline{1100}) h_{1,1}^{−3} h_{2,0}^{2} [3] (i : \overline{1100}) h_{1,1}^{−1} h_{2,1}^{−2} h_{3,0}^{2} [3] (e : \overline{01})(g : \overline{10}) Sq(0,0,1) h_{1,0} h_{1,2}^{−2} [4] (g : \overline{01})(h : \overline{10}) Sq(0,0,1) h_{2,1}^{−1} h_{2,0} h_{1,2}^{−1} [4] (d : \overline{1100}) Sq(1,0,1) h_{1,1}^{2} h_{1,2}^{−3} [5] (d : \overline{1100}) Sq(0,1,1) h_{1,1}^{2} h_{2,1}^{−1} h_{1,2}^{−2} [5] (e : \overline{10})(h : \overline{1100}) h_{1,1}^{−2} h_{2,1} h_{2,0}^{2} h_{1,2}^{−2} [5] (e : \overline{01})(i : \overline{1100}) h_{2,1}^{−1} h_{3,0}^{2} h_{1,2}^{−2} [5] (h : \overline{1000}) h_{1,1}^{−2} h_{1,0}^{−1} h_{2,0}^{3} h_{1,2}^{−1} [5] (h : \overline{10})(i : \overline{1100}) h_{2,1}^{−2} h_{1,0}^{−1} h_{2,0} h_{3,0}^{2} h_{1,2}^{−1} [5] (g : \overline{10})(h : \overline{1100}) Sq(0,0,1) h_{1,1}^{−2} h_{2,1}^{−1} h_{1,0} h_{2,0}^{2} h_{1,2}^{−1} [6] (g : \overline{10})(i : \overline{1100}) Sq(0,0,1) h_{2,1}^{−3} h_{1,0} h_{3,0}^{2} h_{1,2}^{−1} [6] (d : \overline{1000}) Sq(0,1,1) h_{1,1}^{3} h_{1,2}^{−4} [7] (g : \overline{1100}) Sq(0,1,1) h_{1,0}^{2} h_{1,2}^{−3} [7] (h : \overline{1100})(i : \overline{1100}) h_{1,1}^{−2} h_{2,1}^{−2} h_{2,0}^{2} h_{3,0}^{2} h_{1,2}^{−1} [7] (h : \overline{11110000}) h_{1,1}^{−4} h_{2,0}^{4} h_{1,2}^{−1} [7] (i : \overline{11110000}) h_{2,1}^{−4} h_{3,0}^{4} h_{1,2}^{−1} [7] (g : \overline{10})(h : \overline{1100}) Sq(0,0,1) h_{1,1}^{−1} h_{1,0} h_{2,0}^{2} h_{1,2}^{−3} [8] (g : \overline{10})(h : \overline{10})(i : \overline{10}) Sq(0,0,1) h_{2,1}^{−1} h_{1,0} h_{2,0} h_{3,0} h_{1,2}^{−3} [8] (h : \overline{1100})(i : \overline{10}) Sq(0,0,1) h_{2,1}^{−2} h_{2,0}^{2} h_{3,0} h_{1,2}^{−2} [8] (h : \overline{1000}) Sq(0,0,1) h_{1,1}^{−1} h_{2,1}^{−1} h_{2,0}^{3} h_{1,2}^{−2} [8] (e : \overline{10})(g : \overline{10})(h : \overline{1100}) Sq(0,0,1) h_{2,1} h_{1,0} h_{2,0}^{2} h_{1,2}^{−5} [10] ((g : \overline{01})(h : \overline{1000}) + (e : \overline{01})(h : \overline{1000})) Sq(0,0,1) h_{2,0}^{3} h_{1,2}^{−4} [10] (g : \overline{10})(h : \overline{1100})(i : \overline{1100}) Sq(0,0,1) h_{2,1}^{−2} h_{1,0} h_{2,0}^{2} h_{3,0}^{2} h_{1,2}^{−4} [12] (h : \overline{1000})(i : \overline{1100}) Sq(0,0,1) h_{2,1}^{−3} h_{2,0}^{3} h_{3,0}^{2} h_{1,2}^{−3} [12]

Under consideration:

h_{1,1}^{d} h_{2,1}^{e} h_{1,0}^{g} h_{2,0}^{h} h_{3,0}^{i} h_{1,2}^{b} h_{0,0}^{?}

Mon Jun 03 18:01:51 1996

It’s terrible to realize that the program that took me months to write and test out, completed one of the most difficult computations that it’s capable of in no more than 3 seconds! And since Ext_{A(2)}(ℤ_{2}, ℤ_{2}) has been computed almost 30 years ago in [ShI], the result is not even new. (Ext_{A(2)}(ℤ_{2}, ℤ_{2}) is quite messy, and it seems not worthwhile to recompute it from this May resolution.) The last example I shall give is part of a May resolution of the sub Hopf algebra A′ with profile function (3, 3, 2, 1, 0, …), that is “A(3) without Sq(8)”. It took 5\frac{1}{2} minutes to complete. I give only the terms that contribute to the differential δ on \overline{M}, the full resolution is about 2\frac{1}{2} times as long.

Sun Jun 09 14:29:45 1996
Under consideration:

h_{2,2}^{a} h_{1,2}^{b} h_{3,1}^{c} h_{2,1}^{d} h_{1,1}^{e} h_{4,0}^{f} h_{3,0}^{g} h_{2,0}^{h} h_{1,0}^{i} h_{0,0}^{?}

Differential:

(c : \overline{10}) h_{2,2}^{−1} h_{3,1} h_{1,1}^{−1} [1] (d : \overline{10}) h_{1,2}^{−1} h_{2,1} h_{1,1}^{−1} [1] (f : \overline{10}) h_{3,1}^{−1} h_{4,0} h_{1,0}^{−1} [1] (f : \overline{10}) h_{2,2}^{−1} h_{4,0} h_{2,0}^{−1} [1] (g : \overline{10}) h_{1,2}^{−1} h_{3,0} h_{2,0}^{−1} [1] (g : \overline{10}) h_{2,1}^{−1} h_{3,0} h_{1,0}^{−1} [1] (h : \overline{10}) h_{1,1}^{−1} h_{2,0} h_{1,0}^{−1} [1] (c : \overline{1100}) h_{2,2}^{−2} h_{1,2}^{−1} h_{3,1}^{2} [3] (c : \overline{10})(d : \overline{10}) h_{2,2}^{−1} h_{1,2}^{−2} h_{3,1} h_{2,1} [3] (d : \overline{1100}) h_{1,2}^{−3} h_{2,1}^{2} [3] (e : \overline{10})(f : \overline{10}) h_{2,2}^{−1} h_{1,2}^{−1} h_{1,1} h_{4,0} h_{1,0}^{−1} [3] (e : \overline{10})(g : \overline{10}) h_{1,2}^{−2} h_{1,1} h_{3,0} h_{1,0}^{−1} [3] (f : \overline{1100}) h_{3,1}^{−2} h_{1,1}^{−1} h_{4,0}^{2} [3] (f : \overline{10})(g : \overline{10}) h_{3,1}^{−1} h_{2,1}^{−1} h_{1,1}^{−1} h_{4,0} h_{3,0} [3] (f : \overline{10})(h : \overline{10}) h_{3,1}^{−1} h_{1,1}^{−2} h_{4,0} h_{2,0} [3]

(g : \overline{1100}) h_{2,1}^{−2} h_{1,1}^{−1} h_{3,0}^{2} [3]

(g : \overline{10})(h : \overline{10}) h_{2,1}^{−1} h_{1,1}^{−2} h_{3,0} h_{2,0} [3]

(h : \overline{1100}) h_{1,2}^{−1} h_{2,0}^{2} h_{1,0}^{−2} [3]

(h : \overline{1100}) h_{1,1}^{−3} h_{2,0}^{2} [3]

(e : \overline{10})(g : \overline{10})(h : \overline{10}) h_{1,2}^{−2} h_{1,1}^{−1} h_{3,0} h_{2,0} [5]

(e : \overline{01})(f : \overline{10})(g : \overline{10}) h_{2,2}^{−1} h_{1,2}^{−1} h_{2,1}^{−1} h_{4,0} h_{3,0} [5]

(e : \overline{10})(f : \overline{10})(g : \overline{10}) h_{1,2}^{−2} h_{3,1}^{−1} h_{4,0} h_{3,0} [5]

(e : \overline{10})(f : \overline{10})(h : \overline{10}) h_{2,2}^{−1} h_{1,2}^{−1} h_{1,1}^{−1} h_{4,0} h_{2,0} [5]

(f : \overline{1100}) h_{2,2}^{−2} h_{2,1}^{−1} h_{4,0}^{2} [5]

(f : \overline{1100})(h : \overline{10}) h_{1,2}^{−1} h_{3,1}^{−2} h_{4,0}^{2} h_{2,0} h_{1,0}^{−1} [5]

(f : \overline{10})(g : \overline{10})(h : \overline{10}) h_{1,2}^{−1} h_{3,1}^{−1} h_{2,1}^{−1} h_{4,0} h_{3,0} h_{2,0} h_{1,0}^{−1} [5]

(g : \overline{1100}) h_{2,2}^{−1} h_{3,0}^{2} h_{1,0}^{−2} [5]

(g : \overline{1100}) h_{1,2}^{−2} h_{2,1}^{−1} h_{3,0}^{2} [5]

(g : \overline{1100})(h : \overline{10}) h_{1,2}^{−1} h_{2,1}^{−2} h_{3,0}^{2} h_{2,0} h_{1,0}^{−1} [5]

(h : \overline{1000}) h_{1,2}^{−1} h_{1,1}^{−2} h_{2,0}^{3} h_{1,0}^{−1} [5]

(e : \overline{10})(f : \overline{10})(g : \overline{10})(h : \overline{10}) h_{2,2}^{−1} h_{1,2}^{−2} h_{2,1}^{−1} h_{1,1} h_{4,0} h_{3,0} h_{2,0} h_{1,0}^{−1} [7]

(e : \overline{10})(f : \overline{10})(g : \overline{10}) h_{2,2}^{−1} h_{1,2}^{−3} h_{1,1} h_{4,0} h_{3,0} [7]

(e : \overline{10})(f : \overline{10})(g : \overline{10})(h : \overline{10}) h_{1,2}^{−3} h_{3,1}^{−1} h_{1,1} h_{4,0} h_{3,0} h_{2,0} h_{1,0}^{−1} [7]

(e : \overline{10})(f : \overline{1100}) h_{2,2}^{−2} h_{1,2}^{−2} h_{1,1} h_{4,0}^{2} [7]

(e : \overline{10})(g : \overline{1100}) h_{1,2}^{−4} h_{1,1} h_{3,0}^{2} [7]

(f : \overline{11110000}) h_{1,2}^{−1} h_{3,1}^{−4} h_{4,0}^{4} [7]

(f : \overline{1000})(g : \overline{10}) h_{1,2}^{−1} h_{3,1}^{−3} h_{2,1}^{−1} h_{4,0}^{3} h_{3,0} [7]

(f : \overline{1100})(g : \overline{1100}) h_{1,2}^{−1} h_{3,1}^{−2} h_{2,1}^{−2} h_{4,0}^{2} h_{3,0}^{2} [7]

(f : \overline{1100})(h : \overline{1100}) h_{1,2}^{−1} h_{3,1}^{−2} h_{1,1}^{−2} h_{4,0}^{2} h_{2,0}^{2} [7]

(f : \overline{10})(g : \overline{1000}) h_{1,2}^{−1} h_{3,1}^{−1} h_{2,1}^{−3} h_{4,0} h_{3,0}^{3} [7]

(f : \overline{10})(g : \overline{10})(h : \overline{1100}) h_{1,2}^{−1} h_{3,1}^{−1} h_{2,1}^{−1} h_{1,1}^{−2} h_{4,0} h_{3,0} h_{2,0}^{2} [7]

(g : \overline{11110000}) h_{1,2}^{−1} h_{2,1}^{−4} h_{3,0}^{4} [7]

(g : \overline{1100})(h : \overline{1100}) h_{1,2}^{−1} h_{2,1}^{−2} h_{1,1}^{−2} h_{3,0}^{2} h_{2,0}^{2} [7]

(h : \overline{11110000}) h_{1,2}^{−1} h_{1,1}^{−4} h_{2,0}^{4} [7]

(e : \overline{10})(f : \overline{1000})(g : \overline{10}) h_{1,2}^{−3} h_{3,1}^{−3} h_{1,1} h_{4,0}^{3} h_{3,0} [9]

(e : \overline{10})(f : \overline{1000})(g : \overline{10}) h_{2,2}^{−1} h_{1,2}^{−2} h_{3,1}^{−2} h_{2,1}^{−1} h_{1,1} h_{4,0}^{3} h_{3,0} [9]

(e : \overline{10})(f : \overline{10})(g : \overline{1000}) h_{2,2}^{−1} h_{1,2}^{−2} h_{2,1}^{−3} h_{1,1} h_{4,0} h_{3,0}^{3} [9]

(e : \overline{10})(f : \overline{10})(g : \overline{1000}) h_{1,2}^{−3} h_{3,1}^{−1} h_{2,1}^{−2} h_{1,1} h_{4,0} h_{3,0}^{3} [9]

(e : \overline{01})(f : \overline{10})(g : \overline{10})(h : \overline{1100}) h_{2,2}^{−1} h_{1,2}^{−2} h_{2,1}^{−1} h_{1,1}^{−1} h_{4,0} h_{3,0} h_{2,0}^{2} [9]

(e : \overline{01})(f : \overline{10})(g : \overline{10})(h : \overline{1100}) h_{1,2}^{−3} h_{3,1}^{−1} h_{1,1}^{−1} h_{4,0} h_{3,0} h_{2,0}^{2} [9]

(f : \overline{1100})(g : \overline{10}) h_{2,2}^{−3} h_{4,0}^{2} h_{3,0} h_{1,0}^{−1} [9]

(g : \overline{1000}) h_{2,2}^{−1} h_{1,2}^{−2} h_{3,0}^{3} h_{1,0}^{−1} [9]

(f : \overline{1100})(g : \overline{10})(h : \overline{10}) h_{2,2}^{−3} h_{1,1}^{−2} h_{4,0}^{2} h_{3,0} h_{2,0} [11]

(f : \overline{1000})(g : \overline{10}) h_{2,2}^{−3} h_{3,1}^{−1} h_{1,1}^{−1} h_{4,0}^{3} h_{3,0} [11]

(f : \overline{10})(g : \overline{10})(h : \overline{1000}) h_{1,2}^{−4} h_{3,1}^{−1} h_{4,0} h_{3,0} h_{2,0}^{3} h_{1,0}^{−1} [11]

(f : \overline{10})(g : \overline{10})(h : \overline{1000}) h_{2,2}^{−1} h_{1,2}^{−3} h_{2,1}^{−1} h_{4,0} h_{3,0} h_{2,0}^{3} h_{1,0}^{−1} [11]

(f : \overline{10})(g : \overline{1000}) h_{2,2}^{−2} h_{1,2}^{−1} h_{2,1}^{−1} h_{1,1}^{−1} h_{4,0} h_{3,0}^{3} [11]

(g : \overline{1000})(h : \overline{10}) h_{2,2}^{−1} h_{1,2}^{−2} h_{1,1}^{−2} h_{3,0}^{3} h_{2,0} [11]

(e : \overline{01})(f : \overline{1000})(g : \overline{10}) h_{2,2}^{−4} h_{1,2}^{−1} h_{4,0}^{3} h_{3,0} [13]

(e : \overline{01})(f : \overline{10})(g : \overline{1000}) h_{2,2}^{−2} h_{1,2}^{−3} h_{4,0} h_{3,0}^{3} [13] (f : \overline{1000})(g : \overline{10})(h : \overline{10}) h_{2,2}^{−3} h_{1,2}^{−1} h_{3,1}^{−1} h_{4,0}^{3} h_{3,0} h_{2,0} h_{1,0}^{−1} [13] (f : \overline{11110000}) h_{2,2}^{−5} h_{4,0}^{4} [13] (f : \overline{10})(g : \overline{1000})(h : \overline{1100}) h_{2,2}^{−1} h_{1,2}^{−3} h_{2,1}^{−3} h_{4,0} h_{3,0}^{3} h_{2,0}^{2} [13] (f : \overline{10})(g : \overline{1000})(h : \overline{1100}) h_{1,2}^{−4} h_{3,1}^{−1} h_{2,1}^{−2} h_{4,0} h_{3,0}^{3} h_{2,0}^{2} [13] (f : \overline{1000})(g : \overline{10})(h : \overline{1100}) h_{2,2}^{−1} h_{1,2}^{−3} h_{3,1}^{−2} h_{2,1}^{−1} h_{4,0}^{3} h_{3,0} h_{2,0}^{2} [13] (f : \overline{1000})(g : \overline{10})(h : \overline{1100}) h_{1,2}^{−4} h_{3,1}^{−3} h_{4,0}^{3} h_{3,0} h_{2,0}^{2} [13] (f : \overline{10})(g : \overline{1000})(h : \overline{10}) h_{2,2}^{−2} h_{1,2}^{−2} h_{2,1}^{−1} h_{4,0} h_{3,0}^{3} h_{2,0} h_{1,0}^{−1} [13] (f : \overline{1100})(g : \overline{1100}) h_{2,2}^{−3} h_{1,2}^{−2} h_{4,0}^{2} h_{3,0}^{2} [13] (g : \overline{11110000}) h_{2,2}^{−1} h_{1,2}^{−4} h_{3,0}^{4} [13] (e : \overline{10})(f : \overline{1000})(g : \overline{10})(h : \overline{10}) h_{2,2}^{−4} h_{1,2}^{−2} h_{1,1} h_{4,0}^{3} h_{3,0} h_{2,0} h_{1,0}^{−1} [15] (e : \overline{10})(f : \overline{10})(g : \overline{1000})(h : \overline{10}) h_{2,2}^{−2} h_{1,2}^{−4} h_{1,1} h_{4,0} h_{3,0}^{3} h_{2,0} h_{1,0}^{−1} [15] (f : \overline{10})(g : \overline{1000})(h : \overline{1100}) h_{2,2}^{−2} h_{1,2}^{−2} h_{2,1}^{−1} h_{1,1}^{−2} h_{4,0} h_{3,0}^{3} h_{2,0}^{2} [15] (f : \overline{1000})(g : \overline{1000}) h_{2,2}^{−2} h_{1,2}^{−2} h_{3,1}^{−2} h_{2,1}^{−1} h_{4,0}^{3} h_{3,0}^{3} [15] (f : \overline{1000})(g : \overline{10})(h : \overline{1100}) h_{2,2}^{−3} h_{1,2}^{−1} h_{3,1}^{−1} h_{1,1}^{−2} h_{4,0}^{3} h_{3,0} h_{2,0}^{2} [15] (f : \overline{1000})(g : \overline{1000}) h_{2,2}^{−3} h_{1,2}^{−1} h_{3,1}^{−1} h_{2,1}^{−2} h_{4,0}^{3} h_{3,0}^{3} [15] (e : \overline{10})(f : \overline{1000})(g : \overline{1000}) h_{2,2}^{−4} h_{1,2}^{−2} h_{2,1}^{−2} h_{1,1} h_{4,0}^{3} h_{3,0}^{3} [17] (e : \overline{10})(f : \overline{1000})(g : \overline{1000}) h_{2,2}^{−2} h_{1,2}^{−4} h_{3,1}^{−2} h_{1,1} h_{4,0}^{3} h_{3,0}^{3} [17] (e : \overline{01})(f : \overline{10})(g : \overline{1000})(h : \overline{1100}) h_{2,2}^{−2} h_{1,2}^{−4} h_{1,1}^{−1} h_{4,0} h_{3,0}^{3} h_{2,0}^{2} [17] (e : \overline{01})(f : \overline{1000})(g : \overline{10})(h : \overline{1100}) h_{2,2}^{−4} h_{1,2}^{−2} h_{1,1}^{−1} h_{4,0}^{3} h_{3,0} h_{2,0}^{2} [17] (f : \overline{1000})(g : \overline{10})(h : \overline{1000}) h_{2,2}^{−4} h_{1,2}^{−3} h_{4,0}^{3} h_{3,0} h_{2,0}^{3} h_{1,0}^{−1} [19] (f : \overline{10})(g : \overline{10100000}) h_{2,2}^{−2} h_{1,2}^{−6} h_{2,1}^{−1} h_{1,1}^{2} h_{4,0} h_{3,0}^{5} [19] (f : \overline{1000})(g : \overline{1000}) h_{2,2}^{−5} h_{1,2}^{−1} h_{1,1}^{−1} h_{4,0}^{3} h_{3,0}^{3} [19] (f : \overline{10})(g : \overline{1000})(h : \overline{1000}) h_{2,2}^{−2} h_{1,2}^{−5} h_{4,0} h_{3,0}^{3} h_{2,0}^{3} h_{1,0}^{−1} [19] (f : \overline{10})(g : \overline{10100000}) h_{2,2}^{−1} h_{1,2}^{−7} h_{3,1}^{−1} h_{1,1}^{2} h_{4,0} h_{3,0}^{5} [19] (f : \overline{1000})(g : \overline{1000})(h : \overline{1100}) h_{2,2}^{−2} h_{1,2}^{−5} h_{3,1}^{−2} h_{4,0}^{3} h_{3,0}^{3} h_{2,0}^{2} [21] (f : \overline{1000})(g : \overline{1000})(h : \overline{1100}) h_{2,2}^{−4} h_{1,2}^{−3} h_{2,1}^{−2} h_{4,0}^{3} h_{3,0}^{3} h_{2,0}^{2} [21] (f : \overline{1000})(g : \overline{1000})(h : \overline{10}) h_{2,2}^{−5} h_{1,2}^{−2} h_{4,0}^{3} h_{3,0}^{3} h_{2,0} h_{1,0}^{−1} [21] (f : \overline{1000})(g : \overline{1000})(h : \overline{1100}) h_{2,2}^{−5} h_{1,2}^{−2} h_{1,1}^{−2} h_{4,0}^{3} h_{3,0}^{3} h_{2,0}^{2} [23] (f : \overline{10})(g : \overline{10000000}) h_{2,2}^{−3} h_{1,2}^{−8} h_{1,1}^{2} h_{4,0} h_{3,0}^{7} [27] (f : \overline{1000})(g : \overline{10100000}) h_{2,2}^{−5} h_{1,2}^{−6} h_{1,1}^{2} h_{4,0}^{3} h_{3,0}^{5} [27]

Sun Jun 09 14:35:15 1996

A(3) would be the natural next step, and it’s here that the computation becomes too involved: after two days the program runs out of memory, leaving behind lots of pages of useless TEX-output…

4 Concluding remarks / Sins of omission

The greatest appeal of the method used above — apart from giving the entire resolution, not just a part of it — is, that one does not have to compute kernels and images to prove exactness of the resolution; the enormous size of the matrices involved, for example, in the computation of a minimal resolution of ℤ_{2} over A being what limits the range of those computations. Here, this is replaced by application of a Koszul null homotopy, which can be implemented easily. If there were other examples of filtrations of the Steenrod algebra, for which the associated graded algebra had such a manageable minimal resolution, this approach would, of course, carry over, but I know of no such filtrations.

One could hope that it might somehow be possible to construct resolutions indexed on a basis of the May E_{2} term, not the E_{1} term, in a similar fashion. It seems, however, that for this one would need at least a practical explicit description of the isomorphism between H^{∗}(E_{1}, d_{1}) and the E_{2} term as given in [Tan]. This would amount to computing May differentials by embedding into a May resolution, not into the bar resolution as in [Tan], and to give the differential in this May resolution on E_{1} representatives of the E_{2} term only. Somehow I can’t help thinking, that it’s more reasonable to hope for RAM memory to become cheaper, and CPUs faster, so that computations with minimal resolutions can go on, instead.

Well, and then there are the sins of omission. In the case of this article, these include everything truly mathematical. It seems to me that one should take symbolical May resolutions seriously, i.e. set up a proper definition, prove that (some modification of) the algorithm implemented always terminates and find a, maybe algebro-geometrical-formal-group-or-invariant-theory-related, interpretation of the coefficent functions used. Note that the termination of that algorithm would imply that the May spectral sequence terminates at some finite E_{r}, as has recently been proved in [BS]. Or one might consider computing with related algebras instead; these could be simpler, hence more accessible. But since I didn’t think about any of these matters seriously until now, I shall say no more.

References

[BS] A. Bajer and H. Sadofsky, Cohomology of finite-dimensional connected cocommutative Hopf algebras, Journ. Pure and Appl. Alg. 94 (1994) 115-126

[Rav] D. C. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Academic Press 1986

[ShI] N. Shimada and A. Iwai, On the cohomology of some Hopf algebras, Nagoya Math. J. 30 (1967) 103-111

[Tan] M. C. Tangora, On the Cohomology of the Steenrod Algebra, Math. Z. 116, 18-64 (1970)

Current Address: C. Nassau, FB Mathematik, Johann Wolfgang Goethe Universität, 60054 Frankfurt, Germany

E-mail address: nassau@math.uni-frankfurt.de