The May Spectral Sequence

Paul Julian Day Wadham College Oxford

A thesis submitted for the degree of Doctor of Philosophy Hilary Term 1994

Contents

Introduction ii

1 Resolutions 1

1.1 Rings, graded rings and filtered rings . . . . . . . . . . . . . . . . . . 1 1.1.1 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Graded rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.3 Filtered rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.4 The functor gr . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.1 Associated graded resolutions . . . . . . . . . . . . . . . . . . 3 1.2.2 Ungrading resolutions . . . . . . . . . . . . . . . . . . . . . . 5 1.2.3 The bar resolution . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.4 The comparison theorem . . . . . . . . . . . . . . . . . . . . 9 1.2.5 Filtered-projective modules . . . . . . . . . . . . . . . . . . . 10

2 The May spectral sequence 11

2.1 Tensor products, Hom, Tor and Ext . . . . . . . . . . . . . . . . . . 11 2.1.1 The graded case . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.1.2 The filtered case . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 The homological spectral sequence . . . . . . . . . . . . . . . . . . . 16 2.2.1 Construction and convergence . . . . . . . . . . . . . . . . . . 16 2.2.2 Naturality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 The cohomological spectral sequence . . . . . . . . . . . . . . . . . . 19 2.4 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.1 Cup products . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.4.2 Yoneda products . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3 The triple graded case for graded rings 24 3.1 The spectral sequence . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Restricted Lie algebras 27 4.1 Polynomial and truncated polynomial algebras . . . . . . . . . . . . 27 4.2 Ordinary Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 The cohomology of restricted Lie algebras . . . . . . . . . . . . . . . 33

4.3.1 Even characteristic . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3.2 Odd characteristic . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Group algebras 37

5.1 The E_{1}-page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.1.1 The E_{1}-page for Hopf algebras . . . . . . . . . . . . . . . . . 37 5.1.2 The E_{1}-page for group algebras . . . . . . . . . . . . . . . . . 38

5.2 Identifying inflations . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 5.2.1 Even characteristic . . . . . . . . . . . . . . . . . . . . . . . . 39 5.2.2 Odd characteristic . . . . . . . . . . . . . . . . . . . . . . . . 41

6 Group algebra examples 42 6.1 Homogeneous presentations . . . . . . . . . . . . . . . . . . . . . . . 42 6.2 Dihedral, semi-dihedral and quaternion groups . . . . . . . . . . . . 43 6.3 Extraspecial 2-groups . . . . . . . . . . . . . . . . . . . . . . . . . . 46 6.4 The commutative case . . . . . . . . . . . . . . . . . . . . . . . . . . 48 6.5 Groups of order p^{3} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

7 Other examples 53 7.1 An example in which F_{0}/F_{−1} is not semisimple . . . . . . . . . . . . 53 7.2 A counterexample to a Theorem of Grünenfelder . . . . . . . . . . . 55

Appendix 1 - Renumbering and May’s construction 57

Appendix 2 - Diagrammatic methods 61

Introduction

In [19], May constructed a spectral sequence that relates the homology of a locally finite, connected, filtered-graded K-algebra, A, to that of its associated graded algebra, gr(A). This construction depends on a certain filtration of the normalised bar resolution of A, and the resulting spectral sequence is trigraded. Under suitable circumstances, given a filtered-graded left A-module M, this spectral sequence has the form

Tor_{∗,∗,∗}^{gr(A)}(K, gr(M)) ⟹ Tor_{∗,∗}^{A}(K, M).

May also gave a filtration of the cobar resolution that gives rise to a dual cohomological spectral sequence. His original purpose was to use this spectral sequence as a tool for calculating the cohomology of the Steenrod algebra, this cohomology ring being the E_{2}-page of the Adams Spectral Sequence to compute the p-component of the stable homotopy groups of spheres.

We define both homological and cohomological spectral sequences for a filtered ring R. We refer to either one as the May Spectral Sequence, or MSS, for short. These constructions do not depend on the bar resolution, but on a more general filtered resolution, that is strict and filtered-free. Chapter 1 defines these notions for filtered resolutions, and relates them to the associated graded case. Chapter 2 consists of the setting up of our spectral sequence in a bigraded form, which, under suitable conditions is a convergent spectral sequence,

Tor_{∗,∗}^{gr(R)}(gr(N), gr(M)) ⟹ Tor_{∗}^{R}(M, N),

for filtered left R-modules M and N. We show that the spectral sequence we define is independent of the resolution used to define it, and set up appropriate product structures for the corresponding cohomological case. We also note that the homological version of our spectral sequence gives a spectral sequence of Serre, [26], as a special case. Serre’s spectral sequence relates the cohomology of a Noetherian commutative ring filtered I-adically to the cohomology of its associated graded ring, and his work predates that of May, so perhaps we should have called our spectral sequence the Serre-May Spectral Sequence.

We note that Grünenfelder tried to give an abstract categorical generalisation of Serre’s work in [11]. Unfortunately, this work contains many errors, both category theoretic and ring theoretic, so that most of his stated theorems are in fact false. For example, he claims that, with trivial coefficients, the mod-p homology of a finite p-group, P, is isomorphic to the homology of the associated graded algebra corresponding to the case when the mod p group algebra of P is filtered by powers of its radical. Chapter 6 provides many counterexamples to this assertion.

In Chapter 3, we define a triple graded spectral sequence for filtered-graded rings, extending the construction from Chapter 2. A proof that we have indeed obtained the spectral sequence as originally defined by May is given in Appendix 1.

As an immediate application of our spectral sequence, we give a simple construction in Chapter 4 of spectral sequences also due to May, [19], that may be used to calculate the cohomology of the restricted enveloping algebra of a restricted p-Lie algebra. Here we use the fact that the MSS is independent of the resolution used to define it to save us a lot of work: originally, May constructed explicit and lengthy free resolutions for restricted enveloping algebras to obtain his results.

These theorems on restricted Lie algebra cohomology have an application to the MSS for group algebras, since if kP is a modular group algebra of a finite p-group, filtered by powers of its radical, then the Quillen-Jennings Theorem says that gr(kP) is the universal enveloping algebra of a certain restricted Lie algebra that may be explicitly written down. We explore this situation in Chapter 5, observing that if p is 2, we have an ‘edge’ in the MSS.

We also give, in Chapter 6, some specific calculations for certain types of p-group, namely dihedral, semi-dihedral and generalised quaternion 2-groups, extraspecial 2groups, and groups of order p^{3}, for p odd. In general, our methods rely heavily on known results in group cohomology. (Note that for the Steenrod algebra, the calculations that have been done for the MSS are heavily dependent on direct computations using the cobar resolution - which are helped by the fact that the dual of the Steenrod algebra has a particularly nice form - and have only being successfully carried out in low cohomological degrees. They also use known results on stable homotopy. See [25] and [28] for further details.)

We also prove some results about powerful p-groups: here we note that the structure of the E_{1}-page of the MSS is rather easy to work out. For, even using May’s results on restricted Lie algebras and the trigraded Lyndon-Hochschild-Serre Spectral Sequence for central extensions of connected Hopf algebras, it can be difficult to calculate the E_{1}-page in odd characteristic, in general.

Finally, we give two examples of the MSS for non-Hopf algebras in Chapter 7. The first illustrates a spectral sequence having a filtration of a sort not considered by May. The E_{1}-page is ‘highly non-commutative’, being a free algebra on two variables, but we are able to compute d_{1} and show E_{2} = E_{∞}. The second example gives a counterexample to another claim of Grünenfelder. The MSS in this case may be computed using diagrammatic methods, which are described briefly in Appendix 2.

Chapter 1

Resolutions

1.1 Rings, graded rings and filtered rings

In this section we introduce some notation and define three categories that will concern us.

1.1.1 Rings

By a ring we mean an associative ring with a 1. Given a ring R, we have the category, R-mod, of (left) R-modules whose morphisms are just the R-module homomorphisms. This is an abelian category in which the free objects are just direct sums of the left R-module R.

1.1.2 Graded rings

A ring R is said to be graded if there are additive subgroups, (R_{i})_{i∈ℤ}, such that R = ⨁_{i} R_{i} and R_{j}R_{k} ⊆ R_{j+k} for all j and k.

If R is a graded ring, a (left) R-module M is said to be graded if it has additive subgroups (M_{i})_{i∈ℤ} such that M = ⨁_{i} M_{i} and R_{j}M_{k} ⊆ M_{j+k} for all j and k. We define the category R-gr to be the category of graded R-modules in which the morphisms are the gradation-preserving R-homomorphisms (i.e., R-homomorphisms ϕ : M → N such that ϕ(M_{i}) ⊆ N_{i} for all i). It is an abelian category.

If M is a graded R-module, we write M[n] for the graded R-module M with the grading (M[n])_{i} = M_{i−n}. A graded R-module M is said to be graded-free if there is a set J and integers n_{j} for each j ∈ J such that

L ≅ ⨁_{j∈J} R[n_{j}]

as graded R-modules, the grading on the direct sum being defined in the obvious way (i.e., (⨁_{j} R[n_{j}])_{i} = ⨁_{j} R[n_{j}]_{i} = ⨁_{j} R_{i−n_{j}}). The graded-free modules are just the free objects in R-gr, and any graded R-module is the homomorphic image in R-gr of some graded-free module.

1.1.3 Filtered rings

A ring R is said to be filtered if there is an ascending or descending chain of additive subgroups, (F_{i}R)_{i∈ℤ}, such that 1 ∈ F_{0}R, and (F_{i}R)(F_{j}R) ⊆ F_{i+j}R for all i and j. For clarity of exposition, we shall assume that the filtration is ascending (i.e., F_{i}R ⊆ F_{i+1}R for all i), but note that our results are valid for descending filtrations as well, by formally renumbering a descending filtration (F_{i}R)_{i∈ℤ} as the ascending filtration (F′_{i}R)_{i∈ℤ}, where F′_{i}R = F_{−i}R. Also we shall assume throughout that our filtrations are exhaustive, i.e., R = ⋃_{i} F_{i}R, and separated, i.e., 0 = ⋂_{i} F_{i}R.

If R is a filtered ring, a (left) R-module M is said to be filtered if it comes with an ascending chain (F_{i}M)_{i∈ℤ} of additive subgroups of M such that (F_{i}R)(F_{j}M) ⊆ F_{i+j}M for all i and j: again we shall assume that any such filtration is both exhaustive and separated. We define the category R-filt to be the category whose objects are filtered R-modules and whose morphisms are the filtration-preserving R-homomorphisms (i.e., those R-homomorphisms ϕ : M → N such that ϕ(F_{i}M) ⊆ F_{i}N for all i). In general, R-filt is an additive category but it is not abelian: monomorphisms are not necessarily kernels, for instance. However, we define a filtered homomorphism ϕ : M → N to be strict if ϕ(F_{i}M) = ϕ(M) ∩ F_{i}N for all i; then it is clear that kernels are just the strict monomorphisms and cokernels are just the strict epimorphisms.

If M is a filtered R-module with filtration (F_{i}M)_{i∈ℤ}, we shall write M[n] for the R-module M with the filtration F_{i}(M[n]) = F_{i−n}M. A filtered R-module L is said to be filtered-free if there is a set J and integers n_{j} for each j ∈ J such that

L ≅ ⨁_{j∈J} R[n_{j}]

as filtered R-modules, the filtration on the direct sum being defined in the obvious way (i.e., F_{i}(⨁_{j} R[n_{j}]) = ⨁_{j} F_{i}R[n_{j}] = ⨁_{j} F_{i−n_{j}}R). The filtered-free modules are just the free objects in R-filt. Moreover, as we shall see, any filtered R-module is the homomorphic image in R-filt of some filtered-free module.

1.1.4 The functor gr

Given a filtered module M over a filtered ring R, we can form the associated graded module,

gr(M) = ⨁_{i∈ℤ}F_{i}M/F_{i−1}M,

which is a graded module over the associated graded ring,

gr(R) = ⨁_{i∈ℤ}F_{i}R/F_{i−1}R.

It is clear that a filtration-preserving R-module homomorphism, ϕ : M → N induces a graded gr(R)-homomorphism,

gr(ϕ) : gr(M) → gr(N),

so that gr is a functor from R-filt to gr(R)-gr.

We note that any graded ring R can be filtered by defining F_{i}R = ⨁_{j≤i} R_{j}. Moreover, with this filtration, gr(R) ≅ R as graded rings.

Another important example of a filtered ring may be given when we have a (twosided) ideal, I, of an arbitrary ring, R. Then we define the I-adic filtration on R to be

F_{i}R = \{ \begin{matrix}R & if i ≥ 0, \\ I^{−i} & if i < 0. \end{matrix}

Note that we require ⋂_{i} I^{i} = 0 to satisfy our assumption that the filtration must be separated. If necessary, we shall write gr_{I}(R) for gr(R) to distinguish the associated graded ring corresponding this filtration. We also get an induced filtration on any (left) R-module M by setting F_{i}M = (F_{i}R)M, and denote the corresponding associated graded module by gr_{I}(M).

1.2 Resolutions

1.2.1 Associated graded resolutions

All of the definitions given above for modules can be extended to chain complexes. For example, if R is a filtered ring and C = (C, d) = (C_{n}, d_{n}) is a chain complex of R-modules of degree k, we say C is a filtered complex if each C_{n} is a filtered Rmodule and each d_{n} : C_{n} → C_{n+k} is a filtered homomorphism. Letting F_{i}C denote the chain complex (F_{i}C_{n}, d_{n}|F_{i}C_{n}), we define the associated graded complex to be

gr(C) = ⨁_{i} F_{i}C/F_{i−1}C = (⨁_{i} F_{i}C_{n}/F_{i−1}C_{n}, gr(d_{n})).

Moreover, we say that C is strict if each d_{n} is strict and C is filtered-free if each C_{n} is filtered-free. Similar definitions apply to graded complexes.

Given a filtered ring R, we shall now aim at building a filtered-free R-resolution, P, for any filtered R-module, M, in such a way that the associated graded complex, gr(P), is in fact a graded-free gr(R)-resolution of gr(M). First we make the simple observation that if P is any filtered-free R-resolution, then gr(P) is a graded-free gr(R)-complex, since if each P_{n} is filtered-free then each gr(P_{n}) is graded-free. So we only have to ensure exactness: the next result shows why P must be strict.

Lemma 1 Suppose R is a filtered ring and L, M and N are filtered R-modules. Suppose, furthermore, that ϕ : L → M and ψ : M → N are filtered homomorphisms such that ψϕ = 0 and

gr(L) ⟶^{gr(ϕ)} gr(M) ⟶^{gr(ψ)} gr(N)

is an exact sequence of gr(R)-modules. Then ψ : M → N is strict.

Proof ([21, I.4.4.2]) Assume that y ∈ F_{i}N ∩ ψ(M) and y ∉ F_{i−1}N. Then since the filtration is exhaustive (by our definition of a filtered module), we can find x ∈ F_{i+t}M, for some t ≥ 0, with y = ψ(x). Now if t = 0 there is nothing to prove. If t > 0 then gr(ψ)(\overline{x}) = 0, where \overline{x} = x + F_{i+t−1}M. Hence, by exactness, there is some z ∈ F_{i+t}L with gr(ϕ)(\overline{z}) = \overline{x}. Therefore, x − ϕ(z) ∈ F_{i+t−1}M, and y = ψ(x) = ψ(x − ϕ(z)). Repeating this procedure t times, we can construct some u ∈ F_{i}M with y = ψ(u). □

We also have the converse, showing that strict exactness is a sufficient condition.

Lemma 2 Suppose L, M and N are filtered R-modules and ϕ and ψ are strict filtered homomorphisms such that

L ⟶^{ϕ} M ⟶^{ψ} N

is an exact sequence of R-modules. Then

gr(L) ⟶^{gr(ϕ)} gr(M) ⟶^{gr(ψ)} gr(N)

is an exact sequence of gr(R)-modules.

Proof ([21, I.4.4.1], [26, II-2]) It is clear that gr(ψ)gr(ϕ) = 0. Conversely, if x ∈ F_{i}M and gr(ψ)(\overline{x}) = 0 (where \overline{x} denotes x + F_{i−1}M), then

ψ(x) ∈ F_{i−1}N ∩ ψ(M) = ψ(F_{i−1}M).

Therefore, ψ(x) = ψ(y) for some y ∈ F_{i−1}M and

x − y ∈ ker ψ ∩ F_{i}M = ϕ(L) ∩ F_{i}M = ϕ(F_{i}L).

Hence x − y = ϕ(z) for some z ∈ F_{i}L, and so \overline{x} = gr(ϕ)(\overline{z}), as required. □

Corollary 1 Suppose the filtered R-module M has a strict filtered-free R-resolution P. Then gr(P) is a graded-free gr(R)-resolution of gr(M). □

So we now ask when a filtered R-module M has a strict filtered-free resolution. The first answer is very general, but somewhat artificial.

Lemma 3 Every filtered R-module M has a strict filtered-free resolution.

Proof It suffices to construct a filtered-free module P and a strict filtered epimorphism P →^{ϕ} N for any filtered module N ≠ 0.

By our definition of filtered module, we have assumed that our filtrations are exhaustive and separated, so if 0 ≠ x ∈ N, there is a unique integer n_{x} with x ∈ F_{n_{x}}N, but x ∉ F_{n_{x}−1}N. We set

P = ⨁_{0≠x∈N} R[n_{x}]

and let ϕ be the map induced by sending the x^{th} filtered-free basis element, 1_{x}, to x. Then ϕ(F_{i}P) = F_{i}N by construction, so ϕ is a strict filtered epimorphism. □

In the formal proof above, it is not sufficient just to define P as a free module with a basis corresponding to a set of generators for the R-module N since ϕ, although surjective, may not be strict. For example, let

R = k[x, y]/(xy, x^{2} − x, y^{2}) = (k1 ⊕ kx̄) ⊕ kȳ

be the graded algebra over a field k, with deg x̄ = 0 and deg ȳ = −1 (which we can regard as a filtered ring in the usual way). Then the ideal (x̄, ȳ) of R is principal, generated by x̄ + ȳ. But if we give this ideal the filtration defined by its inclusion in R, the corresponding map ϕ : R[0] → (x̄, ȳ), 1 ↦ x̄ + ȳ, is not strict, since

ϕ(F_{−1}R) = 0 ≠ F_{−1}(x̄, ȳ) = kȳ.

However, it is clear that we have a strict epimorphism R[0] ⊕ R[−1] → (x̄, ȳ) given by 1_{0} ↦ x̄ and 1_{−1} ↦ ȳ. In fact, we can prove a general result that we shall need later on.

Lemma 4 Suppose A is a commutative Noetherian ring filtered I-adically, for some ideal I of A, and M is a filtered A-module that is finitely generated (i.e., just as an A-module, ignoring the filtered structures) and is either I-adically filtered or is a submodule of a finitely generated filtered-free A-module. Then there is a finitely generated filtered-free A-module, P, and a strict filtered epimorphism P → M; that is, M is finitely generated as a filtered module.

Proof This is essentially a consequence of the Artin-Rees Lemma, [2, 10.9], which says that the filtration on M is I-stable: i.e., there exists an n such that IF_{i}M = F_{i−1}M for all i < n.

Since M is filtered and finitely generated, we have F_{m}M = M for some m ≥ n. For each r ≥ 0, it is clear that we can find a finitely generated filtered-free module P_{r} and a filtered epimorphism P_{r} → F_{m−r}M. Now when r = n − m, the resulting homomorphism P_{n−m} → F_{n}M is necessarily strict by the Artin-Rees Lemma. Therefore, we can take P = ⨁_{r=n−m}^{n} P_{r} and the induced epimorphism, P → M, is strict. □

1.2.2 Ungrading resolutions

We now show that in many cases we can obtain a strict filtered-free R-resolution by ungrading a graded-free gr(R)-resolution.

Lemma 5 Suppose R is a filtered ring, M is a filtered R-module, and P is a filteredfree R-module. Then for any graded map α : gr(P) → gr(M) there is a filtered map ϕ : P → M such that gr(ϕ) = α.

Proof It suffices to prove this when P = R[n] = R1_{n}. But if α(¯1_{n}) = x̄, choose x ∈ F_{n}M such that x + F_{n−1}M = x̄ and define ϕ by ϕ(1_{n}) = x. □

This lemma shows that any resolution can be ungraded, but the ungraded object may not be an R-resolution. We say that a filtered R-module M is complete if the natural surjections M → M/F_{−n}M induce an isomorphism M = lim_{⟵} M/F_{−n}M. One important example of a complete filtration occurs when M is discrete (i.e., there is an n such that F_{i}M = 0 for all i ≤ n).

Lemma 6 Let N be a complete filtered R-module and let M be a filtered R-module. Then any filtered map ϕ : N → M which induces an epimorphism gr(ϕ) : gr(N) → gr(M) must be a strict epimorphism.

Proof ([21, D III.3.3]) Let x ∈ F_{i}M. We have to show that there is a y ∈ F_{i}N such that ϕ(y) = x. Now, writing \overline{x} = x + F_{i−1}M etc., there is a y_{i} ∈ F_{i}N such that gr(ϕ)(\overline{y_{i}}) = \overline{x} as gr(ϕ) is surjective. Hence x − ϕ(y_{i}) ∈ F_{i−1}M. So there is a y_{i−1} ∈ F_{i−1}N with gr(ϕ)(\overline{y_{i−1}}) = \overline{x − ϕ(y_{i})}. Continuing this process we obtain y_{i−r} ∈ F_{i−r}N such that

x − ϕ(y_{i}) − ϕ(y_{i−1}) − ⋯ − ϕ(y_{i−r}) ∈ F_{i−r−1}M

for all r ≥ 0. Then y = ∑_{r=0}^{∞} y_{i−r} ∈ F_{i}N exists since N is complete: moreover, x − ϕ(y) ∈ ⋂_{r=0}^{∞} F_{i−r}M = 0, so ϕ(y) = x as required. □

We say that a filtered module M over a filtered ring R is filtered-finitely generated if there is a strict epimorphism P → M where P is isomorphic as a filtered R-module to some finite direct sum ⨁_{j=1}^{k} R[n_{j}]: and we say that R is filtered-Noetherian if every left ideal is filtered-finitely generated. So if R is filtered-Noetherian and M is a filtered-finitely generated R-module, it is clear that M has a strict filtered-free resolution P in which each P_{n} is filtered-finitely generated. Applying the functor gr to such a resolution, we obtain a graded-free gr(R)-resolution of gr(M) in which each gr(P_{n}) is finitely generated as a graded-free gr(R)-module. We have the following converse.

Theorem 1 Let R be a complete filtered ring and let M be a filtered R-module. Suppose there is a graded-free gr(R)-resolution, (Q, δ), of gr(M), in which each Q_{n} is finitely generated as a graded-free module. Then there exists a strict filtered-free R-resolution, (P, d), of M, such that gr(P) ≅ Q as gr(R)-resolutions of gr(M).

Proof Choose the filtered-free P_{n} by requiring gr(P_{n}) ≅ Q_{n}. Each P_{n} is finitely generated, so is complete, and hence by Lemma 5, each δ_{n} lifts to some map d_{n} : P_{n} → P_{n−1} so that gr(d_{n}) = δ_{n}. Now d_{0} : P_{0} → M induces an epimorphism δ_{0} : gr(Q_{0}) → gr(M), so must be strict by Lemma 6. Hence

0 → ker d_{0} → P_{0} ⟶^{d_{0}} M → 0

is strict exact, where ker d_{0} has the filtration induced by the inclusion map. Therefore, by Lemma 2,

0 → gr(ker d_{0}) → Q_{0} ⟶^{δ_{0}} gr(M) → 0

is exact. In other words, we may identify ker δ_{0} with gr(ker d_{0}). Repeating this argument, we see that P must be strict and exact. □

1.2.3 The bar resolution

We now give a basic example of an ‘ungraded resolution’ following an observation of J.P.May, [19, p.129].

Let A be an algebra over a field k, and let M be an A-module. We shall use T(V) to denote the tensor algebra on a k-vector space V and use ⊗ to mean ⊗_{k}. In this context, we write V [1] for a copy of V having homological degree one. So V [1]⊗V [1] has homological degree two, and so on. The unnormalised bar resolution for M over A is the free resolution

B(A, M) = A ⊗ T(A[1]) ⊗ M

with differential d_{n} : A ⊗ A^{⊗^{n}} ⊗ M → A ⊗ A^{⊗^{n−1}} ⊗ M given by

d_{n}(a(a_{1}, …, a_{n})m) = aa_{1}(a_{2}, …, a_{n})m +

∑_{i=1}^{n−1}(−1)^{i}a(a_{1}, …, a_{i}a_{i+1}, …, a_{n})m +

(−1)^{n}a(a_{1}, …, a_{n−1})a_{n}m.

The following result is easy to see.

Lemma 7 Let X and Y be filtered vector spaces. Filtering X ⊗ Y by

F_{i}(X ⊗ Y) = ∑_{j+k=i} F_{j}X ⊗ F_{k}Y,

there is an isomorphism of graded vector spaces

gr(X ⊗ Y) ≅ gr(X) ⊗ gr(Y). □

Lemma 8 Given a filtered algebra A over a field k and a filtered k-vector space X, the left A-module, A ⊗ X, filtered by

F_{i}(A ⊗ X) = ∑_{j+k=i} F_{j}A ⊗ F_{k}X,

is filtered-free.

Proof For each k ∈ ℤ, choose a set {x_{k,l}}_{l∈I_{k}} of elements in X such that

\{\overline{x}_{k,l} = x_{k,l} + F_{k−1}X\}_{l∈I_{k}}

is a basis for F_{k}X/F_{k−1}X. Then the x_{k,l}’s form a basis for X and

A ⊗ X ≅ ⨁_{k∈ℤ,l∈I_{k}} A ⊗ x_{k,l} ≅ ⨁_{k∈ℤ,l∈I_{k}} A[k]

as filtered left A-modules. □

Proposition 1 Let A be a filtered algebra over a field and let M be a filtered Amodule. Then the unnormalised bar resolution B(A, M) may be filtered so that it is strict and filtered-free. Moreover, the associated graded complex is isomorphic to the bar resolution B(gr(A), gr(M)).

Proof We set

F_{i}B_{n}(A, M) = ∑_{i_{0}+i_{1}+...+i_{n+1}=i} F_{i_{0}}A ⊗ (F_{i_{1}}A ⊗ … F_{i_{n}}A) ⊗ F_{i_{n+1}}M.

It is clear that this defines a filtered-free filtration on B_{n}(A, M) by Lemma 8. Also it is immediate from the definition of the differential that

d_{n}(F_{i}B_{n}(A, M)) ⊆ F_{i}B_{n−1}(A, M).

By Lemma 7, there is an isomorphism of graded vector spaces

gr(B_{n}(A, M)) ≅ B_{n}(gr(A), gr(M)).

Using this identification, we see that gr(d_{n}) is the differential for the bar resolution of gr(M) over gr(A). Finally, the exactness of B(gr(A), gr(M)) implies that B(A, M) is strict by Lemma 1. □

We may also consider the normalised bar resolution for M over A, that is, the free resolution

\overline{B}(A, M) = A ⊗ T((A/k)[1]) ⊗ M

where A/k denotes the cokernel of the unit map and the differential is given by

\overline{d}_{n} : A ⊗ (A/k)^{⊗^{n}} ⊗ M → A ⊗ (A/k)^{⊗^{n−1}} ⊗ M,

\overline{d}_{n}(a[a_{1} | … | a_{n}]m) = aa_{1}[a_{2} | … | a_{n}]m +

∑_{i=1}^{n−1}(−1)^{i}a[a_{1} | … | a_{i}a_{i+1} | … | a_{n}]m +

(−1)^{n}a[a_{1} | … | a_{n−1}]a_{n}m.

Now if A is filtered, we can filter the vector space A/k by defining

F_{i}(A/k) = (F_{i}A + k)/k.

Then it is clear that gr(A/k) ≅ gr(A)/k, and the proof of Proposition 1 gives the corresponding result for the normalised bar resolution, i.e.,

gr(\overline{B}(A, M)) ≅ \overline{B}(gr(A), gr(M)).

Note that if A is augmented, we can identify A/k with the augmentation ideal I(A) as filtered modules: this is how May originally expressed his observation. Also we can extend Lemma 7 whenever X and Y are filtered modules over a commutative ring k and if either the modules X/F_{i}X and Y/F_{i}Y are flat for all i, or the sequences

\begin{matrix}0 → F_{i}X → X → X/F_{i}X → 0 \\ 0 → F_{i}Y → Y → Y/F_{i}Y → 0 \end{matrix}

are split exact for all i. (See the remarks in [20, pp.240-241].) But this is at the expense of demanding that the filtration on A is discrete in order to prove the corresponding version of Lemma 8. In this case, the obvious generalizations of Proposition 1 are valid. However, it is easy to find an example where this construction fails.

Example Fix an integer m ≥ 2. Let k = ℤ and let A = M = ℤ with the filtration defined by F_{i}A = m^{−i}ℤ for all i ≤ 0. Then B_{n}(A, M) ≅ ℤ^{⊗^{n+2}} ≅ ℤ, so

gr(B_{n}(A, M)) ≅ gr(ℤ) ≅ ℤ/mℤ[x]

where deg x = −1. But

B_{n}(gr(A), gr(M)) ≅ gr(ℤ)^{⊗^{n+2}} ≅ ℤ/mℤ[x_{1}, …, x_{n+2}]

where deg x_{i} = −1. Hence gr(B(A, M)) ≅ B̸(gr(A), gr(M)) in this case. A similar construction (e.g., for A = ℤ[x]/(x^{2}), F_{i}A = m^{−i}A for i ≤ 0, and k = M = ℤ) gives an example where there is no corresponding isomorphism for the normalised bar resolution. □

1.2.4 The comparison theorem

The comparison theorem is a simple but vital tool in homological algebra. Here we extend the basic proof to the filtered category.

Lemma 9 Suppose that P is a filtered-free R-module, and that we are given maps of filtered R-modules d : M → N and ϕ : P → N, so that d is strict and surjective. Then there is a filtered map ψ : P → M such that ϕ = dψ.

Proof It suffices to prove this in the case when P = R[i]. Then ϕ(1_{R}) ∈ F_{i}N = d(F_{i}M), so choosing x ∈ F_{i}M with ϕ(1_{R}) = d(x), we define ψ by setting ψ(1_{R}) = x. □

Proposition 2 (i) Suppose we are given the following complexes of filtered Rmodules and filtered homomorphisms

⋯ → P_{n} →^{d_{n}} P_{n−1} → ⋯ → P_{1} →^{d_{1}} P_{0} →^{d_{0}} M → 0

⋯ → X_{n} →^{d′_{n}} X_{n−1} → ⋯ → X_{1} →^{d′_{1}} X_{0} →^{d′_{0}} N → 0

such that each P_{n} is filtered-free and the bottom complex is strict exact. Then a filtered homomorphism ϕ : M → N extends to a chain map (ϕ_{n}) where each ϕ_{n} is a filtered map. If (ψ_{n}) is another such filtered chain map then there is a filtered chain homotopy (h_{n} : P_{n} → X_{n+1}) between (ϕ_{n}) and (ψ_{n}). Thus, passing by the functor gr to the graded case, we obtain a graded chain map (gr(ϕ_{n})) extending gr(ϕ), which is graded homotopy equivalent to (gr(ψ_{n})).

Proof The maps d′_{0} : X_{0} → N and d′_{n} : X_{n} → ker d′_{n−1} for n ≥ 1 are strict and surjective, so by Lemma 9 we can construct the required chain homotopy inductively. Moreover, given two filtered chain homotopies (ϕ_{n}) and (ψ_{n}), the usual inductive construction (see [4, Vol.1, p.28], for example) carries over to the filtered case: that is, given h_{n−1}, we can construct h_{n} by Lemma 9, on noting that d′_{n}(ϕ_{n}−ψ_{n}−h_{n−1}d_{n}) = 0. □

1.2.5 Filtered-projective modules

Given a filtered ring R and a filtered R-module M, we have already discussed strict filtered-free R-resolutions for M. In fact it makes sense to widen our considerations and look at the more general case of resolutions consisting of filtered-projective modules. These may be defined in the usual universal way to be the projective objects in the category R-filt, and it is straightforward to show that a filtered R-module P is filtered-projective if and only if there is a filtered module Q such that P ⊕ Q is a filtered-free R-module. (See [21, VI.5, p.308], for instance.) Using this second characterisation, we can extend the comparison theorem to this wider context; and there are also obvious generalizations of our results to those relating filtered-projective resolutions to graded-projective resolutions.

We may also define a filtered-projective R-module P to be a filtered-projective cover for a filtered R-module M if

1. there is a strict epimorphism ϕ : P → M, and,

2. any filtered map ψ : N → P for which the composite ϕψ is a strict epimorphism must itself be a strict epimorphism.

This enables us to consider ‘minimal’ resolutions in the filtered category. Indeed, following the usual proof of the existence of projective covers for Noetherian semiperfect rings, it is possible to show that a filtered-finitely generated R-module, M, has a minimal strict filtered-projective cover provided gr(R) is Noetherian and semiperfect (cf. [11, 2.9]).

Note, however, that for a local algebra over a field, filtered by powers of its Jacobson radical, the filtered-projecive modules are just the filtered-free modules. As such algebras are our main concern, and to clarify the exposition, we shall continue to base our constructions on filtered-free resolutions, just briefly mentioning how they generalize to the filtered-projective case where necessary.

The May spectral sequence

2.1 Tensor products, Hom, Tor and Ext

In this chapter, we shall use our results about resolutions to construct the May spectral sequence. However, first we must give some definitions for the graded and filtered categories.

2.1.1 The graded case

Suppose R is a graded ring, M is a graded right R-module and N is a graded left Rmodule. Now if (P, d) is a graded-free R-resolution for N, then we define the group Tor_{n}^{R}(M, N) to be the nth-homology group of the complex (M ⊗_{R}P, 1_{M} ⊗_{R}d). Note, however, that each M ⊗_{R} P_{n} is itself a graded abelian group: indeed, any tensor product of a graded right R-module X and a graded left R-module Y is graded by defining

(X ⊗_{R} Y)_{i} = ∑_{j+k=i} X_{j} ⊗_{R} Y_{k}.

Here we abuse notation, understanding by X_{j} ⊗_{R} Y_{k} the subgroup

⟨x ⊗_{R} y | x ∈ X_{j}, y ∈ Y_{k}⟩

of X ⊗_{R} Y.

In the case when P_{n} = ⨁_{j} R[n_{j}], we have an isomorphism of graded groups M ⊗_{R} P_{n} ≅ ⨁_{j} M[n_{j}]. Moreover, it is clear that the differential 1_{M} ⊗_{R} d preserves this grading, so defining

Tor_{n,i}^{R}(M, N) = H_{n}((M ⊗_{R} P)_{i}),

we have an isomorphism Tor_{n}^{R}(M, N) = ⨁_{i∈ℤ} Tor_{n,i}^{R}(M, N). The standard comparison argument (cf. Proposition 2 or see [1, 2.1, p.27]) shows that these groups are independent of the choice of resolution.

We may also define the graded Ext groups. To do this we assume that both M and N are graded left R-modules and P is a graded-free R-resolution for M. We define

Hom_{R}^{i}(M, N) = {θ ∈ Hom_{R}(M, N) | θ(M_{j}) ⊆ N_{j−i} for all j ∈ ℤ}.

That is, Hom_{R}^{i}(M, N) consists of all those R-module homomorphisms that lower the degree by precisely i. Note that this definition does not give in general a grading on Hom_{R}(M, N). For instance, let R = k be a field with the trivial grading (i.e., k_{0} = k), M = ⨁_{i∈ℤ} k[i] and N = k. Then

Hom_{R}(M, N) ≠ ⨁_{i∈ℤ} Hom_{R}^{i}(M, N)

as can be seen by considering the k-homomorphism, ϕ ∈ Hom_{k}(⨁_{i∈ℤ} k[i], k), given by 1_{i} ↦ 1. Therefore, we define

HOM_{R}(M, N) = ⨁_{i∈ℤ} Hom_{R}^{i}(M, N),

so that, by definition, we obtain a graded abelian group that is graded by the Hom_{R}^{i}(M, N)’s.

Now the map

M ↦ HOM_{R}(M, N)

is a contravariant additive functor from R-gr to the category of graded abelian groups. So we may define EXT_{R}^{n}(M, N) as the corresponding n^{th} right derived functor (see [16, XII.9]). Since we have identities

HOM_{R}(P_{n}, N) = ⨁_{i∈ℤ} Hom_{R}^{i}(P_{n}, N)

that extend to give identities of graded cochain complexes,

HOM_{R}(P, N) = ⨁_{i∈ℤ} Hom_{R}^{i}(P, N)

we also have

EXT_{R}^{n}(M, N) = ⨁_{i∈ℤ} Ext_{R}^{n,i}(M, N),

where Ext_{R}^{n,i}(M, N) = H^{n}(Hom_{R}^{i}(P, N)).

We do not have an isomorphism between EXT_{R}^{n}(M, N) and Ext_{R}^{n}(M, N) in general, where Ext_{R}^{n}(M, N) is the usual Ext group, defined ignoring the graded structure. However, we do have isomorphisms

EXT_{R}^{n}(M, N) ≅ Ext_{R}^{n}(M, N)

in the important case when each P_{n} is finitely generated. This follows from the equivalent observation at the level of homomorphisms. Indeed, we have graded isomorphisms,

HOM_{R}(R[i], N) ≅ Hom_{R}(R[i], N),

so if P_{n} = ⨁_{i=1}^{t} R[r_{i}], we have graded isomorphisms

HOM_{R}(P_{n}, N) ≅ ⨁_{i=1}^{t} HOM_{R}(R[r_{i}], N)

≅ ⨁_{i=1}^{t} Hom_{R}(R[r_{i}], N)
≅ Hom_{R}(P_{n}, N)

which induce an isomorphism of complexes HOM_{R}(P, N) ≅ Hom_{R}(P, N).

2.1.2 The filtered case

We now give some definitions for the filtered category, R-filt. These are motivated by the graded case.

To define filtered Tor, we define the filtration on a tensor product over R of a filtered right R-module X and a filtered left R-module Y by

F_{i}(X ⊗_{R} Y) = ∑_{j+k=i} F_{j}X ⊗_{R} F_{k}Y.

It is clear that this defines a filtration on the abelian group X ⊗_{R} Y that is both exhaustive and separated (because, by definition of a filtration, the filtrations on X and Y are assumed exhaustive and separated). The next result is simple, yet crucial to our constructions.

Lemma 10 Suppose X is a filtered right R-module. Then, filtering X ⊗_{R} R[n] as above, there is an isomorphism X ⊗_{R} R[n] ≅ X[n] of filtered abelian groups.

Proof The map X ⊗_{R}R[n] → X[n] given by x⊗_{R}r ↦ xr is filtered, since if x ∈ F_{i}X and r ∈ F_{j}R[n] = F_{j−n}R then

xr ∈ F_{i}XF_{j−n}R ⊆ F_{i+j−n}X = F_{i+j}X[n].

This has a filtered inverse given by x ↦ x ⊗_{R} 1. □

Now if M is a filtered left R-module, N is a filtered right R-module and P is a strict filtered-free R-resolution of N, then F_{i}(M ⊗_{R} P) is a subcomplex of M ⊗_{R} P, and so we define

F_{i}Tor_{n}^{R}(M, N) = Im (H_{n}(F_{i}(M ⊗_{R} P)) → H_{n}(M ⊗_{R} P)).

It is clear that this defines a filtration of Tor_{n}^{R}(M, N) (which is both exhaustive and separated).

Also given filtered left R-modules M and N, we let

F_{i}Hom_{R}(M, N) = {ϕ ∈ Hom_{R}(M, N) | ϕ(F_{j}M) ⊆ F_{j−i}N for all j ∈ ℤ}.

Then F_{i}Hom_{R}(M, N) ⊇ F_{i+1}Hom_{R}(M, N) and ⋂_{i} F_{i}Hom_{R}(M, N) = 0, but in general

Hom_{R}(M, N) ≠ ⋃_{i} F_{i}Hom_{R}(M, N).

(For example, regard the graded example given in 2.1.2 as a filtered example.) Therefore, we define the group of filtered homomorphisms as

HOM_{R}(M, N) = ⋃_{i} F_{i}Hom_{R}(M, N)

which has an exhaustive and separated descending filtration given by

F_{i}HOM_{R}(M, N) = F_{i}Hom_{R}(M, N).

We shall need the following basic identity.

Lemma 11 Suppose R is filtered ring and N is a filtered R-module. Then

F_{i}HOM_{R}(R[n], N) = F_{i}Hom_{R}(R[n], N) ≅ F_{n−i}N.

Proof If ϕ ∈ F_{i}Hom_{R}(R[n], N), then ϕ(1_{n}) ∈ F_{n−i}N. This defines a map which has an inverse given by x ↦ (r1_{n} ↦ rx). □

We now suppose that P is a strict filtered-free R-resolution of M. Then we define the n^{th} filtered Ext group as

EXT_{R}^{n}(M, N) = H^{n}(HOM_{R}(P, N))

which has the filtration

F_{i}EXT_{R}^{n}(M, N) = Im (H^{n}(F_{i}HOM_{R}(P, N)) → H^{n}(HOM_{R}(P, N))).

The strictness of P is crucial, since we need to invoke Proposition 2 to show that these groups are independent of the resolution used to define them.

Equivalently, we can consider the functor M ↦ HOM_{R}(M, N) as a functor from the abelian category in which the objects are filtered R-modules and the morphisms are strict filtered maps to the category of filtered abelian groups; then we can construct EXT_{R}^{n}(M, N) as the corresponding n^{th} right derived functor.

We have a natural map

EXT_{R}^{n}(M, N) → Ext_{R}^{n}(M, N)

induced by the inclusion HOM_{R}(P_{n}, N) → Hom_{R}(P_{n}, N); moreover, the proof from the graded case carries over to show that both these maps are isomorphisms in the case when P_{n} is a finitely generated filtered-free module for all n.

We can also define products on our EXT groups.

Lemma 12 Suppose R is a filtered S-algebra where S is a filtered commutative ring (i.e., the unit map S → R is a homomorphism of filtered rings). Then for filtered R-modules M_{1}, M_{2} and M_{3}, there is a filtered Yoneda product

EXT_{R}^{n}(M_{2}, M_{3}) ⊗_{S} EXT_{R}^{m}(M_{1}, M_{2}) → EXT_{R}^{m+n}(M_{1}, M_{3})

which is compatible with ordinary Yoneda composition: that is,

EXT_{R}^{n}(M_{2}, M_{3}) ⊗_{S} EXT_{R}^{m}(M_{1}, M_{2}) ⟶ EXT_{R}^{m+n}(M_{1}, M_{3})
↓  ↓
Ext_{R}^{n}(M_{2}, M_{3}) ⊗_{S} Ext_{R}^{m}(M_{1}, M_{2}) ⟶ Ext_{R}^{m+n}(M_{1}, M_{3})

is a commutative diagram.

Proof Suppose (P, d) and (Q, d′) are strict filtered-free R-resolutions for M_{1} and M_{2} respectively. We can think of an element in F_{i}HOM_{R}(P_{m}, M_{2}) as a filtered morphism ϕ : P_{m} → M_{2}[i]. Moreover, if it represents an element in EXT_{R}^{m}(M_{1}, M_{2}) then it is a cocycle and ϕd_{m+1} = 0. Therefore, by Proposition 2, we have an induced map of filtered complexes

… → P_{m+n} → … → P_{m+1} → P_{m}  →^{ϕ} M_{2}[i] → 0
↓ Ω^{n}ϕ ↓ Ω^{1}ϕ ↓ Ω^{0}ϕ ∥
… → Q_{n}[i] → … → Q_{1}[i] → Q_{0}[i] → M_{2}[i] → 0.

If ψ ∈ F_{j}HOM_{R}(Q_{n}, M_{3}) is a cocycle representing an element in F_{j}EXT_{R}^{n}(M_{2}, M_{3}), the required product is now given by

{ψ} ⊗_{S} {ϕ} ↦ {ψ[i].Ω^{n}ϕ}

where ψ[i] : Q_{n}[i] → M_{3}[i + j] is the obvious map induced by ψ. This product is well defined by Proposition 2 and, by its definition, it is compatible with the usual Yoneda product (cf. [4, Vol.1, pp.36-7], for instance). □

Similarly, under suitable conditions, we may define cup products. Suppose Λ is a filtered Hopf algebra over a commutative filtered ring S (i.e., unit, counit, multiplication and comultiplication are all filtered in an appropriate way) so that Λ is filtered-free as an S-module. Let M, M′, N and N′ be filtered Λ-modules that are filtered-free as S-modules. Then we have a filtered cup product

EXT_{Λ}^{m}(M, N) ⊗_{S} EXT_{Λ}^{n}(M′, N′) → EXT_{Λ}^{m+n}(M ⊗_{S} M′, N ⊗_{S} N′)

that is compatible with the usual cup product (see [4, Vol.1, pp.51-2]). This is defined as follows. If P and Q are strict filtered-free Λ-resolutions for M and N respectively, and ϕ : P_{m} → N[i] and ψ : Q_{n} → N′[j] are cocycles representing elements of F_{i}EXT_{Λ}^{m}(M, N) and F_{j}EXT_{Λ}^{n}(M′, N′) respectively, then we extend the composite

P_{m} ⊗_{S} Q_{n} ⟶^{ϕ⊗_{s}ψ} N[i] ⊗_{S} N′[j] ↪ (N ⊗_{S} N′)[(i + j)]

by zero to give a cocycle

(P ⊗_{S} Q)_{m+n} → (N ⊗_{S} N′)[(i + j)]

that represents the required product in F_{i+j}EXT_{Λ}^{m+n}(M ⊗_{S} M′, N ⊗_{S} N′). (Here our conditions ensure P ⊗_{S} Q is a strict filtered-free Λ-resolution.) Again, it is straightforward to check that this product is well-defined and compatible with the ordinary cup product.

We note that it is possible to give corresponding definitions for Yoneda and cup products in the case of graded rings. These constructions are detailed in [1, 2.1 & 2.2], for example.

2.2 The homological spectral sequence

2.2.1 Construction and convergence

Let M be a filtered right R-module, N a filtered left R-module and P a strict filteredfree R-resolution of N. We filter M ⊗_{R} P as in 2.1.2: then, if P_{n} = ⨁_{j} R[n_{j}], we have, by Lemma 10, graded isomorphisms

gr(M ⊗_{R} P_{n}) ≅ gr(M ⊗_{R} (⨁_{j} R[n_{j}]))
≅ gr(⨁_{j} M[n_{j}])
≅ ⨁_{j} gr(M[n_{j}])
≅ ⨁_{j} gr(M) ⊗_{gr(R)} (gr(R)[n_{j}])
≅ gr(M) ⊗_{gr(R)} (⨁_{j} gr(R)[n_{j}])
≅ gr(M) ⊗_{gr(R)} gr(P_{n}).

Moreover,

gr(1_{M} ⊗_{R} d_{n}) ≅ 1_{gr(M)} ⊗_{gr(R)} gr(d_{n})

with this identification. But by Corollary 1,

(gr(P), gr(d))

is a graded-free gr(R)-resolution of gr(N), so the (graded) homology of gr(M ⊗_{R} P) is just Tor_{∗,∗}^{gr(R)}(gr(M), gr(N)).

Therefore, we have a filtered complex and hence a corresponding spectral sequence, which we will refer to as the (homological) May Spectral Sequence or MSS for short. Under suitable convergence conditions, this spectral sequence is of the form

Tor_{n,i}^{gr(R)}(gr(M), gr(N)) ⟹ Tor_{n}^{R}(M, N).

So we now ask, “What are suitable conditions for convergence?”

Perhaps the easiest case is just to demand that the filtration of M ⊗_{R} P_{n} is bounded for all n, that is, for each n there are s(n), t(n) ∈ ℤ with

0 = F_{s(n)}(M ⊗_{R} P_{n}) ⊆ … ⊆ F_{t(n)}(M ⊗_{R} P_{n}) = M ⊗_{R} P_{n}.

This automatically guarantees convergence (see [16, XI.3.1]).

Theorem 2 Suppose R be a filtered ring and M and N are, respectively, filtered right and left R-modules, such that the filtrations on R, M and N are exhaustive and separated. Suppose further that N has a strict filtered-free R-resolution, P, such that M ⊗_{R} P_{n} is bounded for all n. Then there is a convergent spectral sequence

E_{n,i}^{1} = Tor_{n,i}^{gr(R)}(gr(M), gr(N)) ⟹ Tor_{n}^{R}(M, N).

Here i denotes the filtration degree and n denotes the homological degree, and the differential is of the form d^{r} : E_{n,i}^{r} → E_{n−1,i−r}^{r}. Using the filtration defined in 2.1.2, we have isomorphisms

E_{n,i}^{∞} ≅ F_{i}Tor_{n}^{R}(M, N)/F_{i−1}Tor_{n}^{R}(M, N).

Furthermore, for every n we can find an r(n) such that E_{n,i}^{r(n)} ≅ E_{n,i}^{∞} for all i. □

Of course, it is more usual to renumber spectral sequences using the filtration degree, p = i, and complementary degree, q = n − i, as x- and y-axes, respectively. In such a context, we can also formally renumber the graded Tor groups, so that we write Tor_{p,q} for the ‘old’ Tor_{p+q,p}. We discuss further numbering conventions in Appendix 1. Generally, in what follows, we will stick to the convention of using indices (n, i) for the homological degree/filtration degree bigraded case, and indices (p, q) for filtered/complementary bidegrees. The next result is a restatement of Theorem 2.

Theorem 3 Suppose R, M, N and P are as in Theorem 2. Then there is a convergent spectral sequence

E_{p,q}^{1} = Tor_{p,q}^{gr(R)}(gr(M), gr(N)) ⟹ Tor_{p+q}^{R}(M, N).

Here p = i denotes the filtration degree and q denotes the complementary degree, so that p + q = n is the homological degree, and the differential is of the form d^{r} : E_{p,q}^{r} → E_{p−r,q+r−1}^{r}. Using the filtration defined in 2.1.2, we have isomorphisms

E_{p,q}^{∞} ≅ F_{p}Tor_{p+q}^{R}(M, N)/F_{p−1}Tor_{p+q}^{R}(M, N). □

The conditions required for convergence clearly hold if the filtrations on R, M and N are finite. It is also possible to gives a series of definitions for the filtered category to give conditions on the filtrations R, M and N to imply that M ⊗_{R} P_{n} has a finite filtration for all n (cf. [11]). As these seem a little contorted and are not entirely relevant to what follows, we omit them.

Another example in which the spectral sequence converges in a ‘strong’ sense applies to commutative Noetherian rings filtered by powers of an ideal.

Theorem 4 (Serre, [26, II-17]) Suppose A is a commutative Noetherian ring, I is an ideal in A such that ⋂_{t=1}^{∞} I^{t} = 0, and M and N are finitely generated A-modules. Then if A, M and N are all given the I-adic filtration, we have a convergent spectral sequence

E_{p,q}^{1} = Tor_{p,q}^{gr_{I}(A)}(gr_{I}(M), gr_{I}(N)) ⟹ Tor_{p+q}^{A}(M, N)

where for every n we can find an r(n) such that E_{p,q}^{r(n)} ≅ E_{p,q}^{∞} whenever p + q = n.

Proof By Lemma 4, N has a strict filtered-free A-resolution, P, in which each P_{n} is finitely generated.

The condition that the convergence is as stated can be reformulated as the statement that for every n there is a positive integer s(n) with

F_{p−s(n)}(M ⊗_{R} P_{n}) ⋂ d(M ⊗_{R} P_{n+1}) ⊆ d(F_{p−1}(M ⊗_{R} P_{n+1}))

for all p (see [26, II-15]). But because all the modules involved are finitely generated and filtered I-adically, this condition follows immediately from the Artin-Rees Lemma, [2, 10.10]. □

Of course, if we just require the spectral sequence to be convergent in a more general sense, we can impose suitably weakened conditions. For example, weak convergence as defined in [16, p.327] is guaranteed by [16, XI, Prop. 3.2] if each M ⊗_{R} P_{n} is bounded below (i.e., there is an integer s(n) with F_{s(n)}(M ⊗_{R} P_{n}) = 0).

2.2.2 Naturality

Suppose R and S are filtered rings and α : R → S is a filtered ring homomorphism (i.e., α(F_{i}R) ⊆ F_{i}S for all i). Then we may regard filtered S-modules as filtered R-modules via α. Also any strict exact filtered complex of S-modules is, a fortiori, strict exact as a complex of filtered R-modules, since strictness depends only on the filtration on the modules in the complex which is unaffected by whether we regard them as S modules or R-modules via α.

Now suppose P is a strict filtered-free R-resolution of the filtered left R-module N_{1} and Q is a strict filtered-free S-resolution of the filtered left S-module N_{2}. Then, any map ψ : N_{1} → N_{2} which is a homomorphism of filtered R-modules via α must extend to give a filtered chain map of R-modules, ψ_{∗} : P → Q, by Proposition 2: moreover, this chain map is unique up to chain homotopy. So given a filtered right R-module M_{1}, a filtered right S-module, and a filtered R-module homomorphism, ϕ : M_{1} → M_{2}, we obtain a filtered chain map

M_{1} ⊗_{R} P ⟶^{ϕ⊗_{R}ψ_{∗}} M_{2} ⊗_{R} Q → M_{2} ⊗_{S} Q

which gives rise to a map of the corresponding May Spectral Sequences.

This construction allows us to regard the MSS as a functor from a category in which the objects are triples, (R, M_{1}, N_{1}), and the morphisms are suitably defined triples, (α, ϕ, ψ), to the category of ‘spectral sequences’: because of the uniqueness up to homotopy of the chain maps, this functor is well-defined (see [16, XII, Prop.3.5]) and natural in each variable. In particular, in the case when R = S, α = 1, M_{1} = M_{2}, ϕ = 1_{M_{1}}, N_{1} = N_{2} and ψ = 1_{N_{1}}, we get the following result.

Lemma 13 The (homological) MSS is independent of the filtered-free resolution used to define it. □

2.3 The cohomological spectral sequence

Let M and N be filtered left R-modules, and let (P, d) be a strict filtered-free R-resolution of M. Recall that if we filter Hom_{R}(P_{n}, N) as in 2.1.2, the filtration is not necessarily exhaustive. Therefore, we filter HOM_{R}(P_{n}, N). If P_{n} = ⨁_{j} R[n_{j}], we have the following isomorphisms, by Lemma 11.

F_{i}HOM_{R}(P_{n}, N) = F_{i}(Hom_{R}(P_{n}, N))
= F_{i}(Hom_{R}(⨁_{j} R[n_{j}], N))
≅ F_{i}(∏_{j} Hom_{R}(R[n_{j}], N))
≅ ∏_{j} F_{i}Hom_{R}(R[n_{j}], N)
≅ ∏_{j} F_{n_{j}−i}N

Therefore,

gr(HOM_{R}(P_{n}, N))_{i} ≅ \frac{∏_{j} F_{n_{j}−i}N}{∏_{j} F_{n_{j}−i−1}N}
≅ ∏_{j} \frac{F_{n_{j}−i}N}{F_{n_{j}−i−1}N}
= ∏_{j} gr(N)_{n_{j}−i}
≅ ∏_{j} Hom_{gr(R)}^{i}(gr(R)[n_{j}], gr(N))
≅ Hom_{gr(R)}^{i}(⨁_{j} gr(R)[n_{j}], gr(N))
≅ Hom_{gr(R)}^{i}(gr(P_{n}), gr(N)).

Hence, we have a graded isomorphism,

gr(HOM_{R}(P_{n}, N) ≅ HOM_{gr(R)}(gr(P_{n}), gr(N))

for all n. Of course, it is possible to prove this isomorphism directly, by showing that the map given by

f ↦ (x + F_{j−1}M ↦ f(x) + F_{j−i−1}N)

for f ∈ F_{i}Hom_{R}(P_{n}, N) and x ∈ F_{j}M has F_{i+1}Hom_{R}(P_{n}, N) as its kernel and is surjective. (We need to use the fact that P_{n} is filtered-free to prove surjectivity here.) As these isomorphisms are compatible with the appropriate boundary maps, we therefore have a filtered complex HOM_{R}(P, N) for which

H^{∗}(HOM_{R}(P, N)) = EXT_{R}^{∗}(M, N)

and

H^{∗,∗}(gr(HOM_{R}(P, N))) ≅ H^{∗,∗}(HOM_{gr(R)}(gr(P), gr(N)))
= EXT_{gr(R)}^{∗,∗}(gr(M), gr(N)).

Therefore, this filtered complex gives rise to the (cohomological) May Spectral Sequence, which under suitable convergence conditions is of the form

E_{1}^{p,q} = EXT_{gr(R)}^{p,q}(gr(M), gr(N)) ⟹ EXT_{R}^{p+q}(M, N).

It is now possible to analyse convergence conditions in a similar fashion as for the homological MSS. We shall just give a fairly straightforward case.

Theorem 5 Suppose R be a filtered ring and M and N are filtered left R-modules, such that the filtrations on R, M and N are exhaustive and separated. Suppose further that M has a filtered-free R-resolution, P, such that Hom_{R}(P_{n}, N) has a bounded filtration for all n. Then there is a convergent spectral sequence

E_{1}^{p,q} = Ext_{gr(R)}^{p,q}(gr(M), gr(N)) ⟹ Ext_{R}^{p+q}(M, N).

Here p = i denotes the filtration degree and q denotes the complementary degree, so that p + q = n is the (co)homological degree, and the differential is of the form d_{r} : E_{r}^{p,q} → E_{r}^{p+r,q−r+1}. Using the filtration defined in 2.1.2, we have isomorphisms

E_{∞}^{p,q} ≅ F_{p}Ext_{p+q}^{R}(M, N)/F_{p+1}Ext_{p+q}^{R}(M, N).

Furthermore, for every n we can find an r(n) such that E_{r(n)}^{p,q} ≅ E_{∞}^{p,q} whenever p + q = n. □

Proof Note that the condition on the filtration ensures that Hom_{R}(P_{n}, N) = HOM_{R}(P_{n}, N) and Ext_{R}^{∗}(M, N) = EXT_{R}^{∗}(M, N) for all n, with similar isomorphisms for the graded modules over gr(R). Convergence is clear. □

Remarks As for the homological MSS, the comparison theorem (Proposition 2) shows that cohomological MSS is natural in its variables and is independent (i.e., up to a natural spectral sequence isomorphism) of the choice of the filtered resolution used to define it.

Although we have stuck to strict filtered-free resolutions for ease of exposition, it is straightforward to carry through both the homological and cohomological spectral sequence constructions above using strict filtered-projective resolutions: essentially, we just need the fact that a filtered-projective module is just a filtered direct summand of a filtered-free module.

2.4 Products

It is possible to introduce both Yoneda and cup product structures into the cohomological MSS under suitable conditions.

2.4.1 Cup products

We shall first deal with cup products in the case of a filtered Hopf algebra Λ: the advantage here is that comultiplication allows us to define a pairing of MSS’s independently of the resolutions chosen to define them. This is generally not the case for Yoneda composition. We shall write E_{r}^{∗,∗}(M, N) for the r^{th} page of the cohomological MSS corresponding to filtered Λ modules M and N.

Lemma 14 Suppose Λ is a filtered Hopf algebra over the commutative filtered ring S, and assume that Λ is filtered-free as an S-module. Let M, M′, N and N′ be filtered Λ-modules that are filtered-free as S-modules. Then there is a pairing of spectral sequences

\begin{matrix}E_{r}^{p,q}(M, N) ⊗_{S} E_{r}^{p′,q′}(M′, N′) & → & E_{r}^{p+p′,q+q′}(M ⊗_{S} M′, N ⊗_{S} N′) \\ x ⊗_{S} y & ↦ & xy \end{matrix}

(r = 1, 2, 3, …) such that (i) d_{r}(xy) = d_{r}(x)y + (−1)^{p+q}xd_{r}(y); (ii) when r = 1 this pairing is just the usual cup product on the graded EXT

groups;

(iii) if all the spectral sequences involved converge, this pairing induces a pairing on the E_{∞} pages that equals the pairing induced by the filtered cup product

EXT_{Λ}^{m}(M, N) ⊗_{S} EXT_{Λ}^{n}(M′, N′) → EXT_{Λ}^{m+n}(M ⊗_{S} M′, N ⊗_{S} N′)

via the isomorphisms

E_{∞}^{p,q}(X, Y) ≅ F_{p}Ext_{p+q}^{R}(X, Y)/F_{p+1}Ext_{p+q}^{R}(X, Y).

Proof This all follows (see [4, Vol.2, 3.9] or [17, 7]) from a suitable pairing of filtered chain complexes

HOM_{R}(P, N) ⊗_{S} HOM_{R}(Q, N′) → HOM_{R}(P ⊗_{S} Q, N ⊗_{S} N′).

But we can just take the chain map we defined in 2.1.2 when we defined the filtered cup product. □

2.4.2 Yoneda products

To define Yoneda composition, we shall assume that A is a filtered augmented algebra over a field k. We shall take a specific filtered-free resolution to define a pairing of spectral sequences, namely the bar construction, B(A) = B(A, k). We shall write ⊗ for ⊗_{k}.

Now given a filtered k-vector space V, we define

F_{i}(V^{∗}) = {ϕ ∈ V^{∗} | ϕ(V_{j}) ⊆ V_{j−i} for all j ∈ ℤ}.

We shall abuse notation, and write V^{∗} for HOM_{k}(V, k) = ⋃_{i} F_{i}(V^{∗}). Then we have filtered isomorphisms

HOM_{A}(B_{n}(A), k) ≅ HOM_{A}(A ⊗ A[1]^{⊗^{n}}, k)
≅ HOM_{k}(A[1]^{⊗^{n}}, k)
≅ (A[1]^{⊗^{n}})^{∗}
≅ A^{∗}[1]^{⊗^{n}}.

Thus we obtain the filtered cobar complex, in which the differential is given by

d^{n}[α_{1}|α_{2}| ⋯ |α_{n−1}] = ∑_{1≤r≤n; u}(−1)^{r}[α_{1}| ⋯ |α′_{r,u}|α′′_{r,u}| ⋯ |α_{n−1}]

where we write ∑_{u} α′_{r,u} ⊗ α′′_{r,u} for the image of α_{r} under the comultiplication A^{∗} → A^{∗} ⊗ A^{∗} given by the dual of multiplication. The cohomology of this complex is of course just EXT_{A}^{∗}(k, k); and the associated graded complex is just the cobar complex for gr(A).

There is a cup product of cochains in the cobar complex

A^{∗}[1]^{⊗^{m}} ⊗ A^{∗}[1]^{⊗^{n}} → A^{∗}[1]^{⊗^{(m+n)}}
[α_{1}| ⋯ |α_{m}] ⊗ [β_{1}| ⋯ |β_{n}] ↦ [α_{1}| ⋯ |α_{m}|β_{1}| ⋯ |β_{n}]

which satisfies d(xy) = d(x)y+(−1)^{m}d(y) for all x ∈ A^{∗}[1]^{⊗^{m}} and y ∈ A^{∗}[1]^{⊗^{n}} where d is the coboundary map. Moreover, this product induces the Yoneda product on cohomology. (See [1, p.33] for the proofs if we ignore the filtered structure. What is crucial here, is to observe that all these identifications and maps respect the filtrations involved so the proofs carry over directly to the filtered case.) This pairing of cobar constructions therefore gives the following result.

Lemma 15 Suppose A is a finite-dimensional filtered augmented algebra over a field k. Then there is a Yoneda product structure on the cohomological MSS whose first page is EXT_{gr(A)}^{∗,∗}(k, k), such that

(i) the r^{th}-page, E_{r}, is a differential algebra with respect to d_{r};

(ii) the product on the first page is just Yoneda composition on EXT_{gr(A)}^{∗,∗}(k, k); (iii) if the spectral sequence converges, E_{∞} ≅ gr(EXT_{A}^{∗}(k, k)) as bigraded k-

algebras. □

Remark Although we have use the bar resolution to define this product structure, we recall that the MSS does not depend on the choice of strict filtered-free resolution used to define it, so we may always impose this underlying product structure.

2.4.3 Duality

Finally, we have one more pairing that induces a duality between homological and cohomological MSS’s.

Lemma 16 If A is a finite-dimensional filtered algebra over a field k, and M and N are finite-dimensional filtered left A-modules, then the cohomological MSS

Ext_{gr(A)}^{∗,∗}(gr(M), gr(N)) ⟹ Ext_{A}^{∗}(M, N)

is dual to the homological MSS

Tor_{∗,∗}^{gr(A)}(gr(N^{∗}), gr(M)) ⟹ Tor_{∗}^{A}(N^{∗}, M).

Proof Given any strict filtered-free A-resolution, P, of M, the duality detween the spectral sequences is induced by the pairing of filtered cochain complexes

(N^{∗} ⊗_{A} P) ⊗_{k} HOM_{A}(P, N) → k

given by

(α ⊗_{A} x_{m}) ⊗_{k} ϕ_{n} ↦ \{ \begin{matrix}(−1)^{mn}α(ϕ_{n}(x_{m})) & if m = n \\ 0 & if m ≠ n \end{matrix}

where α ∈ N^{∗}, x_{m} ∈ P_{m} and ϕ_{n} ∈ Hom_{A}(P_{n}, N). □

The triple graded case for graded rings

May constructed his spectral sequence as a triple graded spectral for graded rings. In this chapter, we define a triple graded spectral sequence, extending our previous constructions. (We prove in Appendix 1 that this definition does indeed give, as a special case, the spectral sequence originally defined by May in [19].)

3.1 The spectral sequence

Suppose R = ⨁_{t∈ℤ} R_{t} is a graded ring which has a filtration (F_{i}R)_{i∈ℤ}. Writing F_{i}R_{t} for F_{i}R ∩ R_{t}, we say R is a filtered-graded ring if F_{i}R = ⨁_{t∈ℤ} F_{i}R_{t} for all i. For such a ring we have

gr(R) = ⨁_{i∈ℤ} \frac{F_{i}R}{F_{i−1}R} = ⨁_{i∈ℤ}(⨁_{t∈ℤ} \frac{F_{i}R_{t}}{F_{i−1}R_{t}}),

so gr(R) is a bigraded ring. Filtered-graded R-modules, filtered-graded R-module homomorphisms, and so on, are defined in the obvious way.

Now suppose P is a strict exact filtered-graded-free R-resolution of a filteredgraded left R-module N. So each P_{n} is isomorphic as a filtered-graded module to some direct sum of the form ⨁_{i} R[m_{i}, n_{i}], where

(F_{i}R[m, n])_{t} = F_{i−m}R_{t−n},

and the differentials are graded maps that are strict as filtered R-module homorphisms. (It is possible to construct such resolutions by extending the methods in Chapter 1 in the obvious way: indeed, if R is a filtered-graded algebra over a field, then we can take P to be the bar resolution for N - cf. Proposition 1.) We see that gr(P) is a bigraded-free R-resolution of gr(N). So if M is filtered-graded right R-module, we can form the filtered-graded complex

M ⊗_{R} P = ⨁_{n≥0} ⨁_{t∈ℤ}(M ⊗_{R} P_{n})_{t}.

Now for fixed i and t, (F_{i}(M ⊗_{R} P))_{t} is a filtered subcomplex. Defining

F_{i}Tor_{n,t}^{R}(M, N) = Im (H_{n}((F_{i}(M ⊗_{R} P))_{t}) → H_{n}((M ⊗_{R} P)_{t}),

we have an isomorphism

F_{i}Tor_{n}^{R}(M, N) = ⨁_{t∈ℤ} F_{i}Tor_{n,t}^{R}(M, N).

Now the bigraded isomorphism

gr(M ⊗_{R} P) ≅ gr(M) ⊗_{gr(R)} gr(P)

induces graded isomorphisms

gr((M ⊗_{R} P)_{t}) ≅ (gr(M) ⊗_{gr(R)} gr(P))_{t},

for each fixed t. Hence, for each t, we obtain a spectral sequence of the form

E_{n,i}^{1} = Tor_{n,i,t}^{gr(R)}(gr(M), gr(N)) ⟹ Tor_{n,t}^{R}(M, N)

under suitable convergence conditions (e.g., the filtrations of each (M ⊗_{R} P_{n})_{t} are bounded). These spectral sequences are usually written together in the form of a triple graded spectral sequence.

Theorem 6 (J.P.May) Suppose R be a filtered graded ring and M and N are, respectively, filtered graded right and left R-modules. Suppose further that N has a strict filtered-free graded R-resolution, P, such that (M ⊗_{R} P_{n})_{t} has a bounded filtration for all n and t. Then there is a convergent triple graded spectral sequence

E_{p,q,t}^{1} = Tor_{p,q,t}^{gr(R)}(gr(M), gr(N)) ⟹ Tor_{p+q,t}^{R}(M, N).

Here p = i denotes the filtration degree and q denotes the complementary degree, so that p + q = n is the homological degree, and the differential is of the form d^{r} : E_{p,q,t}^{r} → E_{p−r,q+r−1,t}^{r}. We have isomorphisms

E_{p,q,t}^{∞} ≅ F_{p}Tor_{p+q,t}^{R}(M, N)/F_{p−1}Tor_{p+q,t}^{R}(M, N).

Furthermore, for every n and t we can find an r(n, t) such that E_{p,q,t}^{r(n,t)} ≅ E_{p,q,t}^{∞} whenever p + q = n. □

Now suppose M and N are filtered-graded left R-modules. We define

F_{i}Hom_{R}^{t}(M, N) = {ϕ ∈ Hom_{R}(M, N) | ϕ(F_{j}M_{s}) ⊆ F_{j−i}N_{s−t} for all j, s ∈ ℤ},

and then define

HOM_{R}^{t}(M, N) = ⋃_{i} F_{i}Hom_{R}^{t}(M, N),

HOM_{R}(M, N) = ⨁_{t} HOM_{R}^{t}(M, N).

We may use these functors and a strict filtered-graded-free R-resolution for M, to define the appropriate filtered-graded EXT groups (cf. 2.1).

Similarly, for a bigraded modules M and N over a bigraded ring R, we define

Hom_{R}^{i,t}(M, N) = {ϕ ∈ Hom_{R}(M, N) | ϕ(M_{j,s}) ⊆ N_{j−i,s−t} for all j, s ∈ ℤ},

and then set

HOM_{R}^{t}(M, N) = ⨁_{i} Hom_{R}^{i,t}(M, N),

HOM_{R}(M, N) = ⨁_{t} HOM_{R}^{t}(M, N);

and then define bigraded EXT groups in the obvious way, using a bigraded-free resolution for N.

To construct our triple graded cohomological spectral sequence, we assume R is a filtered-graded ring, M and N are filtered-graded R-modules, and P is a filteredgraded-free R-resolution for M. Filtering HOM_{R}(P, N) in the way defined above, we have isomorphisms

gr(Hom_{R}^{t}(P_{n}, N))_{i} ≅ Hom_{gr(R)}^{i,t}(gr(P_{n}), gr(N))

inducing an isomorphism of bigraded complexes,

gr(HOM_{R}(P, N)) ≅ HOM_{gr(R)}(gr(P), gr(N)).

Therefore, assuming each Hom_{R}^{t}(P_{n}, N) has a bounded filtration for all n and t, to guarantee convergence, we get the following result.

Theorem 7 (J.P.May) Under the assumptions and notation listed above, we have a convergent triple graded spectral sequence

E_{1}^{p,q,t} = EXT_{gr(R)}^{p,q,t}(gr(M), gr(N)) ⟹ EXT_{R}^{p+q,t}(M, N),

where d_{r} : E_{r}^{p,q,t} → E_{r}^{p+r,q−r+1,t}. □

Remarks Under suitable circumstances, it is straightforward to introduce a Yoneda or cup product structure into the triple graded cohomological MSS.

If we give a filtered ring R and filtered R-modules M and N the trivial grading (i.e., R = R_{0}, M = M_{0} and N = N_{0}) we effectively just get the MSS corresponding to the ungraded case (i.e., as in Theorem 3 or Theorem 5) as the t = 0 part of the corresponding triple graded MSS. (For t ≠ 0, the terms in the triple graded spectral sequence are of course just all zero.)

See Appendix 1 for a proof that the spectral sequences defined by May in [19] are special cases of Theorems 6 and 7.

Restricted Lie algebras

In this chapter, we apply the MSS to give easy proofs of some basic results in the cohomology of restricted Lie algebras. In turn, these results are important for understanding the MSS of a (finite-dimensional) local Hopf algebra over a field of non-zero characteristic, which is filtered by powers of its maximal ideal.

4.1 Polynomial and truncated polynomial algebras

First we gather together some notation and basic results that we shall need.

Given a (graded) vector space V over a field k, we shall denote the (graded) symmetric algebra on V by

S(V) = T(V)/(x ⊗ y − y ⊗ x | x, y ∈ V),

and the (graded) alternating algebra on V by

Λ(V) = T(V)/(x ⊗ x | x ∈ V).

We shall denote the divided power algebra on V by Γ(V). If V has a basis, x_{1}, …, x_{n}, we may identify this with the algebra Γ(x_{1}) ⊗ ⋯ ⊗ Γ(x_{n}), where

Γ(x) = k ⊗_{ℤ} ℤ[x, x^{2}/2!, x^{3}/3!, …, x^{r}/r!, …],

and we may define γ_{r}(x) to be 1 ⊗_{ℤ} x^{r}/r!. We note that we have natural isomorphisms, Λ^{m}(V)^{∗} ≅ Λ^{m}(V^{∗}) and Γ^{m}(V)^{∗} ≅ S^{m}(V^{∗}).

In what follows below, we shall often write a free resolution for a filtered or graded A-module in the form A ⊗ V where V is, respectively, a filtered- or gradedfree k-vector space. In many cases, we shall write V as a graded algebra (cf. the bar resolution), but this should be regarded as a convenient way of labelling elements rather than implying extra structure in the resolution (which can be imposed - see [19], for example - but is unnecessary for our purposes).

Lemma 17 Let k[x] be a graded polynomial ring on one variable of degree t over a field k. Let V = kx be the graded vector space spanned by x, and write the graded dual as V^{∗} = ky. Then we have a minimal graded-free k[x]-resolution of k of the form k[x] ⊗ Λ(V [1]), such that the natural product in Λ(V [1]^{∗}) induces an isomorphism H^{∗,∗}(k[x]) ≅ Λ(y), where the filtered/complementary bidegree of y is (t, 1 − t).

Proof We have a minimal graded-free resolution of k,

k[x][t] →^{x} k[x][0] → k. □

Lemma 18 Let k[x]/(x^{2}) be the graded truncated polynomial algebra of dimension two over a field k, where the degree of x is t. Then the trivial k[x]/(x^{2})-module, k, has a minimal graded-free resolution of the form k[x]/(x^{2}) ⊗ Γ(V [1]), where V = kx. Furthermore, the natural product in Γ(V [1])^{∗} = S(V [1]^{∗}) = k[y] induces an isomorphism of bigraded algebras

H^{∗,∗}(k[x]/(x^{2})) ≅ k[y]

where the filtered/complementary bidegree of y is (t, 1 − t)

Proof Let A = k[x]/(x^{2}). Then we have a minimal graded-free A-resolution of the trivial A-module k of the form

⋯ → A[4t] →^{x} A[3t] →^{x} A[2t] →^{x} A[t] →^{x} A[0] →^{ϵ} k.

The graded Yoneda structure and the bidegrees of y follow by direct calculation. □

Given a graded vector space, V, we shall use V^{[n]} to denote the graded vector space on the same underlying vector space as V, but with the graded degrees multiplied by n. That is,

(V^{[n]})_{i} = \{ \begin{matrix}V_{j} & if i = nj \\ 0 & otherwise . \end{matrix}

Lemma 19 Let k[x]/(x^{n}) be the graded truncated polynomial ring over a field k, where n > 2 and the degree x is t. Then the trivial k[x]/(x^{n})-module, k, has a minimal graded-free resolution of the form

k[x]/(x^{n}) ⊗ (Λ(V [1]) ⊗ Γ(V^{[n]}[2])),

where V = kx. Furthermore, the natural product in

(Λ(V [1]) ⊗ Γ(V^{[n]}[2]))^{∗} ≅ Λ(V [1])^{∗} ⊗ Γ(V^{[n]}[2])^{∗} ≅ Λ(u) ⊗ k[v],

induces an isomorphism of bigraded algebras

H^{∗,∗}(k[x]/(x^{n})) ≅ Λ(u) ⊗ k[v]

where the filtered/complementary bidegrees are given by

bideg u = (t, 1 − t), bideg v = (nt, 2 − nt).

Proof Let A = k[x]/(x^{n}). Then we have a minimal graded-free A-resolution of the trivial A-module k of the form

⋯ → A[2nt] →^{x^{n−1}} A[(n + 1)t] →^{x} A[nt] →^{x^{n−1}} A[t] →^{x} A[0] →^{ϵ} k.

The remaining identifications are straightforward. □

Lemma 20 Suppose R and S are finite-dimensional local k-algebras over a field k, having maximal ideals I and J, respectively, so that R/I = S/J = k. Then if R⊗_{k} S is filtered by powers of its radical ideal, we have an isomorphism of graded k-algebras

gr(R ⊗ S) ≅ gr(R) ⊗ gr(S).

Proof Since rad (R ⊗ S) = I ⊗ S + R ⊗ J, we have

rad^{i} (R ⊗ S) = ∑_{r=1}^{i} I^{r} ⊗ J^{i−r},

where I^{0} = R and J^{0} = S. The result now follows from Lemma 7. □

Corollary 2 If P and Q are finite p-groups, and k is a field of characteristic p, then

gr(k(P × Q)) ≅ gr(kP) ⊗ gr(kQ).

Moreover, if A is an abelian p-group, there is an isomorphism of augmented kalgebras,

gr(kA) ≅ kA.

Proof We have an isomorphism k(P ×Q) ≅ kP ⊗kQ and so the first remark follows from the lemma above. The second remark then follows from the observation

kC_{p^{n}} ≅ k[x]/(x^{p^{n}}) ≅ gr(kC_{p^{n}}). □

Lemma 21 If R and S are Hopf algebras over a field k, then there is an isomorphism of graded algebras

H^{∗}(R ⊗ S) ≅ H^{∗}(R) ⊗ H^{∗}(S)

if we adopt the usual sign convention for the tensor product of cohomology rings by defining

(x_{1} ⊗ y_{1})(x_{2} ⊗ y_{2}) = (−1)^{(deg x_{2})(deg y_{1})}x_{1}x_{2} ⊗ y_{1}y_{2}

for homogeneous x_{i} ∈ H^{∗}(R) and y_{i} ∈ H^{∗}(S). In the case when R and S are filtered (respectively graded) Hopf algebras and R ⊗ S is filtered (respectively graded) via the filtrations (respectively gradings) on R and S, this isomorphism is one of filtered graded (respectively bigraded) algebras.

Proof For the ordinary case, we take a free R-resolutions, P, of the trivial Rmodule k and a free S-resolution, Q, of the trivial S-module k. Then by the Künneth theorem (cf. [4, p.62, Vol.1]) P ⊗ Q is a free R ⊗ S-resolution of k and there is an isomorphism of complexes

Hom_{R⊗S}(P ⊗ Q, k) ≅ Hom_{R}(P, k) ⊗ Hom_{S}(Q, k)

that induces an isomorphism of cohomology groups. This isomorphism of groups is in fact an isomorphism of rings, because of the definition of the cup product via diagonal maps

P →^{Δ_{1}} P ⊗ P

Q →^{Δ_{2}} Q ⊗ Q, and

P ⊗ Q ⟶^{Δ_{1}⊗Δ_{2}} (P ⊗ P) ⊗ (Q ⊗ Q) ⟶^{1⊗τ⊗1} (P ⊗ Q) ⊗ (P ⊗ Q).

For the filtered case, we assume that P and Q are strict filtered-free resolutions. Then P ⊗ Q is also strict and filtered-free as an R ⊗ S-resolution. Moreover, by Proposition 2, we can ensure that all the diagonal maps are filtered. Interpreting H^{∗}(R) as EXT_{R}^{∗}(k, k), the result now follows from the isomorphism of filtered complexes,

HOM_{R⊗S}(P ⊗ Q, k) ≅ HOM_{R}(P, k) ⊗ HOM_{S}(Q, k).

The graded case is similar. □

4.2 Ordinary Lie algebras

Let 𝔤 be a finite dimensional Lie algebra over a field k. We define its universal enveloping algebra to be

U𝔤 = T(𝔤)/(x ⊗ y − y ⊗ x − [x, y] | x, y ∈ 𝔤).

This is a Hopf algebra with comultiplication induced by the map

𝔤 → U𝔤 ⊗ U𝔤,
x ↦ x ⊗ 1 + 1 ⊗ x.

Letting π : T(𝔤) → U𝔤 be the natural surjection, we define the Lie filtration of U𝔤 as

F_{i}(U𝔤) = \{ \begin{matrix}π(k + 𝔤 + … + 𝔤^{⊗^{i}}) & if i ≥ 0 \\ 0 & if i < 0. \end{matrix}

This filtration gives U𝔤 the structure of a filtered Hopf algebra, since multiplication and comultiplication are both filtered homomorphisms. (Clearly, both unit and counit are filtered.)

The proof of the Poincaré-Birkhoff-Witt Theorem ([13, p.159]) implies that there is an isomorphism of graded algebras

gr(U𝔤) ≅ S(𝔤)

where we regard 𝔤 as a graded vector space of degree one.

The trivial S(𝔤)-module has a minimal (graded-free) resolution of the form S(𝔤)⊗ Λ(𝔤[1]), by Lemma 17 and the proof of Lemma 21. Since the filtration on U𝔤 is bounded below, it is certainly complete, so we may apply Theorem 1 and ungrade this resolution to obtain a strict filtered-free resolution of the trivial U𝔤-module of the form U𝔤 ⊗ Λ(𝔤[1]). Thus we can obtain the standard resolution for Lie algebras ([12, VII 4.2]). However, in order to compute cohomology, we need not ungrade the entire resolution, merely the first few terms:

… → S(𝔤) ⊗ Λ^{2}(𝔤) →^{δ_{2}} S(𝔤) ⊗ Λ^{1}(𝔤) →^{δ_{1}} S(𝔤) → k.

If we write ⟨x_{1} ∧ … ∧ x_{n}⟩ to denote 1 ⊗ (x_{1} ∧ … ∧ x_{n}) ∈ S(𝔤) ⊗ Λ^{n}(𝔤), δ_{1} and δ_{2} are determined by δ_{1}(⟨x⟩) = x and δ_{2}(⟨x ∧ y⟩) = x⟨y⟩ − y⟨x⟩. It is easy to ungrade these two maps to give the first few terms of a filtered-free resolution,

… → U𝔤 ⊗ Λ^{2}(𝔤) →^{δ′_{2}} U𝔤 ⊗ Λ^{1}(𝔤) →^{δ′_{1}} U𝔤 → k,

where δ′_{1}(⟨x⟩) = x and δ′_{2}(⟨x ∧ y⟩) = x⟨y⟩ − y⟨x⟩ − ⟨[x, y]⟩.

Now, again by Lemmas 17 and 21, we have an isomorphism of bigraded algebras

H^{∗,∗}(S(𝔤)) ≅ Λ(𝔤^{∗}[1]) ≅ Λ(y_{1}, …, y_{n})

where y_{1}, …, y_{n} is a dual basis for 𝔤, each y_{i} having filtered/complementary bidegree (1, 0). Thus, in the corresponding MSS to compute H^{∗}(U𝔤), all the non-zero elements lie on the line q = 0. We therefore have

E_{2} = E_{∞} = H^{∗}(U𝔤).

Since the spectral sequence has a multiplicative structure (in fact, both cup product and Yoneda product give multiplicative structures, but they are equal) it suffices to calculate the differential d_{1}^{1,0} : E_{1}^{1,0} → E_{1}^{2,0} in terms of the product structure to calculate H^{∗}(U𝔤). To work out how the product structure relates to our partial resolution, we could, of course, calculate and then ungrade a diagonal map,

gr(U𝔤) ⊗ Λ(𝔤) → (gr(U𝔤) ⊗ Λ(𝔤)) ⊗ (gr(U𝔤) ⊗ Λ(𝔤)),

for homological dimensions 0, 1 and 2. However, we can also compare the cochain complex

Hom_{U𝔤}(U𝔤 ⊗ Λ(𝔤), k)

with the cobar construction in low dimensions, and thus deduce that the natural product

Λ^{1}(𝔤^{∗}) ⊗ Λ^{1}(𝔤^{∗}) → Λ^{2}(𝔤^{∗})

is in fact just the cup product in the spectral sequence. To do this we begin to construct a chain map, (ϕ_{n}) of filtered chain complexes, as in the proof of Proposition 2, defining ϕ_{0}, ϕ_{1} and ϕ_{2} as follows.

… → U𝔤 ⊗ Λ^{2}(𝔤) → U𝔤 ⊗ Λ^{1}(𝔤) → U𝔤 → k
↓ ϕ_{2} ↓ ϕ_{1} ↓ ϕ_{0} ∥
… → U𝔤 ⊗ T^{2}(U𝔤) → U𝔤 ⊗ T^{1}(U𝔤) → U𝔤 → k

In the diagram above, the top row is the start of our unfiltered resolution and the bottom row consists of the first few terms in the bar resolution for U𝔤: we define filtered U𝔤-module homomorphisms by

ϕ_{0}(1) = 1,
ϕ_{1}(⟨x⟩) = [x],
ϕ_{2}(⟨x ∧ y⟩) = [x|y] − [y|x].

It is easy to check this diagram commutes. Hence, applying the functor

Hom_{U𝔤}(−, k),

we have the following commutative diagram.

⋯ ← Λ^{2}(𝔤)^{∗} ← Λ^{1}(𝔤)^{∗} ← k
↑ ϕ_{2}^{∗} ↑ ϕ_{1}^{∗} ↑ ϕ_{0}^{∗}
⋯ ← T^{2}(U𝔤)^{∗} ← T^{1}(U𝔤)^{∗} ← k.

But we have natural isomorphisms T^{n}(V^{∗}) ≅ T^{n}(V)^{∗} and Λ^{n}(V^{∗}) ≅ Λ^{n}(V)^{∗} for all n. For example, when n = 2, the second isomorphism is given by

ϕ ∧ ψ ↦ (x ∧ y ↦ (ϕ(x)ψ(y) − ϕ(y)ψ(x))).

With these identifications, it is straightforward to check that the cup product is as required.

Finally, since we have a duality between the cohomological and homological MSS’s by Lemma 16, it suffices to calculate

d^{1} : Λ^{2}(𝔤) ≅ E_{2,0}^{1} → E_{1,0}^{1} ≅ Λ^{1}(𝔤).

But this is just x ∧ y ↦ −[x, y], as can be seen immediately from the explicit calculation of δ′_{2} given above. Thus we have proved the following well-known result.

Theorem 8 If 𝔤 is a (finite-dimensional) Lie algebra over a field k, then

H^{∗}(U𝔤) ≅ H(Λ(𝔤^{∗}), d)

where d is a graded derivation with d_{1} given by

Λ^{1}(𝔤^{∗}) ≅ (Λ^{1}(𝔤))^{∗} →^{d^{2}^{∗}} (Λ^{2}(𝔤))^{∗} ≅ Λ^{2}(𝔤^{∗})

where d^{2}(x ∧ y) = −[x, y]. □

4.3 The cohomology of restricted Lie algebras

We now adapt the methods of the previous section to consider the cohomology of restricted Lie algebras.

A restricted p-Lie algebra is a Lie algebra, 𝔤, over a field k of characteristic p ≠ 0, which has the additional structure of a p^{th}-power operation, x ↦ x^{[p]}, satisfying

(i) (λx)^{[p]} = λ^{p}x^{[p]},
(ii) (x + y)^{[p]} = x^{[p]} + y^{[p]} + ∑_{i=1}^{p−1} s_{i}(x, y), and
(iii) [x, y^{[p]}] = x(ad y)^{p},

for all λ ∈ k and x, y ∈ 𝔤, where ad is the the adjoint map and is_{i}(x, y) is the coefficient of t^{i−1} in x(ad (tx + y))^{p−1}. We define the restricted enveloping algebra of such a 𝔤 as

V𝔤 = T(𝔤)/(x ⊗ y − y ⊗ x − [x, y], x^{⊗^{p}} − x^{[p]} | x, y ∈ 𝔤).

This is a Hopf algebra with comultiplication induced by the map

𝔤 → V𝔤 ⊗ V𝔤,
x ↦ x ⊗ 1 + 1 ⊗ x.

Thus there is clearly a natural epimorphism of Hopf algebras U𝔤 → V𝔤.

Now letting π : T(𝔤) → V𝔤 be the natural surjection, we have the Lie filtration on V𝔤 given by

F_{i}(V𝔤) = \{ \begin{matrix}π(k + 𝔤 + … + 𝔤^{⊗^{i}}) & if i ≥ 0 \\ 0 & if i < 0. \end{matrix}

For this case, there is also a ‘Poincaré-Birkhoff-Witt Theorem’. Given a restricted Lie algebra, 𝔤, we let 𝔤^{♯} be the abelian restricted Lie algebra having the same underlying vector space as 𝔤 with restriction zero (i.e., x^{[p]} = 0 for all x ∈ 𝔤^{♯}). Then by the universal property of the the restricted enveloping algebra, the map 𝔤^{♯} → gr(V𝔤) given by x ↦ x + k ∈ gr(V𝔤)_{1} induces a map of Hopf algebras

V(𝔤^{♯}) → gr(V𝔤).

This is a map of graded Hopf algebras if elements in 𝔤^{♯} are given graded degree one.

Lemma 22 If the restricted enveloping algebra of a restricted Lie algebra, 𝔤, is filtered by the Lie filtration, there is a natural isomorphism of graded Hopf algebras V(𝔤^{♯}) ≅ gr(V𝔤).

Proof This is proved by Milnor and Moore in [20, Thm. 6.20]. □

4.3.1 Even characteristic

We now assume that 𝔤 is a finite-dimensional restricted Lie algebra over a field k of characteristic 2. If we chose a basis {x_{1}, …, x_{t}} for 𝔤, it is clear that there is an isomorphism of graded algebras,

V(𝔤^{♯}) ≅ k[x_{1}, …, x_{t}]/(x_{1}^{2}, …, x_{t}^{2}) ≅ k(\underbrace{C_{2} × ⋯ × C_{2}}_{t times}).

So by Lemmas 18 and 21 there is a graded-free minimal resolution for the trivial V(𝔤^{♯})-module k of the form

(V(𝔤^{♯}) ⊗ Γ(𝔤[1]), δ).

Ungrading the first few terms to obtain a filtered-free V𝔤-resolution of k, we get

… → V(𝔤) ⊗ Γ^{2}(𝔤) →^{δ′_{2}} V(𝔤) ⊗ Γ^{1}(𝔤) →^{δ′_{1}} V(𝔤) → k,

where

δ′_{1}(⟨γ_{1}(x)⟩) = x,
δ′_{2}(⟨γ_{1}(x)γ_{1}(y)⟩) = x⟨γ_{1}(y)⟩ + y⟨γ_{1}(x)⟩ + ⟨γ_{1}([x, y])⟩,
δ′_{2}(⟨γ_{2}(x)⟩) = x⟨γ_{1}(x)⟩ + ⟨γ_{1}(x^{[2]})⟩.

Moreover, by Lemmas 22, 18 and 21, we have isomorphisms of bigraded algebras

H^{∗,∗}(gr(V𝔤)) ≅ H^{∗,∗}(V𝔤^{♯}) ≅ S(𝔤^{∗}[1]).

Hence in the corresponding MSS to compute H^{∗}(V𝔤), all the non-zero elements lie on the line q = 0. We therefore have

E_{2} = E_{∞} = H^{∗}(V𝔤).

To calculate the product structure in low dimensions and

d_{1} : E_{1}^{1,0} → E_{1}^{2,0}

in the spectral sequence, we again find a partial chain map,

… → V𝔤 ⊗ Γ^{2}(𝔤) → V𝔤 ⊗ Γ^{1}(𝔤) → V𝔤 → k
↓ ϕ_{2} ↓ ϕ_{1} ↓ ϕ_{0} ∥
… → V𝔤 ⊗ T^{2}(V𝔤) → V𝔤 ⊗ T^{1}(V𝔤) → V𝔤 → k

where ϕ_{0}, ϕ_{1} and ϕ_{2} are given by

ϕ_{0}(1) = 1,
ϕ_{1}(⟨γ_{1}(x)⟩) = [x],
ϕ_{2}(⟨γ_{1}(x)γ_{1}(y)⟩) = [x|y] + [y|x],
ϕ_{2}(⟨γ_{2}(x)⟩) = [x|x].

By applying Hom_{V𝔤}(−, k), we see that the natural product in S(𝔤^{∗}) ≅ Γ(𝔤)^{∗} is the cup product. To check this we note the isomorphism

S^{2}(𝔤^{∗}) ≅ Γ^{2}(𝔤)^{∗}

is given by

ϕψ ↦ \{ \begin{matrix}γ_{1}(x)γ_{1}(y) & ↦ & ϕ(x)ψ(y) + ϕ(y)ψ(x) \\ γ_{2}(x) & ↦ & ϕ(x)ψ(x) \end{matrix}

for ϕ, ψ ∈ S^{1}(𝔤^{∗}) ≅ 𝔤^{∗}.

Theorem 9 (May) If 𝔤 is a finite-dimensional restricted Lie algebra over a field k of characteristic 2, then

H^{∗}(V𝔤) ≅ H(S(𝔤^{∗}, d))

where d is a graded derivation determined as the dual of the map d^{2} : Γ^{2}(𝔤) → Γ^{1}(𝔤) which is given by

d^{2}(γ_{1}(x)γ_{1}(y)) = γ_{1}([x, y]), d^{2}(γ_{2}(x)) = γ_{1}(x^{[2]}). □

Corollary 3 If 𝔤 is a finite dimensional restricted Lie algebra over a field of characteristic 2, then H^{∗}(V𝔤) is a finitely generated commutative k-algebra. □

4.3.2 Odd characteristic

We keep the same notation as above, except that we shall now assume that the underlying characteristic, p, is odd.

In this case, we have isomorphisms of graded algebras,

gr(V𝔤) ≅ V(𝔤^{♯}) ≅ k[x_{1}, …, x_{t}]/(x_{1}^{p}, …, x_{t}^{p}) ≅ k(\underbrace{C_{p} × ⋯ × C_{p}}_{t times}).

By Lemmas 19 and 21, there is a graded-free minimal resolution for the trivial V(𝔤^{♯})-module k of the form

(V(𝔤^{♯}) ⊗ (Λ(𝔤[1]) ⊗ Γ(𝔤^{[p]}[2])), δ).

Again, we may ungrade the first few terms to obtain the start of a filtered-free V𝔤-resolution of k:

… → V(𝔤) ⊗ (Λ(𝔤) ⊗ Γ(𝔤))_{2} →^{δ′_{2}} V(𝔤) ⊗ (Λ(𝔤) ⊗ Γ(𝔤))_{1} →^{δ′_{1}} V(𝔤),

where

δ′_{1}(⟨x ⊗ 1⟩) = x,
δ′_{2}(⟨(x ∧ y) ⊗ 1⟩) = x⟨y ⊗ 1⟩ − y⟨x ⊗ 1⟩ − ⟨[x, y] ⊗ 1⟩,
and δ′_{2}(⟨1 ⊗ γ_{1}(x)⟩) = x^{p−1}⟨x ⊗ 1⟩.

Now, using Lemmas 19 and 21 again, we have isomorphisms of bigraded algebras

H^{∗,∗}(gr(V𝔤)) ≅ Λ(𝔤^{∗}[1]) ⊗ S(𝔤^{[p]∗}[2]).

Thus the E_{1}-page of the corresponding MSS is generated as an algebra by elements with filtered/complementary bidegrees (1, 0) and (p, 2 − p).

We can identify d_{1} in this spectral sequence as follows. By embedding the start of our resolution in the bar resolution, it is straightforward to check that the natural product

Λ^{1}(𝔤^{∗}) ⊗ Λ^{1}(𝔤^{∗}) → Λ^{2}(𝔤^{∗})

is just the cup product. Now the natural projection ρ : U𝔤 → V𝔤 is a filtered homomorphism if we give both U𝔤 and V𝔤 the Lie filtration. Thus by Proposition 2, we get a chain map

ψ : U𝔤 ⊗ Λ(𝔤[1]) → V𝔤 ⊗ (Λ(𝔤[1]) ⊗ Γ(𝔤^{[p]}[2]))

which in small degrees is given by

ψ_{0} = ρ,
ψ_{1}(⟨x⟩) = ⟨x ⊗ 1⟩,
ψ_{2}(⟨x ∧ y⟩) = ⟨(x ∧ y) ⊗ 1⟩.

Hence we see that the derivation d_{1} restricted to E_{1}^{∗,0} is just the same as that given in Theorem 8. Thus we have the following result.

Theorem 10 (May) If 𝔤 is a finite-dimensional restricted Lie algebra over a field k of odd characteristic, then there is a convergent spectral sequence

E_{2}^{∗,∗} = H^{∗}(U𝔤) ⊗ S(𝔤^{[p]∗}) ⟹ H^{∗}(V𝔤)

where elements in H^{n}(U𝔤) have filtered/complementary bidegree (n, 0), and elements of S^{m}(𝔤^{[p]∗}) have filtered/complementary bidegree (pm, (2 − p)m). □

Corollary 4 If 𝔤 is a finite dimensional restricted Lie algebra over a field of odd characteristic, then H^{∗}(V𝔤) is a finitely generated graded commutative k-algebra. □

Remarks If 𝔤 is a graded ordinary or restricted Lie algebra, then the appropriate enveloping algebra inherits a natural grading so that all the constructions for Theorems 8, 9 and 10 carry through for the graded case, the spectral sequences being the corresponding triple graded MSS’s.

We can also renumber the spectral sequence of Theorem 10, by rewriting

E_{r(p−2)+1}^{(p−1)i+j,−(p−2)i}

as E_{r}^{i,j}. The renumbered spectral sequence has E_{2}-page,

E_{2}^{i,j} = \{ \begin{matrix}S^{i/2}(𝔤^{∗}) ⊗ H^{j}(U𝔤) & if i is even \\ 0 & if i is odd \end{matrix}

with a differential d_{2} : E_{2}^{i,j} → E_{2}^{i+2,j−1}.

Group algebras

In this chapter, we examine the cohomological MSS,

E_{1}^{∗,∗} = H^{∗,∗}(gr(kP)) = Ext_{gr(kP)}^{∗,∗}(k, k) ⟹ H^{∗}(P) = Ext_{kP}^{∗} (k, k),

where P is a finite p-group, k is a field of characteristic p, and kP is filtered by powers of its radical ideal. By the Quillen-Jennings Theorem, gr(kP) is the restricted enveloping algebra of a restricted Lie algebra, so we can use our results on restricted Lie algebra cohomology to prove some results about this spectral sequence.

5.1 The E_{1}-page

5.1.1 The E_{1}-page for Hopf algebras

Given a Hopf algebra Λ over a field k, we say that x ∈ Λ is primitive if

Δ(x) = x ⊗ 1 + 1 ⊗ x

where Δ is the comultiplication. If the characteristic, p, of k is non-zero, it is straightforward to check that the set of primitive elements in Λ, P (Λ), forms a restricted Lie algebra with Lie product and p^{th}-power maps defined by [x, y] = xy − yx and x^{[p]} = x^{p}, respectively. We say that Λ is primitively generated if it is generated as an algebra by P (Λ).

Lemma 23 Suppose Λ is a primitively generated Hopf algebra over a field of characteristic p ≠ 0. Then Λ ≅ V(P (Λ)).

Proof This is Theorem 6.11(2) in [20]. □

Now suppose Λ is a local Hopf algebra over a field k of non-zero characteristic, with augmentation ideal I = ker ϵ. We give Λ the I-adic filtration.

For any x ∈ I,

(1 ⊗ ϵ)(Δ(x) − x ⊗ 1 − 1 ⊗ x) = (ϵ ⊗ 1)(Δ(x) − x ⊗ 1 − 1 ⊗ x) = 0

since (Λ, Δ, ϵ) is a coalgebra. Therefore,

Δ(x) = x ⊗ 1 + 1 ⊗ x + ∑_{i=1}^{t} y_{i} ⊗ z_{i},

for some t, and y_{i}, z_{i} ∈ I. Since gr(Λ) inherits the structure of a graded Hopf algebra from Λ we see that

Δ(\overline{x}) = \overline{x} ⊗ 1 + 1 ⊗ \overline{x}

for all \overline{x} = x + I^{2} ∈ gr(Λ)_{−1}. Hence each such \overline{x} is primitive, and since the set of all such elements generates gr(Λ), we see that gr(Λ) is primitively generated. So, by Lemma 23, gr(Λ) ≅ V(P (Λ)). Hence under suitable conditions (e.g.,when Λ is finite dimensional over k) we have a convergent MSS

H^{∗,∗}(gr(Λ)) ⟹ H^{∗}(Λ)

whose E_{1}-page may be computed as the cohomology of a certain graded restricted Lie algebra. In particular, if Λ is finite dimensional, the the E_{1}-page is a finitely generated graded commutative algebra by Corollaries 3 and 4.

Remark In the finite dimensional case, suppose the MSS stops (i.e., E_{r} = E_{∞} for some r < ∞). Then it follows by a standard argument, cf. [8, Prop.2.1], that the cohomology ring H^{∗}(Λ) is a finitely generated k-algebra.

5.1.2 The E_{1}-page for group algebras

We now consider the special case when Λ = kP, the group algebra of a finite pgroup over a field of characteristic p. Filtering kP by powers of its radical, I, the Quillen-Jennings Theorem identifies the set of primitive elements in gr(kP), to give an explicit realisation of gr(kP) as a restricted enveloping algebra.

We write ℧^{n}(P) for the subgroup ⟨x^{p^{n}} | x ∈ P ⟩ of P and γ_{i}(P) for the i^{th} term in the lower central series of P.

The Jennings series of P, (Γ_{r}(P))_{r≥0}, is the series of subgroups defined inductively by Γ_{1}(P) = P and

Γ_{r}(P) = [Γ_{r−1}(P), P]℧^{1}(Γ_{m}(P))

for r > 1, where m is the least integer such that pm ≥ r. Equivalently, we have

Γ_{r}(P) = ∏_{ip^{k}≥r} ℧^{k}(γ_{i}(P)).

Each Γ_{r}(P)/Γ_{r+1}(P) is an elementary abelian p-group, which we may regard as an F_{p}-vector space. Then

Jen_{∗}(P) = ⨁_{r≥1} k ⊗_{F_{p}} Γ_{r}(P)/Γ_{r+1}(P)

is a restricted Lie algebra, the Lie bracket given by the commutator, and the p^{th} power operation being induced by taking the p^{th} power of elements of P. This is a graded restricted Lie algebra if we let the degree of Γ_{r}(P)/Γ_{r+1}(P) be −r. Defining the r^{th} dimension subgroup of P as

D_{r}(P) = {x ∈ P | x − 1 ∈ I^{r}},

we see that Γ_{r}(P) ⊆ D_{r}(P), so we can define a vector space homomorphism

\begin{matrix}Jen_{∗}(P) & ⟶^{ϕ} & gr(kP) \\ \overline{g} & ⟼ & (g − 1) + I^{r+1} \end{matrix}

which is in fact a homomorphism of restricted Lie algebras, inducing the required isomorphism, [22].

Lemma 24 (Quillen, Jennings) Let P be a finite p-group. Then the dimension subgroups, D_{r}(P), are equal to the Jennings subgroups, Γ_{r}(P), and the map ψ : V(Jen_{∗}(P)) → gr(kP) induced by the map ϕ given above is an isomorphism of graded k-algebras. □

5.2 Identifying inflations

5.2.1 Even characteristic

if P is a finite 2-group, and k is a field of characteristic 2, we can identify an ‘edge’ in the corresponding MSS, and an ‘edge’ homomorphism, even though the spectral sequence does not arise from a double complex. This is analogous to the horizontal edge homomorphism in the LHSSS (Lyndon-Hochschild-Serre Spectral Sequence),

E_{2}^{p,q} = H^{p}(P/Φ(P), H^{q}(Φ(P))) ⟹ H^{p+q}(P),

which is just the usual inflation map,

inf_{P/Φ(P),P} : H^{n}(P/Φ(P)) = E_{2}^{n,0} → E_{∞}^{n,0} ↪ H^{n}(P),

in group cohomology (see [9, 7.2.2]).

Proposition 3 Let P be a finite 2-group, and k a field of characteristic 2. Then in the MSS

H^{p,q}(gr(kP)) ⟹ H^{p+q}(P)

all the elements on the line 2p + q = 0 are permanent cocycles, and we have maps

H^{n}(P/Φ(P)) ≅ F_{−n}H^{n}(P/Φ(P))/F_{−n+1}H^{n}(P/Φ(P))

≅ H^{−n,2n}(gr(k(P/Φ(P)))) ⟶^{inf} H^{−n,2n}(gr(kP)) → E_{∞}^{−n,2n}

≅ F_{−n}H^{n}(P)/F_{−n+1}H^{n}(P) ≅ F_{−n}H^{n}(P) ↪ H^{n}(P)

such that the composite is just inf_{P/Φ(P),P}.

Proof The natural surjection P → P/Φ(P) induces a surjection of filtered algebras, kP → k(P/Φ(P)), a surjection of graded Lie algebras,

𝔤 = Jen_{∗}(P) → 𝔥 = Jen_{∗}(P/Φ(P))

and hence a surjection of graded algebras

gr(kP) → gr(k(P/Φ(P))).

Therefore, by naturality, we get a map of MSS’s

H^{∗,∗}(gr(k(P/Φ(P)))) ⟹ H^{∗}(P/Φ(P))
↓  ↓
H^{∗,∗}(gr(kP))  ⟹  H^{∗}(P)

which we shall call inf. On the E_{1}-page, we get the map

H^{−n,2n}(gr(k(P/Φ(P)))) ⟶^{inf} H^{−n,2n}(gr(kP))

as given in the statement of the Proposition. Also, again using naturality, the induced map on the E_{∞} pages is just gr(inf_{P/Φ(P),P}), where inf_{P/Φ(P)} is the usual inflation map in group cohomology.

Now in the MSS for P/Φ(P),

E_{1} = S(𝔥^{∗}) ≅ E_{∞} ≅ H^{∗}(P/Φ(P)),

by Corollary 2. The generators for this E_{1}-page have filtered/complementary bidegree (−1, 2). Thus we have isomorphisms,

H^{−n,2n}(gr(k(P/Φ(P)))) ≅ F_{−n}H^{n}(P/Φ(P))/F_{−n+1}H^{n}(P/Φ(P))
≅ H^{n}(P/Φ(P)).

We claim that H^{p,q}(gr(kP)) = H^{p,q}(V𝔤) is 0 whenever 2p + q > 0. This follows from examining the degrees of elements in the graded complex S(𝔤^{∗}) whose graded cohomology is just H^{∗,∗}(V𝔤). Since 𝔤_{i} is zero for i ≥ 0, the graded vector space (𝔤^{∗})_{i} is also zero for i ≥ 0 as (𝔤^{∗})_{i} ≅ (𝔤_{i})^{∗}. Hence, (S^{n}(𝔤^{∗}))_{i} = 0 for i > −n, so H^{n,i}(V𝔤) = 0 for i > n. Using filtered/complementary bidegrees and rewriting this in terms of p = i and q = n − i, establishes our claim.

Therefore the E_{1}-page of the MSS for kP is zero in the region 2p + q > 0: so it is clear that elements on the line 2p + q = 0 are permanent cocycles since d_{r} : E_{r}^{−n,2n} → E_{r}^{−n+r,2n−r+1} and

2(−n + r) + (2n − r + 1) = r + 1 > 0,

and F_{r}H^{n}(P) = 0 for r > −n. Hence the remaining maps in the statement of the Proposition are well-defined, and the restriction of inf : E_{∞} → E_{∞} to the line 2p + q = 0 is just the usual inflation map via our identifications. □

5.2.2 Odd characteristic

The corresponding case for an odd prime p is not quite so tidy, but we can prove a similar result.

If P is a p-group, and k has characteristic p, we again have a surjection

gr(kP) → gr(k(P/Φ(P)))

of graded algebras induced by the natural map P → P/Φ(P). In the MSS for P/Φ(P),

E_{1} = Λ(𝔥^{∗}) ⊗ S(𝔥^{∗}) ≅ E_{∞} ≅ H^{∗}(P/Φ(P)),

where the generators of the alternating algebra have homological/filtered bidegree (1, −1) and the polynomial generators have bidegree (2, −p). By naturality, the images under the inflation map, inf, of the elements in H^{∗}(gr(k(P/Φ(P)))) are permanent cocycles in the spectral sequence

H^{∗,∗}(gr(kP)) ⟹ H^{∗}(P).

Moreover, the induced map on E_{∞}-pages is gr(inf_{P/Φ(P),P}). So we can identify the image of this graded inflation map as a subring of the E_{∞}-page of the MSS for P, lying between the lines n + i = 0 and pn + 2i = 0.

Remark For the case above, Anetta Bajer [3] has ‘identified’ a further element in the MSS: namely an element of H^{∗}(P) in the MSS,

H^{∗,∗}(gr(kP)) ⟹ H^{∗}(P),

that restricts to a non-zero element of H^{∗}(Z), where Z is any cyclic (necessarily central) subgroup contained in the last non-trivial subgroup in the Jennings series for P. This enables her to prove that the MSS for a finite p-group stops, following and using the results of Evens [8] and Wilkerson [29].

Group algebra examples

In this chapter, we will calculate some specific examples of the MSS.

6.1 Homogeneous presentations

Perhaps the simplest example of a spectral sequence is when all the differentials are zero, and the spectral sequence stops on the first page (i.e., E_{1} = E_{∞}). As we have seen, this is the case for the group algebra, kP, of a finite abelian p-group over a field of characteristic p filtered by powers of its augmentation ideal, since kP ≅ gr(kP). Here we try to generalize this observation.

Suppose A is a finitely generated algebra over a field k, having a presentation

A = k⟨x_{1}, …, x_{n}⟩/J

where

J = (r_{j} | j ∈ S) ⊲ k⟨x_{1}, …, x_{n}⟩

and 0 ≠ r_{j} ∈ (x_{1}, …, x_{n}) for all j. Let

I = (x_{1}, …, x_{n})/J ⊲ A.

We can write each r_{j} as r_{j} = l_{j} + m_{j} where l_{j} is the smallest non-zero homogeneous component of the polynomial r_{j} and deg l_{j} = t_{j} ≥ 1.

Now consider the morphism of graded algebras

ϕ : k⟨x_{1}, …, x_{n}⟩ → gr_{I}(A)

given by

ϕ(x_{i}) = \overline{x}_{i} = (x_{i} + J) + I^{2} ∈ gr_{I}(A)_{−1}.

(So we are assuming that each x_{i} has graded degree −1.) Then

ϕ(l_{j}(x_{1}, …, x_{n})) = l_{j}(\overline{x}_{1}, …, \overline{x}_{n})
= (l_{j}(x_{1}, …, x_{n}) + m_{j}(x_{1}, …, x_{n}) + J) + I^{t_{j}+1}
= (r_{j}(x_{1}, …, x_{n}) + J) + I^{t_{j}+1}
= (0 + J) + I^{t_{j}+1}
= 0

since m_{j} + J ∈ I^{t_{j}+1}. Hence we have a surjective map of graded algebras,

k⟨x_{1}, …, x_{n}⟩/(l_{j} | j ∈ S) →^{\overline{ϕ}} gr_{I}(A).

Clearly, this is not always an isomorphism: for instance, consider the case when J = (x, x − y^{2}) and A = k⟨x, y⟩/J. However, if each r_{j} is homogeneous (i.e., r_{j} = l_{j}), then by counting the dimensions of each graded component, we see that \overline{ϕ} must be an isomorphism.

Lemma 25 Suppose k is a field and A = k⟨x_{1}, …, x_{n}⟩/J ≠ 0 where J is generated by homogeneous polynomials in the x_{i}’s. Writing I = (x_{1}, …, x_{n})/J, there is an isomorphism of augmented k-algebras

A ≅ gr_{I}(A). □

Corollary 5 If A is a finite dimensional local algebra over a field k with maximal ideal M such that A/M = k and M^{3} = 0. Then A ≅ gr_{M}(A).

Proof It suffices to show that A can be presented homogeneously. So choose elements y_{1}, …, y_{n} ∈ M which form a basis for M/M^{2} and define a surjective algebra homomorphism ϕ : k⟨x_{1}, …, x_{n}⟩ → A by ϕ(x_{i}) = y_{i}. Suppose 0 ≠ r ∈ ker ϕ. Then r has degree at least two, and since ker ϕ contains all polynomials of degree 3 or more, we see that the quadratic part of r must also be in ker ϕ. Thus ker ϕ is generated by homogeneous elements. □

6.2 Dihedral, semi-dihedral and quaternion groups

In this section we shall consider the dihedral, semidihedral and generalised quaternion 2-groups, with the following presentations:

\begin{matrix}D_{2^{m}} = ⟨x, y | x^{2^{m−1}} = y^{2} = 1, x^{y} = x^{−1}⟩ & m ≥ 3, \\ SD_{2^{m}} = ⟨x, y | x^{2^{m−1}} = y^{2} = 1, x^{y} = x^{−1+2^{m−2}}⟩ & m ≥ 4, \\ Q_{2^{m}} = ⟨x, y | x^{2^{m−1}} = 1, x^{2^{m−2}} = y^{2}, x^{y} = x^{−1}⟩ & m ≥ 3. \end{matrix}

Letting P denote any one of these groups we have

Γ_{i}(P) = \{ \begin{matrix}P & i = 1, \\ ⟨x^{2^{r}}⟩ & 2^{r−1} < i ≤ 2^{r}, for 2 ≤ r ≤ m − 2 \\ 1 & i > 2^{m−2}. \end{matrix}

Hence in each case there is an isomorphism of (ordinary) Lie algebras

Jen_{∗}(P) ≅ (k\overline{x} ⊕ k\overline{y}) ⊕ k\overline{x^{2}} ⊕ k\overline{x^{4}} ⊕ … ⊕ k\overline{x^{2^{m−2}}},

where deg \overline{x^{2^{i}}} = −2^{i}, deg \overline{y} = −1, [\overline{x^{2^{i}}}, \overline{x^{2^{j}}}] = 0, [\overline{x}, \overline{y}] = \overline{x^{2}}, and [\overline{x^{2^{i}}}, \overline{y}] = 0 for i ≥ 1.

We now consider the power structure in these Lie algebras. First suppose that we are given two p-mappings of a Lie algebra L in characteristic p which make it restricted. Then to show that they are equal, it suffices to show that they agree on a basis (see [13, p.189]).

It is clear that \overline{x^{2^{i}}}^{[2]} = \overline{x^{2^{i+1}}} in all cases, and that \overline{y}^{[2]} = 0 in all cases except P = Q_{8}. For Q_{8}, we have

\overline{x}^{[2]} = \overline{y}^{[2]} = (\overline{x} + \overline{y})^{[2]} = \overline{x^{2}}.

Choosing any λ ∈ k, λ ≠ 0, we may take the basis \{\overline{x}, λ\overline{x} + \overline{y}\} for Jen_{−1}(Q_{8}). Then

\begin{aligned}(λ\overline{x} + \overline{y})^{[2]} & = & λ^{2}\overline{x}^{[2]} + \overline{y}^{[2]} + [λ\overline{x}, \overline{y}] \\ & = & (1 + λ + λ^{2})\overline{x^{2}}. \end{aligned}

It follows that Jen_{∗}(Q_{8}) ≅ Jen_{∗}(D_{8}) as restricted Lie algebras if and only if k ⊇ F_{4}. For if there exists an ω in k with ω^{3} = 1 and ω ≠ 1, then an isomorphism is determined by the maps \overline{x} ↦ \overline{x} and ω\overline{x} + \overline{y} ↦ \overline{y}. Conversely, if k does not contain ω then the restricted Lie algebras are not isomorphic: this is because Jen_{∗}(D_{8}) is generated by two elements, one whose square is zero, whereas the square of any member of a two element generating set of Jen_{∗}(Q_{8}) cannot be zero. In these remarks we have proved the following result.

Lemma 26 If k is any field of characteristic 2 and m ≥ 4 then

Jen_{∗}(D_{2^{m}}) ≅ Jen_{∗}(SD_{2^{m}}) ≅ Jen_{∗}(Q_{2^{m}})

as restricted Lie algebras. There is an isomorphism

Jen_{∗}(D_{8}) ≅ Jen_{∗}(Q_{8})

if and only if k ⊇ F_{4}. □

Lemma 27 There is an algebra isomorphism

gr(kD_{2^{m}}) ≅ kD_{2^{m}}.

Proof The algebra homomorphism given by X ↦ xy + 1 and Y ↦ y + 1 induces an isomorphism of algebras,

k⟨X, Y ⟩/(X^{2}, Y^{2}, (XY)^{2^{m−2}} + (Y X)^{2^{m−2}}) ≅ kD_{2^{m}}.

The result now follows from Lemma 25. □

Corollary 6 For m ≥ 4 there is an algebra isomorphism

gr(kD_{2^{m}}) ≅ gr(kQ_{2^{m}}) ≅ gr(kSD_{2^{m}}) ≅ kD_{2^{m}};

also gr(kQ_{8}) is isomorphic to gr(kD_{8}) ≅ kD_{8} if k ⊇ F_{4}. □

We therefore can calculate the cohomology of the dihedral groups as the cohomology of the corresponding restricted Lie algebras.

Proposition 4 We have an isomorphism

H^{∗,∗}(gr(kD_{2^{m}})) ≅ k[u, v, w]/(uv)

where cohomological/filtered bidegrees are given by

bideg u = bideg v = (1, −1), bideg w = (2, −2^{m−1}).

Proof Taking the basis \{\overline{x}, \overline{x^{2}}, …, \overline{x^{2^{m−2}}}, \overline{y}\} for 𝔤 = Jen_{∗}(D_{2^{m}}) given above, let

{α_{0}, α_{1}, …, α_{m−2}, β}

be the corresponding dual basis. By Theorem 9, we compute the cohomology as

H(k[α_{0}, …, α_{m−2}, β], d)

where d_{1} is dual to the map d^{1} : Γ^{2}(𝔤) → Γ^{1}(𝔤) given by

d^{1}(γ_{2}(\overline{x^{2^{i}}})) = \{ \begin{matrix}γ_{1}(\overline{x^{2^{i+1}}}) & for 0 ≤ i < m − 2 \\ 0 & for i = m − 2 \end{matrix}

d^{1}(γ_{2}(\overline{y})) = 0
d^{1}(γ_{1}(\overline{x^{2^{i}}})γ_{1}(\overline{x^{2^{j}}})) = 0

d^{1}(γ_{1}(\overline{x^{2^{i}}})γ_{1}(\overline{y})) = \{ \begin{matrix}γ_{1}(\overline{x^{2}}) & for i = 0 \\ 0 & for 0 < i ≤ m − 2. \end{matrix}

Hence

d_{1}(α_{i}) = \{ \begin{matrix}0 & for i = 0 \\ α_{0}^{2} + α_{0}β & for i = 1 \\ α_{i−1}^{2} & for i > 1 \end{matrix}

d_{1}(β) = 0

and so

H(S(𝔤^{∗})) ≅ k[α_{0}, β, α_{m−2}^{2}]/(α_{0}^{2} + α_{0}β).

Now we just put u = α_{0}, v = α_{0} + β and w = α_{m−2}^{2}. □

For the quaternion and semidihedral groups, we do have non-zero differentials in the MSS. In what follows, when dealing with Q_{8}, we shall assume that k ⊇ F_{4}.

The first possibility for a non-zero differential is d_{2^{m−1}−3}, which is determined by d_{2^{m−1}−3}(w). To identify this differential, we can identify the terms on the line 2p+q = 0 for E_{∞} as inflations from P/Φ(P), by Proposition 3. So, for the quaternion groups, we must have (by adjusting w by a non-zero scalar if necessary)

d_{2^{m−1}−3}(w) = u^{3} + v^{3}.

For otherwise, we get the wrong product stucture from inflations in low dimensions. (The calculations of the mod 2 cohomology for the generalised quaternion groups can be found in [27, pp.30–34].) Alternatively, identification of this differential is forced by just knowing that dim_{k} H^{2}(Q_{2^{m}}) = 2 and that the Krull dimension of H^{∗}(Q_{2^{m}}) is one (i.e., it equals the rank of a maximal elementary abelian subgroup, a theorem of Quillen, [24, Corollary 3.3]). Also, we can even compute the differential directly, by constructing the first few terms of a strict filtered-free resolution (cf. Appendix 2).

It is clear that all subsequent differentials are zero (for otherwise, H^{4}(Q_{2^{m}}) = 0, a contradiction) and so

E_{2} = E_{∞} = k[u, v, z]/(uv, u^{3} + v^{3}),

where bideg z = (−2^{m}, 4 + 2^{m}).

Remark The ungrading of E_{∞} is clearly trivial and

H^{∗}(Q_{2^{m}}) ≅ k[u, v, z]/(uv, u^{3} + v^{3}).

When m = 3 and k = F_{2}, this is not true. In particular, we may deduce that gr(F_{2}Q_{8}) is not isomorphic to gr(F_{2}D_{8}).

For the semidihedral groups, similar remarks apply, and we may assume (by adjusting w by a non-zero scalar multiple, and swapping around u and v if necessary) that

d_{2^{m−1}−3}(w) = u^{3}.

(The computations for H^{∗}(SD_{2^{m}}) are given in [10].) Moreover,

E_{2} = E_{∞} = k[u, v, z, ζ]/(uv, u^{3}, uz, z^{2} + ζv^{2})

where bideg z = (−2^{m−1} − 1, 4 + 2^{m−1}) and bideg ζ = (−2^{m}, 4 + 2^{m}).

Alternatively, to prove this without using known calculations, consider the case when k = F_{2}. Then we may prove that dim_{k} H^{2}(SD_{2^{m}}) ≠ 3 either using diagrammatic methods to construct the first three terms of a free resolution for the trivial module (see Appendix 2) or by a standard universal coefficient argument (cf. [9, p.85] or [6]). This forces d_{2^{m−1}−3}(w) ≠ 0. We have three choices for this image, u^{3}, v^{3} and u^{3} + v^{3}. The first two are equivalent, by symmetry in u and v, and the last gives us an E_{2} that is too small (i.e., we get a contradiction to Quillen’s theorem cited above). In this case E_{2} = E_{∞}, as any further non-zero differentials would again give a contradiction to Quillen’s theorem.

6.3 Extraspecial 2-groups

We say P is an extraspecial 2-group if there is an extension

1 → N → P → E → 1

where N = Z(P) = P′ is cyclic of order two and E is an elementary abelian 2-group, of order 2^{m}, say. We have a quadratic form q : E → N defined by taking q(e) to be the square of a preimage ẽ, and the corresponding associated symmetric bilinear form b : E × E → N is given by taking the commutator of preimages.

Taking a basis {e_{1}, …, e_{m}} for E, regarded as an F_{2}-vector space and 1 ≠ n ∈ N, we have

𝔤 = Jen_{∗}(P) = E ⊕ N

where E has degree −1 and N has degree −2. The restricted Lie algebra structure is given by

[e_{i}, e_{j}] = b(e_{i}, e_{j})n,
e_{i}^{[2]} = q(e_{i})n.

Therefore, the map d^{1} : Γ^{2}(𝔤) → Γ^{1}(𝔤) of Theorem 9 is given by

γ_{1}(e_{i})γ_{1}(e_{j}) ↦ b(e_{i}, e_{j})γ_{1}(n),
γ_{1}(e_{i})γ_{1}(n) ↦ 0,
γ_{2}(e_{i}) ↦ q(e_{i})γ_{1}(n),
γ_{2}(n) ↦ 0.

Letting {u_{1}, …, u_{m}} and {v} be dual bases for E and N, respectively, the dual map

d_{1} : S^{1}(𝔤^{∗}) → S^{2}(𝔤^{∗})

is given by

v ↦ ∑_{i=1}^{m} q(e_{i})u_{i}^{2} + ∑_{1≤i<j≤m} u_{i}u_{j},

u_{i} ↦ 0.

Therefore,

H^{∗,∗}(gr(kP)) ≅ k[u_{1}, …, u_{m}, z]/(q(v))

where z = v^{2} and we write q(v) for d_{1}(v) given above. To calculate the rest of the MSS, we need state Quillen’s computation for the cohomology ring of P.

Lemma 28 (Quillen) Let P be an extraspecial 2-group as given above. Then

H^{∗}(P) = k[u_{1}, …, u_{m}]/(q(v), b(v, F (v)), …, b(v, F^{h−1}(v)) ⊗ k[ζ]

where u_{1}, …, u_{m} have cohomological degree one and are inflations from H^{∗}(E). The element ζ has degree 2^{h}, where the integer 2^{h} equals the index in P of a maximal elementary abelian subgroup. It restricts to give a non-zero element in H^{∗}(N). We denote by F the Frobenius map, the k-linear homomorphism that sends u_{i} to u_{i}^{2}. The elements q(v), b(v, F (v)), …, b(v, F^{h−1}(v))) then form a regular sequence in k[u_{1}, …, u_{m}].

Proof See [23] or [4, Vol.2, 5.5]. □

Thus the E_{1}-page of the MSS is ‘like’ the E_{3}-page of the LHSSS,

H^{p}(E) ⊗ H^{q}(N) ⟹ H^{p+q}(P).

Moreover, the result above and Proposition 3 determine all differentials in the MSS. That is, the differential d_{r} on E_{r} is zero unless r = 2^{n} − 1 for n satisfying 1 ≤ n ≤ h − 1. The non-zero differentials are determined by

d_{2^{n}−1}(\overline{u}_{i}) = 0,
d_{2^{n}−1}(z^{2^{n−1}}) = \overline{b(v, F^{n}(v))},

and E_{2^{h}} = E_{∞}.

6.4 The commutative case

We now examine a class of p-groups for which the E_{1}-page of the MSS is easy to compute. For although we have the complex of Theorem 9 to calculate the E_{1}-page in even characteristic, the differentials in the spectral sequence of Theorem 10 seem difficult to calculate in general, and even if we can work out all of these differentials, we still have to ungrade the E_{∞}-page.

First suppose that p is odd. A finite p-group, P, is said to be powerful if P′ ⊆ ℧^{1}(P). For a 2-group this condition is always satisfied and so we define a finite 2-group to be powerful if P′ ⊆ ℧^{2}(P). These definitions and an investigation of powerful p-groups appear in [15]. Examples of powerful p-groups, for p odd, include all metacyclic p-groups.

Lemma 29 Suppose N is a normal subgroup of the nilpotent group G. If there exists an n ≥ 3 with G′ ⊆ γ_{n}(G)N, then G′ ⊆ N.

Proof We have

γ_{n}(G) ⊆ γ_{3}(G) = [G′, G] ⊆ [γ_{n}(G)N, G] = [γ_{n}(G), G][N, G] ⊆ γ_{n+1}(G)N,

so G′ ⊆ γ_{n}(G)N ⊆ γ_{n+1}(G)N. Therefore, G′ ⊆ γ_{m}(G)N for all m. But G is nilpotent, so γ_{r}(G) = 1 for some r, and hence G′ ⊆ N. □

Proposition 5 For any finite p-group P and field k of characteristic p, gr(kP) is commutative if and only if P is powerful.

Proof Let p be an odd prime. Since gr(kP) is generated as a k-algebra by rad kP/rad^{2} kP, gr(kP) is commutative

⟺ [rad kP, rad kP] ⊆ rad^{3} kP
⟺ (1 − x)(1 − y) − (1 − y)(1 − x) ∈ rad^{3} kP, for all x, y ∈ P
⟺ 1 − [x, y] ∈ rad^{3} kP, for all x, y ∈ P
⟺ P′ ⊆ Γ_{3}(P) = γ_{3}(P)℧^{1}(P)
⟺ P′ ⊆ ℧^{1}(P)

by Lemma 24 and Lemma 29.

Now suppose p = 2. Arguing as above we see that gr(kP) is commutative if and only if

P′ ⊆ Γ_{3}(P) = ℧^{2}(P)γ_{3}(P)℧^{1}(P′).

If P is powerful it is clear that this condition is satisfied.

Conversely, this condition on the derived subgroup implies

P′ ⊆ Γ_{3}(P) = ℧^{2}(P)℧^{1}(P′),

by Lemma 29. So, setting N = ℧^{2}(P) and Q = P/N, we have

Q′ = \frac{P′N}{N} ⊆ \frac{℧^{2}(P)℧^{1}(P′)N}{N} = \frac{℧^{1}(P′)N}{N} = ℧^{1}(Q′).

Therefore, Q′ = 1 and so P′ ⊆ N = ℧^{2}(P) as required. □

Lemma 30 (Lubotsky & Mann) If P is powerful then Γ_{r}(P) = ℧^{i}(P) where r > 1 and i satisfies p^{i−1} < r ≤ p^{i}.

Proof This is Theorem 3.1 (for p odd) and Theorem 4.1.3 (for p = 2) of [15]. □

Proposition 6 Suppose P is a powerful p-group. Then there is an abelian p-group, A_{P}, such that gr(kP) is isomorphic as an augmented k-algebra to the group algebra kA_{P}.

Proof Since P is powerful, gr(kP) is commutative and Jen_{∗}(P) is an abelian Lie algebra. Moreover, by Lemma 30,

Jen_{∗}(P) ≅ ⨁_{r≥0} Jen_{−p^{r}}(P),

and the p^{th}-power map Jen_{−p^{r}}(P) → Jen_{−p^{r+1}}(P) is onto for all r. Choose a k basis x_{1}, …, x_{n} of Jen_{−1}(P) as follows. Suppose t is minimal with Jen_{−p^{t+1}}(P) = 0. Pick x_{1}, …, x_{n_{1}} such that x_{1}^{[p^{t}]}, …, x_{n}^{[p^{t}]} form a k-basis for Jen_{−p^{t}}(P); then choose x_{n_{1}+1}, …, x_{n_{2}} such that x_{n_{1}+1}^{[p^{t}]} = … = x_{n_{2}}^{[p^{t}]} = 0 and x_{1}^{[p^{t−1}]}, …, x_{n_{2}}^{[p^{t−1}]} form a kbasis for Jen_{−p^{t−1}}(P). This can be done since for abelian restricted Lie algebras the p^{th}-power map satisfies x^{[p]} + y^{[p]} = (x + y)^{[p]}. Continuing in this way, we obtain a basis x_{1}, …, x_{n} for Jen_{−1}(P). Then if we set

m_{i} = min{j | x_{i}^{[p^{j}]} = 0},

and let A_{P} = C_{p^{m_{1}}} × ⋯ × C_{p^{mn}} we have

Jen_{∗}(P) ≅ Jen_{∗}(A_{P})

as restricted Lie algebras. Hence, we have isomorphisms of augmented k-algebras,

gr(kP) ≅ V(Jen_{∗}(P)) ≅ V(Jen_{∗}(A_{P})) ≅ gr(kA_{P}) ≅ kA_{P} . □

6.5 Groups of order p^{3}

Let p be an odd prime and let k be field of characteristic p. Up to isomorphism, we have two non-abelian groups of order p^{3}, namely,

P_{1} = ⟨x, y | x^{p^{2}} = y^{p} = 1, x^{y} = x^{1+p}⟩,
P_{2} = ⟨x, y, z | x^{p} = y^{p} = z^{p} = [x, z] = [y, z] = 1, [x, y] = z⟩.

The first group, P_{1}, is metacyclic and powerful. Clearly,

gr(kP) ≅ k(C_{p} × C_{p^{2}}),

so the first page of the MSS for P_{1} is given by

E_{1} = k[α_{1}, α_{2}] ⊗ Λ(β_{1}, β_{2})

where filtered/complementary bidegrees are given by

bideg α_{1} = (−p, 2 + p), bideg α_{2} = (−p^{2}, 2 + p^{2}),
bideg β_{1} = bideg β_{2} = (−1, 2).

It is clear that α_{1}, β_{1} and β_{2} represent permanent cocycles in the spectral sequence, by considering there bidegrees. Now dim_{k} H^{2}(P_{1}) = 3, [9, p.85], so we must have

d_{p^{2}−p−1}(α_{2}) = (λβ_{1} + μβ_{2})α_{1}

for some λ, μ ∈ k, not both zero. By replacing (β_{1}, β_{2}) by (λβ_{1} + μβ_{2}, β_{2}) if λ ≠ 0 or (μβ_{2}, β_{1}) otherwise, we do not affect the multiplicative relations of the generators of E_{1}, so we may assume that

d_{p^{2}−p−1}(α_{2}) = β_{1}α_{1}.

It now follows, from the known cohomology of P_{1}, [9, pp.83–86], that E_{p^{2}−p} = E_{∞}, and there is no ungrading necessary, so H^{∗}(P_{1}) ≅ E_{∞}.

The mod p cohomology of P_{2} has been calculated by Leary, [14]. We shall use his results to make some comments about the MSS in this case.

We first compute the E_{1}-page. The Jennings Lie algebra is just

Jen_{∗}(P_{2}) = (k\overline{x} ⊕ k\overline{y}) ⊕ k\overline{z}

where deg \overline{x} = deg \overline{y} = −1, deg \overline{z} = −2, [\overline{x}, \overline{y}] = \overline{z}, and

[\overline{x}, \overline{z}] = [\overline{y}, \overline{z}] = \overline{x}^{[p]} = \overline{y}^{[p]} = \overline{z}^{[p]} = 0.

We have two approaches to calculating the E_{1}-page for this case. Namely, we can use the reindexed trigraded spectral sequence of Theorem 10 or we can use the trigraded LHSSS,

H^{∗,∗}(V(𝔤/𝔫)) ⊗ H^{∗,∗}(V𝔫) ⟹ H^{∗,∗}(V𝔤).

6.5. GROUPS OF ORDER P^{3} 51

In fact, there is a slight advantage for using the method of Theorem 10 here, because it is slightly easier to describe the generators of the resulting cohomology ring. First we compute the ordinary cohomology using Theorem 8:

H^{∗,∗}(U𝔤) = k[α_{1}, α_{2}, β_{1}, β_{2}]/(α_{1}^{2}, α_{2}^{2}, α_{1}α_{2}, β_{1}^{2}, β_{2}^{2}, β_{1}β_{2}, α_{1}β_{1}, α_{2}β_{2}, α_{1}β_{2} − α_{2}β_{1}).

Here the cohomological/graded bidegrees of α_{1} and α_{2} are (1, −1) and the the bidegrees of β_{1} and β_{2} are (2, −3). Thus we get a triple graded spectral sequence

E_{2} = H^{∗}(U𝔤) ⊗ k[u, v, w] ⟹ H^{∗}(V𝔤)

where

deg u = deg v = (2, 0, −p),
deg v = (2, 0, −2p)

and the generators α_{1}, α_{2}, β_{1}, and β_{2} of H^{∗}(U𝔤) have degrees given by

deg α_{1} = deg α_{2} = (0, 1, −1),
deg β_{1} = deg β_{2} = (0, 2, −3).

It is immediate that all the differentials in this spectral sequence are zero, by considering the bidegrees of these elements. Furthermore, there is no problem ‘ungrading’ the elements α_{1}, α_{2}, u, v and w.

Now suppose p > 3. Then it is clear by the trigrading, that we can choose a unique ζ_{i} ∈ H^{2,−3}(V𝔤) which restricts to β_{i} ∈ H^{2,−3}(U𝔤), for i = 1 and 2, and the multiplicative structure on H^{∗}(V𝔤) is formally the same as on the spectral sequence, with the ζ_{i}’s replacing the β_{i}’s.

Modulo the relations given above, the E_{1}-page is generated as

E_{1} = k[α_{1}, α_{2}, ζ_{1}, ζ_{2}, u, v, w].

By virtue of their bidegrees on the E_{1}-page, α_{1}, α_{2}, ζ_{1}, ζ_{2}, u and v are permanent cycles in the spectral sequence. Moreover, we may identify α_{1}, α_{2}, u and v as inflations from P_{2}/Φ(P_{2}) (see 5.2.2). (These elements correspond to y, y′, x and x′ in Leary’s notation, [14, p.48].) The first possibility for a non-zero differential is d_{p−1}. Moreover, this is determined by d_{p−1}(w). In fact, we must have

d_{p−1}(w) = α_{1}v − α_{2}u,

up to a non-zero scalar multiple, so that the relation α_{1}v = α_{2}u holds in E_{∞}, which it must do because of our identifications and Leary’s relation xy′ = x′y in H^{∗}(P_{2}). There are other non-zero differentials in this spectral sequence, but they seem rather complicated to calculate using our methods. For example, we can see that the elements α_{1}w^{2} and α_{2}w^{2} survive to E_{p}, but cannot survive to E_{∞}.

For p = 3, the ungrading of the spectral sequence used to calculate the E_{1}- page of the MSS is non-trivial. Defining ζ_{i} as we did above for p > 3, we see that the bidegrees of ζ_{1}, ζ_{2}, u, v are all (2, −3). In particular, the ζ_{i}’s are not uniquely defined by this method. Yet Massey products of any three elements in H^{1}(V𝔤) are well-defined, and so we can set

ζ_{1} = ⟨α_{1}, α_{1}, α_{2}⟩,
ζ_{2} = ⟨α_{2}, α_{2}, α_{1}⟩,

and then show ζ_{1}ζ_{2} = uv, and so on, analogous calculations within the context of restricted Lie algebras to those of [14, p.47] for p-groups.

Unfortunately, in the corresponding MSS, even though many generators for the E_{1}-page lie on the line 5p + 3q = 0, there are still several non-zero ‘internal’ differentials that appear difficult to work out.

Other examples

In this chapter, we give some examples of the MSS for non-Hopf algebras.

7.1 An example in which F_{0}/F_{−1} is not semisimple

Let k be a field, A = k[u]/(u^{3}), and I = rad A = (u). Consider gr_{I^{2}}(A). This is isomorphic to the graded algebra, Λ = k[v, w]/(v^{2}, w^{2}, vw) where deg v = 0 and deg w = −1. It is easy to see that the minimal resolution of the trivial Λ-module, k, is isomorphic to the corresponding normalised bar resolution, and so H^{∗,∗}(Λ) ≅ k⟨x, y⟩ where bideg x = (0, 1) and bideg y = (−1, 2). Thus we have a 2^{nd}-quadrant MSS.

Now if d_{1}(y) = 0 in this spectral sequence, then 0 = d_{1} = d_{2} = … and so E_{1} = E_{∞}, which is patently not the case. Therefore, by adjusting y by a non-zero scalar if necessary, we may assume d_{1}(y) = x^{2}. (Alternatively, this differential may be calculated directly, using the cobar construction.) In order to calculate E_{2}, we need the following lemma.

Lemma 31 Let A = k⟨x, y⟩ be the free graded algebra on two variables x, y of degree one over a field k. Let d : A → A be the (graded) derivation of degree one on A defined by d(x) = 0 and d(y) = x^{2}. Then

H^{∗}(A, d) = k[w] ⊗_{k} Λ(z) = k[w, z]/(z^{2})

where deg w = 2 and deg z = 1. Indeed, we can take w = (xy + yx) + Im d and z = x + Im d.

Proof By the definition of a derivation on a graded algebra

d(uv) = d(u)v + (−1)^{deg u}ud(v)

for all homogeneous u, v ∈ A. Hence d(xy +yx) = −x^{3} +x^{3} = 0. But Im d ⊆ (x^{2}) ⊲ A, so

(xy + yx)^{n}, x(xy + yx)^{n} ∈ Ker d \ Im d

for n = 0, 1, 2, … Therefore, it suffices to show that any homogeneous element t ∈ Ker d is equal modulo Im d to a scalar multiple of (xy + yx)^{n} or x(xy + yx)^{n} for some n. We prove this claim by induction on m = deg t.

For m = 0, the result is clear since A_{0} = k1 = k(xy + yx)^{0} and d(1) = 0.

For m > 0, we can write t = xp + yq where p, q ∈ A_{m−1}. First, we suppose m = 2n + 1 is odd. Then

0 = d(t) = d(xp + yq) = −xd(p) + x^{2}q − yd(q).

But any identity xf + yg = 0 in A implies f = g = 0 since A is free on x and y. Hence d(q) = 0 and we may assume, by the induction hypothesis, that there exist λ ∈ k and u ∈ A_{m−2} with

q = λ(xy + yx)^{n} + d(u).

(If m = 1, we define A_{−1} = 0, so the above calculation is valid with u = 0.) Therefore,

t = xp + λy(xy + yx)^{n} + yd(u)
= xp′ + λy(xy + yx)^{n} + d(u′)

where p′ = p + xu and u′ = −yu, since yd(u) = x^{2}u − d(yu). But using the fact that d(t) = 0 again, we have

−xd(p′) + λx^{2}(xy + yx)^{n} = 0.

Thus d(p′) = λx(xy + yx)^{n}. But Im d ⊆ Ax^{2}A and x(xy + yx)^{n} ∉ Ax^{2}A, so we are forced to have λ = 0 and d(p′) = 0. Hence, by the induction hypothesis again, there are μ ∈ k and v ∈ A_{m−2} with p′ = μ(xy + yx)^{n} + d(v), so

t = μx(xy + yx)^{n} + xd(v) + d(u′)
= μx(xy + yx)^{n} + d(−xv + u′).

Thus we have proved the induction step for m odd.

Now suppose m = 2n ≠ 0. Again write t = xp + yq. Then d(t) = 0 implies d(q) = 0 as before: so, by induction, there are λ ∈ k and u ∈ A_{m−2} such that

q = λx(xy + yx)^{n−1} + d(u).

Therefore,

t = xp + λyx(xy + yx)^{n−1} + yd(u)
= λ(xy + yx)^{n} + x(−λy(xy + yx)^{n−1} + p + xu) − d(yu)
= λ(xy + yx)^{n} + xp′ + d(u′)

where p′ = −λy(xy + yx)^{n−1} + p + xu and u′ = −yu. Note d(t) = 0 implies d(xp′) = −xd(p′) = 0, and hence d(p′) = 0. So, by induction, there are μ ∈ k and v ∈ A_{m−2} with p′ = μx(xy + yx)^{n−1} + d(v), and hence

t = λ(xy + yx)^{n} + μx^{2}(xy + yx)^{n−1} + xd(v) + d(u′)
= λ(xy + yx)^{n} + d(μy(xy + yx)^{n−1} − xv + u′).

This completes the induction step for m even. □

Therefore, going back to our example, we see that

E_{2} = E_{∞} = k[x, z]/(x^{2})

where x has bidegree (0, 1) and z has bidegree (−1, 3). Since we only have one generator in each homological degree, we get H^{∗}(A) ≅ E_{∞}.

7.2 A counterexample to a Theorem of Grünenfelder

In [11, Theorem 3.4], Grünenfelder claims that H_{∗}(gr(A)) ≅ H_{∗}(A) for any Noetherian local commutative ring, A, filtered by powers of its maximal ideal, M. We give a counterexample to this statement which also serves as a counterexample to [11, Theorems 3.3(ii) and 3.3′].

We look for a counterexample of small dimension over k. Suppose dim_{k} A ≤ 4. Then either M^{3} = 0, in which case gr(A) ≅ A by Corollary 5, or M^{3} ≠ 0, in which case A ≅ k[x]/(x^{4}) and hence gr(A) ≅ A. Therefore, a counterexample must have dimension at least 5.

Proposition 7 Let k be a field and filter the local k-algebra

A = k[x, y]/(xy, x^{2} − y^{3})

by powers of its radical. Then H_{∗}(gr(A)) ≅ H̸_{∗}(A).

Proof Let B = gr(A) ≅ k[x, y]/(xy, x^{2}, y^{4}). Then the trivial B-module k has a minimal (graded-free) resolution that starts

⋯ → B[−2] ⊕ B[−2] ⊕ B[−2] ⊕ B[−4] →^{d_{2}} B[−1] ⊕ B[−1] →^{d_{1}} B[0] → k → 0

where

d_{1}(a, b) = ax + by,
d_{2}(a, b, c, d) = (ax + by, cx + dy^{3}).

Hence, dim_{k} H_{2}(gr(A)) = 4.

However, the trivial A-module k has a minimal resolution that starts

⋯ → A ⊕ A ⊕ A →^{δ_{2}} A ⊕ A →^{δ_{1}} A → k → 0

where

δ_{1}(a, b) = ax + by,
δ_{2}(a, b, c) = (ay + bx, cx − by^{2}).

(This is, of course, not a filtered-free resolution.) Therefore, dim_{k} H_{2}(A) = 3. □

In fact, it is possible to write out a full graded-free gr(A)-resolution for k, and hence compute H^{∗,∗}(gr(A)). This may be done directly or by diagrammatic methods (cf. Appendix 2). We find

H^{∗,∗}(gr(A)) ≅ k⟨u, v, w⟩/(v^{2}, vw)

where the complementary/filtered degrees are

bideg u = bideg v = (−1, 2),

bideg w = (−4, 6).

In the corresponding MSS, d_{1}(u) = d_{1}(v) = 0 and d_{1}(w) = u^{2}v − vu^{2}: the value of d_{1}(w) can be checked by ungrading the graded-free resolution to obtain a filtered-free A-resolution for k. Moreover, a straightforward inductive argument (on the lines of Lemma 31, but simpler) shows d_{1}(z) ≠ 0 for all z ∈ (w). Hence,

E_{2} = ker d_{1}/Im d_{1} = k⟨u, v⟩/(v^{2}, u^{2}v − vu^{2}).

As all the non-zero elements in E_{2} lie on the line 2p + q = 0, E_{2} = E_{∞} = H^{∗}(A).

We can, however, prove H_{∗}(gr(A)) ≅ H_{∗}(A) for commutative Noetherian regular local rings.

Lemma 32 Let A be a commutative regular local ring of dimension d, with maximal ideal M and residue field k. Then

gr_{M}(A) ≅ k[x_{1}, …, x_{d}]

where the x_{i}’s are independent indeterminants of degree −1.

Proof See [2, Theorem 11.22]. □

Proposition 8 Let A be a commutative regular local ring, with maximal ideal M with k = A/M. Then in the cohomological MSS,

E_{1}^{∗,∗} = H^{∗,∗}(gr_{M}(A)) ⟹ H^{∗}(A),

we have E_{1} = E_{∞}.

Proof By Lemma 32,

E_{1}^{∗,∗} ≅ H^{∗,∗}(k[x_{1}, …, x_{d}]) ≅ Λ(y_{1}, …, y_{d}),

where bideg y_{i} = (−1, 2) for all i. As the differential d_{r} has bidegree (r, 1 − r), all the differentials must be zero. □

Remarks Note that the filtration by powers of the maximal ideal for a Noetherian local ring is always separated (i.e., ∩_{i}M^{i} = 0) by Krull’s Theorem, [2, Corollary 10.18].

The Proposition above gives examples of rings for which the MSS stops on the first page without having an isomorphism A ≅ gr(A). Indeed, by Lemma 32, we only have such an isomorphism when A = k.

Appendix 1 - Renumbering and May’s construction

Here we discuss different numbering conventions. Then we relate May’s original construction to our version of the MSS.

Renumbering

We have indicated two ways of numbering the spectral sequence (cf. Theorems 2 and 3). Perhaps the most widely accepted is to number the spectral sequence starting on the E_{1}-page, having the ‘filtration degree’ and ‘complementary degree’ as x and y axes, respectively. This follows the notation given in [16], for instance. Unfortunately, there are no standard conventions in the published literature on the MSS. Serre, [26], constructs the spectral sequence of Theorem 4 with the convention of decreasing filtrations, labels the spectral sequence with ‘filtration degree’ and ‘homological degree’ as axes, and therefore has differentials of the form d^{r} : E_{p,n}^{r} → E_{p+r,n−1}^{r}. In topological applications of the MSS to calculating the cohomology of the Steenrod algebra, it is most common to renumber the triple graded MSS so that the x and y axes are t − n and n respectively, since it is the terms E_{∞}^{n,t} with fixed t − n which are associated with the stable (t − n)-stem in the homotopy groups of spheres (see [25, Ch.1]). Even so, there are several variations on renumbering the axes in this case ([19], [25], [28]).

Another form of renumbering is to change the page-numbering in a formal way which we will define below. In [19], May defines his triple spectral sequence so that the E_{2}-page gives the (co)homology of the associated graded case (i.e., it ‘corresponds’ in some sense to the E_{1}-page in our version). So formally, May’s homological and cohomological spectral sequences are not isomorphic as spectral sequences to those constructed above. However, they are ‘equivalent up to page-numbering’.

Given a filtered chain complex (P, d), having an ascending filtration with deg d = −1, we define a new filtered chain complex (P′, d′), by setting P_{n}′ = P_{n}[n] and d_{n}′ = d_{n}. So F_{i}P′_{n} = F_{i−n}P_{n}, and

d′_{n}(F_{i}P′_{n}) = d_{n}(F_{i−n}P_{n}) ⊆ F_{i−n}P_{n−1} = F_{i−1}P′_{n−1} ⊆ F_{i}P′_{n−1}.

Hence (P′, d′) is indeed a filtered chain complex. Now let {E_{p,q}^{r}} and {E′_{p,q}^{r}} and denote the spectral sequences corresponding to to P and P′, respectively, with filtered/complementary numbering. Then d induces the zero map on E′^{0} = gr(P). Thus E′^{1} = H(E′^{0}, d′^{0}) = E′^{0}. Furthermore,

E′_{p,q}^{1} = \frac{F_{p}P′_{p+q}}{F_{p−1}P′_{p+q}} = \frac{F_{−q}P_{p+q}}{F_{−q−1}P_{p+q}} = E_{−q,p+2q}^{0},

and we therefore have a commutative diagram

E′_{p,q}^{1} = E_{−q,p+2q}^{0}
↓ d′^{1}  ↓ d^{0}
E′_{p−1,q}^{1} = E_{−q,p+2q−1}^{0}

where we write d^{r} and d′^{r} to denote the maps in the spectral sequence induced by d and d′, respectively. Thus we obtain commutative diagrams

E′_{p,q}^{r}  =  E_{−q,p+2q}^{r−1}
↓ d′^{r}  ↓ d^{r−1}
E′_{p−r,q+r−1}^{r} = E_{−q−r+1,p+2q+r−2}^{r−1}

for all n ≥ 1. In such a case we see that the spectral sequences are essentially identical, although the numbering is different and they ‘start’ on different pages. A similar argument applies to chain complexes which have a descending filtration and differential of degree 1.

May’s homological construction

We now relate the original constructions of May, to the triple graded spectral sequence of Theorem 6.

First we note that in May’s setting, his definition of a ‘graded algebra’ is the usual ‘topological’ one, as defined in [20], for instance. We have, however, adopted ‘the algebraic’ convention. So, as a specific example, we would define a graded algebra A to be commutative if xy = yx for all homogeneous elements; but for the ‘topological setting’, A is said to be commutative if xy = (−1)^{mn}yx for all x and y, where m = deg x and n = deg y. It is straightforward to make the sign adjustments between the two conventions as required. In particular, the topological setting introduces extra signs in the bar resolution.

Bearing in mind the comments above, we now work through each stage in May’s construction, as given in [19, pp.126–128].

May starts with a commutative ring of coefficients, K, and assumes that all the K-modules that arise are free of finite type over K. He then takes a filtered-graded augmented K-algebra, A, (having an ascending filtration and augmentation ideal I) and a filtered-graded left A-module, M, such that either

F_{0}A = A, F_{−1}A = I, ⋂_{i} F_{i}A = 0,

F_{0}M = M and ⋂_{i} F_{i}M = 0,

or

F_{−1}A = 0, F_{0}A = K, ⋃_{i} F_{i}A = A,

F_{−1}M = 0, and ⋃_{i} F_{i}M = M,

hold. Note that in both cases his conditions imply that the filtrations are exhaustive and separated.

May then takes the normalised filtered bar resolution for M over A,

\overline{B}(A, M) = A ⊗ T (I[1]) ⊗ M

and defines a filtration on K ⊗_{A} \overline{B}_{n}(A, M) which gives the filtered module

K ⊗_{A} \overline{B}_{n}(A, M)[n],

in our terminology (using our definition of the filtration of the normalised bar resolution given in 1.2.3). It is this filtration that gives rise to his spectral sequence. In other words, he takes the normalised bar resolution, which is a strict filtered-graded free A-resolution, then filters K ⊗_{A} \overline{B}(A, M) in the way we used to obtain Theorem 6, but adjusts the filtration (see the section above) so that the spectral sequence ‘starts’ on the E_{2}-page.

There is one further minor complication concerning numbering. May starts off bigrading associated graded modules in a filtered/complementary way: i.e., he defines

gr(M)_{p,q} = (F_{p}M/F_{p−1}M)_{p+q}.

Since E_{0} = E_{1} for his spectral sequence, he therefore defines E_{p,q,t}^{1} as

E_{p,q,t}^{1} = (\frac{F_{p}X_{p+q}[p + q]}{F_{p−1}X_{p+q}[p + q]})_{t} = (\frac{F_{−q}X_{p+q}}{F_{−q−1}X_{p+q}} )_{t} = (Y_{p+q})_{−q,q+t}

where

X_{n} = K ⊗_{A} \overline{B}_{n}(A, M),
Y_{n} = K ⊗_{gr(A)} \overline{B}_{n}(gr(A), gr(M)),

and the term (Y_{p+q})_{−q,q+t} in May’s notation is just (Y_{p+q})_{−q,t} in our notation. In fact, at this point in his proof, he regrades the spectral sequence to our notation, and therefore obtains a spectral sequence (in our filtered/complementary notation)

E_{p,q,t}^{2} = Tor_{−q,2p+q,t}^{gr(A)}(K, gr(M)) ⟹ Tor_{p+q,t}^{A}(K, M).

This spectral sequence converges, since the filtration conditions specified by May ensure that the filtration on each X_{n,t} is finite.

May’s cohomological construction

May constructs his spectral sequence for cohomology by filtering a cobar complex. We could give a direct argument, in a similar way to the homological case, that this spectral sequence is just a special case of the spectral sequence of Theorem 7 renumbered so that it starts on E_{2}-page. However, it is easier to note that May proves that his cohomological spectral sequence,

E_{2}^{∗,∗,∗} = EXT_{gr(A)}^{∗,∗,∗}(gr(M), K) ⟹ EXT_{A}^{∗,∗}(M, K)

is dual to the homological spectral sequence given above, [19, Theorem 4(iv)]. Also the homological spectral sequence given in Theorem 6 is dual to an appropriate cohomological spectral sequence as defined in Theorem 7, by an extension the proof of Lemma 16. Hence, we see that May’s cohomological construction is just a renumbered case of Theorem 7.

Appendix 2 - Diagrammatic methods

Although we have avoided direct use of diagrammatic methods in our examples, they provide, in somewhat restricted circumstances, a useful tool for calculating filtered-free resolutions for finite-dimensional local algebras over a field k.

The definitions of a module diagram, representation for a module diagram and so on, are given in [5]. To relate the filtered case to the associated graded one, we shall assume that A is a finite-dimensional local algebra over a field k, with maximal ideal I, so that A/I = k. We shall write gr(A) for gr_{I}(A).

Suppose A is the representation of an A-module diagram, D(X, f). Then we define a gr(A)-module diagram, D(gr(X), gr(f)), as follows.

The vertices of D(gr(X), gr(f)) are the same as those of D(X, f). Given an edge e = e(x, y) in D(X, f), suppose

x ∈ Rad^{i} X, x ∉ Rad^{i+1} X,
y ∈ Rad^{j} X, x ∉ Rad^{j+1} X.

Then we define the length of e to be j − i. (See [5, Section 4], for the definition of the radical of a diagram.) Then we define the edges of D(gr(X), gr(f)) to be the edges of D(X, f) of length one. The function gr(f) is defined by assigning the trivial gr(A)-module, k, to each vertex. To each edge e(x, y) in D(gr(X), gr(f)), gr(f) assigns the extension class

θf(e(x, y)) ∈ Ext_{gr(A)}^{1}(k, k)

where θ : Ext_{A}^{1}(k, k) ≅ Ext_{gr(A)}^{1}(k, k) is the isomorphism induced by mapping the extension

0 → k → M → k → 0

to

0 → k → gr_{I}(M) → k → 0.

Lemma 33 Let A be as above. If D(X, f) is an A-module diagram with representation A, then gr(A) is a representation of the gr(A)-module diagram D(gr(X), gr(f)) provided every open set (defined uniquely by its set of vertices) in gr(X) corresponds to a open set in X.

Proof We give A the I-adic filtration. Given a left ideal, M, of A, we define a filtration on M by its inclusion in A, i.e., F_{i}M = M ∩ I^{−i} for i ≤ 0.

Since D(X, f) is an A-module diagram with representation A, we have a function from open sets in X to left ideals of A, which we denote U ↦ M_{U}, satisfying appropriate conditions, [5, Definition 2.2]. Since an open set is determined entirely by its set of vertices, and we have assumed every open set in gr(X) corresponds to an open set in X, we define a representation of D(gr(X), gr(f)) by U ↦ gr(M_{U}). This map satisfies the the four conditions given as the definition of an module diagram representation. □

Examples of group algebras satisfying the conditions of the Lemma include the mod 2 group algebras of dihedral 2-groups (although as noted in Chapter 6, gr(kP) ≅ kP here, and the diagram is unchanged by applying gr). We can write down the appropriate module diagram for kP from the isomorphism given in Lemma 27.

Also for semi-dihedral 2-groups, we have an isomorphism

kSD_{2^{m}} ≅ k⟨X, Y ⟩/I

where

I = ((XY)^{r} − (Y X)^{r}, X^{2} − (Y X)^{r−1}Y − λ(XY)^{r}, Y^{2} − μ(Y X)^{r}, (XY)^{r}X)

over any algebraically closed field k of characteristic 2, for some λ, μ ∈ k, where r = 2^{m−2}, [7, III.13]. Thus we are able to write down the module diagram (cf. [5, Section 15] - here the presentation given for the group algebra of the semi-dihedral group is wrong, but the diagrammatic arguments that follow are correct).

For the generalised quaternion 2-groups, we have a similar isomorphism

kQ_{2^{m}} ≅ k⟨X, Y ⟩/I

where I is generated by

(XY)^{r} − (Y X)^{r}, X^{2} − (Y X)^{r−1}Y − λ(XY)^{r},
Y^{2} − (XY)^{r−1}X − μ(XY)^{r}, (XY)^{r}X, (Y X)^{r}Y

for some λ, μ ∈ k with r = 2^{m−2} and k algebraically closed, [7, III.13]. If m = 3, we can assume that λ = μ = 0, but note that by our calculations of cohomology in Chapter 6, this is not true if k = F_{2}.

We also note that Lemma 33 only gives an idea of what is possible with diagrammatic methods. For example, it can be impossible to ungrade graded-free resolutions, even when R and gr(R) are both representations of module diagrams. On the other hand, it is easy to write down module diagrams for the algebra given in Propostion 7 and its associated graded algebra, and write down appropriate freeresolutions of the trivial module, even though they do not satisfy the conditions of the Lemma, since we do not have the correspondence of open sets in this case.

Bibliography

[1] J.F.Adams, On the non-existence of elements of Hopf invariant one, Ann. Math. (1) 72 (1960), 20–104.

[2] M.F.Atiyah and I.G.Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.

[3] A.M.Bajer, The May spectral sequence for a finite p-group stops, pre-print.

[4] D.J.Benson, Representations and Cohomology, Vols.1 & 2, C.U.P., 1991.

[5] D.J.Benson and J.F.Carlson, Diagrammatic methods for modular representations and cohomology, Comm. in Algebra 15 (1987), 53–121.

[6] T.Diethelm, The mod p cohomology rings of the non-abelian split metacyclic p-groups, Arch. Math. 44 (1985), 29–38.

[7] K.Erdmann, Blocks of Tame Representation Type and Related Algebras, Lecture Notes in Mathematics 1428, Springer-Verlag, 1990.

[8] L.Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224–239.

[9] L.Evens, The Cohomology of Groups, Oxford Mathematical Monographs, OUP, 1991.

[10] L.Evens and S.Priddy, The cohomology of the semi-dihedral group, Conference on Algebraic Topology in honour of Peter Hilton, ed. R. Piccinini and D. Sjerve, Contemp. Math. 37, AMS, 1985.

[11] L.Grünenfelder, On the homology of filtered and graded rings, J. Pure Appl. Algebra 14 (1979), 21–37.

[12] P.J.Hilton and U.Stammbach, A Course in Homological Algebra, GTM 4, Springer-Verlag, 1971.

[13] N.Jacobson, Lie Algebras, Wiley-Interscience, 1962.

[14] I.J.Leary, The cohomology of certain finite groups, Ph.D. Thesis, University of Cambridge, 1990.

[15] A.Lubotsky and A.Mann, Powerful p-groups: I Finite groups, J. Algebra 105 (1987), 484–505.

[16] S.Mac Lane, Homology, Springer-Verlag, 1963.

[17] W.S.Massey, Products in exact couples, Ann. Math. 59 No. 3 (1954), 558–569.

[18] H.Matsumura, Commutative Ring Theory, C.U.P., 1986.

[19] J.P.May, The cohomology of restricted Lie algebras and of Hopf algebras, J. Algebra 3 (1966), 123–146.

[20] J.W.Milnor and J.C.Moore, On the structure of Hopf algebras, Ann. Math. (2) 81 (1965), 211–264.

[21] C.Nǎstǎsescu and F. van Oystaeyen, Graded Ring Theory, North-Holland, 1982.

[22] D.G.Quillen, On the associated graded ring of a group ring, J. Algebra 10 (1968), 411–418.

[23] D.G.Quillen, The mod 2 cohomology rings of extra-special 2-groups and the spinor groups, Math. Ann. 194 (1971), 197–212.

[24] D.G.Quillen, A cohomological criterion for p-nilpotence, J. Pure Appl. Algebra 1 (1971), 361–372.

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

[26] J.-P.Serre, Algèbre Locale - Multiplicités, Lecture Notes in Mathematics 11, Springer-Verlag, Second edition, 1965.

[27] V.P.Snaith, Topological Methods in Galois Representation Theory, J.Wiley & Sons, 1989.

[28] M.C.Tangora, On the cohomology of the Steenrod Algebra, Math. Z. 116 (1970), 18–64.

[29] C.Wilkerson, The cohomology algebras of finite dimensional Hopf algebras, Trans. Amer. Math. Soc. 264 (1981), 137–150.