BUILDING THE BEILINSON MOTIVE AS A CELL MODULE

BERTRAND GUILLOU

0. Introduction

Let X = ℙ_{k}^{1}−{0, 1, ∞}. It is known ([D],[DG]), at least if k is a number field, how to consider ℚ[π_{1}(X, x)]/I^{n+1} as a mixed Tate motive, for x ∈ X(k) and n ≥ 0 and where I denotes the augmentation ideal. We follow ([KM], §IV) in defining rational mixed Tate motives to be a certain subcategory of the derived category 𝒟_{A} of the Bloch cycle dga A = A(Spec k). Recall that one model for the derived category of a commutative dga is given by the homotopy category of cell (=semi-free) modules.

Actually, it will be more convenient to consider homotopy classes of paths rather than loops. Thus let a, b ∈ X(k). Let _{b}P_{a} denote the set of homotopy classes of paths from a to b and note that it is a π(X, b) − π_{1}(X, a)-bimodule. Deligne and Goncharov ([DG],§3) show that (the dual of) ℚ[_{b}P_{a}]/I^{n+1} is given by

A(X^{n}) → A(a × X^{n−1}) ⊕ A(Δ_{X} × X^{n−2}) ⊕ ⋯ ⊕ A(X^{n−1} × b) → ⋯ → A^{⊕^{n+1}}.

The maps in this complex are given by restriction with the approriate sign. We will be interested in producing an eplicit cellular approximation to this complex.

The first three sections serve as background and fix some notations. In §4, we begin to consider the problem in the case n = 2. Here we see that the Steinberg relation in motivic cohomology allows one to build the cell module. This study is continued in §6, where we consider the case n = 3. Already here we see that a higher Steinberg relation is required to hold–namely, certain Massey products must contain zero. For this reason, we briefly review Massey products in §5. Finally, we state and prove the general answer in § 7; the main result is that the obstruction to building the cell module is given by certain n-fold Massey products in the motivic cohomology of X. Strictly speaking, sections 4 and 6 are not necessary in order to understand the results in §7, though the reader may find it helpful to have these lower dimnesional examples worked out explicitly. At the moment, the question of the vanishing of relevant Massey products in H^{∗,∗}(ℙ^{1} − {0, 1, ∞}) is not addressed.

We are very grateful to Spencer Bloch for bringing this question to our attention and for several helpful discussions. This work owes much to the letter [B1]; indeed, much of section 4 comes directly from that source. We also thank Peter May for much guidance and support.

1. Bloch’s cycle complex

Recall the algebraic standard cosimplicial scheme Δ^{•}: we define Δ^{n} ⊆ 𝔸^{n+1} to be the closed subvariety defined by

Δ^{n} = {(x_{0}, …, x_{n}) ∈ 𝔸^{n+1} | ∑_{i} x_{i} = 1}.

Date: August 3, 2007.

The usual formulae for the face maps and degeneracy maps from topology make Δ^{•} into a cosimplicial scheme. Also, note that Δ^{n} ≅ 𝔸^{n}.

Definition 1.1. For Y a quasi-projective variety over a field k, Bloch’s cycle complex is defined as follows. Let z^{q}(Y, n) be the free abelian group on the codimension q subvarieties of Y × Δ^{n} which meet all faces

Y × Δ^{m} ⊆ Y × Δ^{n}

properly, meaning that the intersection is either empty or of codimension q. For fixed q, the groups z^{q}(Y, •) form a simplicial abelian group and thus a chain complex in the usual way. Writing

N^{p}(Y)(q) = z^{q}(Y, 2q − p),

this makes N^{∗}(Y)(q) into a cochain complex. The cohomology groups of this cochain complex, denoted CH^{q}(Y, p), are called the higher Chow groups of Y of codimension q.

1.1. Cubical variant. For the purpose of defining products, it is more convenient to work with a cubical variant. Let □^{n} = (𝔸^{1})^{n} ≅ 𝔸^{n}. For 1 ≤ j ≤ n and ϵ ∈ {0, 1}, there is an inclusion i_{j,ϵ} : □^{n−1} ↪ □^{n} given by inserting ϵ in the jth coordinate. In addition, for each 1 ≤ j ≤ n − 1, there is a projection π_{j} : □^{n} → □^{n−1} which omits the jth coordinate.

Definition 1.2. Let Ñ^{p}(Y)^{c}(q) be the free abelian group on the codimension q subvarieties of Y × □^{2q−p} which meet all faces

Y × □^{m} ⊆ Y × □^{n}

properly. Let

D^{p}(Y)(q) ⊆ Ñ^{p}(Y)^{c}(q)

be the subgroup of “degenerate cycles”, i.e., the sum of the images of the projections π_{j}. Then we define the cubical Bloch complex to be

N^{p}(Y)^{c}(q) = Ñ^{p}(Y)^{c}(q)/D^{p}(Y)(q).

The following proposition says that the cubical cycle complexes also compute the higher Chow groups.

Proposition 1.3 ([L1], Thm. 4.7; [BK], Prop. 5.1). There is a canonical quasiisomorphism

N^{p}(Y)^{c}(q) ⟶^{∼} N^{p}(Y)(q).

1.2. Products. If W ⊆ X × □^{m} and Z ⊆ Y × □^{n} meet all faces properly, then the product

W × Z ⊆ X × □^{m} × Y × □^{n} ≅ X × Y × □^{m+n}

also meets all faces properly. Thus we get a product

N^{m}(X)^{c}(p) ⊗ N^{n}(Y)^{c}(q) → N^{m+n}(X × Y)^{c}(p + q)

inducing a product

CH^{p}(X, m) ⊗ CH^{q}(Y, n) → CH^{p+q}(X × Y, m + n).

Although the cycle complexes only have contravariant functoriality with respect to flat maps, the higher Chow groups nevertheless have contravariant functoriality with respect to all morphisms when the codomain is smooth ([B2], Thm. 4.1).

Thus, if X is smooth, one can pull back along the diagonal X ↪ X × X to make CH^{∗}(X, ∗) into a bigraded ring. Moreover, this product is graded commutative ([B2], Cor. 5.7) with respect to the homological, or cubical, grading (as opposed to the grading by codimension).

1.3. Alternating cycle complex. When Y = Spec k, the above cubical complex N^{p}(Y)^{c}(q) yields a homotopy-commutative dga. After tensoring with ℚ, one can produce an honest cdga by restricting to alternating cycles, as we explain below.

Note that the symmetric group Σ_{n} acts on □^{n} and therefore on N^{p}(Y)^{c}(q). One can then consider the subgroup of cycles which are alternating with respect to this action, that is, the cycles Z such that

σ(Z) = sgn(σ) · Z

for all σ ∈ Σ_{n}. In fact, one can show ([B3], Lemma 1.1) that the alternating cycles are closed under the differential, so that the alternating cycles form a subcomplex. However, the product of alternating cycles need no longer be alternating. One can rectify this, by projecting onto the subspace of alternating cycles, but it is necessary to pass to ℚ coefficients. Let alt_{n} ∈ ℚ[Σ_{n}] be the element

alt_{n} = \frac{1}{n!} ∑_{σ∈Σ_{n}} sgn (σ) · σ.

Then, letting

A^{p}(Y)(q) ⊆ N^{p}(Y)^{c}(q) ⊗ ℚ

be the subspace of alternating cycles, mutliplication with alt_{2q−p} gives a projection

alt_{2q−p} : N^{p}(Y)^{c}(q) ⊗ ℚ → A^{p}(Y)(q).

We now define a product by the composition

A^{m}(X)(p) ⊗ A^{n}(Y)(q) → N^{m+n}(X × Y)(p + q) ⟶^{alt} A^{m+n}(X × Y)(p + q).

When X = Y = Spec k, this makes A^{∗}(Spec k)(∗) into a commutative dga by construction.

2. Friedlander-Suslin-Voevodsky variant

As we have already mentioned, the higher Chow groups have good functoriality for maps between smooth schemes, but the cycle complexes we have introduced do not. In this section, we discuss cycle complexes of Friedlander-Suslin-Voevodsky with good functoriality.

Definition 2.1. Given smooth schemes X and Y Let z_{equi}(Y, r)(X) denote the free abelian group on the closed and irreducible subvarieties Z ⊆ X × Y which are dominant and equidimensional of relative dimension r over X.

As is discussed in ([MVW], 16.1), for fixed Y and r this defines a presheaf in X.

Definition 2.2. We will write

Z^{p}(Y)(q) = C_{2q−p}(z_{equi}(𝔸^{q}, 0))(Y).

Again, this is the free abelian group on the (codimension q) closed, irreducible subvarieties of Δ^{2q−p} × Y × 𝔸^{q} which are quasifinite and dominant over Δ^{2q−p} × Y. For fixed q, this defines a cochain complex.

Since C_{2q−p}(z_{equi}(𝔸^{q}, 0)) determines a presheaf on the category of smooth kschemes Sm_{k} (and in fact an étale sheaf), we get functoriality of these complexes with respect to all maps between smooth schemes.

Note that any codimension q cycle Z ⊆ Δ^{2q−p} × Y × 𝔸^{q} which is quasifinite over Δ^{2q−p} × Y must meet the faces Δ^{m} × Y × 𝔸^{q} properly, so we have an inclusion

Z^{p}(Y)(q) ↪ N^{p}(Y × 𝔸^{q})(q).

Moreover, homotopy invariance of the higher Chow groups ([B2], Thm. 2.1) says that the pullback along the projection Y × 𝔸^{q} → Y induces a quasi-isomorphism

N^{p}(Y)(q) ⟶^{∼} N^{p}(Y × 𝔸^{q})(q).

Thus, in order to show that our new complexes compute the higher Chow groups, it remains to show that the inclusion Z(Y) ↪ N(Y × 𝔸^{q}) is a quasi-isomorphism. For Y = Spec k, this is given by the following theorem of Suslin.

Theorem 2.3 ([S], Thm. 2.1). The inclusion

Z^{∗}(Spec k)(q) ↪ N^{∗}(𝔸^{q})(q)

is a quasi-isomorphism.

Note that, generalizing what we discussed above, there is an inclusion of groups

C_{2q−p}(z_{equi}(Y × A^{q}, r))(Spec k) ↪ N^{p−2 dim Y +2r}(Y × 𝔸^{q})(q + dim Y − r)

Specializing to r = dim Y, this gives

C_{2q−p}(z_{equi}(Y × A^{q}, dim Y))(Spec k) ↪ N^{p}(Y × 𝔸^{q})(q)

Assuming resolution of singularities, Suslin also proved

Theorem 2.4 ([S], Thm. 3.2). If k satisfies resolution of singularities then for any equidimensional quasiprojective scheme Y, the natural inclusion

C_{2q−p}(z_{equi}(Y × 𝔸^{q}, dim Y))(Spec k) ↪ N^{p}(Y × 𝔸^{q})(q)

is a quasi-isomorphism.

The duality theorem of Friedlander and Voevodsky reads

Theorem 2.5 ([FV], Thm. 7.4). If k satisfies resolution of singularities, Y is a smooth quasiprojective k-scheme, and X is a k-scheme if finite type, then for any r the inclusion

z_{equi}(X, r)(Y) ↪ z_{equi}(Y × X, r + dim Y)(Spec k)

induces a quasi-isomorphism

C_{∗}(z_{equi}(X, r))(Y) ⟶^{∼} C_{∗}(z_{equi}(Y × X, r + dim Y))(Spec k).

Combining the previous two theorems gives the desired quasi-isomorphism

Z^{p}(Y)(q) ⟶^{∼} N^{p}(Y × 𝔸^{q})(q).

2.1. Products. Now that we have our nice functorial cycle complexes, we want to modify them to yield cdga’s.

As above, the first step is to pass from simplicial complexes to cubical complexes. Again ([L2], Lemma 5.26.1), one can show that the cubical complexes Z^{∗}(Y)^{c}(∗) are quasi-isomorphic to the simplicial ones.

At this point, we have functorial dga’s computing the higher Chow groups (assuming resolution of singularities). It only remains to make the dga’s commutative. To clarify, the products we are considering are

Z^{m}(Y)^{c}(p) ⊗ Z^{n}(Y)^{c}(q) → Z^{m+n}(Y × Y)^{c}(p + q) ⟶^{Δ^{∗}} Z^{m+n}(Y)^{c}(p + q).

Recall that Z^{m}(Y)(p) is a subgroup of the group of cycles on □^{2p−m} × Y × 𝔸^{p}. There are thus two sources of noncommutativity: the box coordinate m and the affine coordinate p. As above, we tensor our dga’s with ℚ and consider the subspaces of cycles which are alternating with respect to permutation of the box coordinates. It can be shown ([L2], Lemma 5.26.2) that the result is quasi-isomorphic.

However, this does not yet produce a cdga, since we have the extra affine coordinates. If we now restrict to the cycles which are invariant under permutation of the affine coordinates, then we finally obtain a cdga. Once again, this restriction does not change the quasi-isomorphism type ([L2], Lemma 5.27) (this essentially follows from homotopy invariance of the higher Chow groups). We will denote our resulting presheaf of cdga’s by A^{p}(−)(q).

3. The map k^{∗} → CH^{1}(Spec k, 1)

It is known ([NS], [T]) that

CH^{n}(Spec k, n) ≅ K_{n}^{M}(k),

where the groups on the right are the Milnor K-groups. In particular, this says that

CH^{1}(Spec k, 1) ≅ k^{∗},

the units in the field (this appeared already in [B2]), and that CH^{2}(Spec k, 2) is the quotient of k^{∗} ⊗ k^{∗} by the subgroup generated by elements of the form a ⊗ (1 − a), with a ∈ k − {0, 1} (these relations are known as the Steinberg relations).

We will be interested in producing an explicit map

O^{∗}(Spec k) = k^{∗} → A^{1}(Spec k)(1)

as well as finding explicit cycles in A^{1}(Spec k)(2) realizing the Steinberg relations. This, and much more, was done in [T] for the cubical Bloch complex. In fact, it suffices to work with Z^{∗}(Spec k)^{c}(∗) since we have an explicit projection Z⊗ℚ → A.

The map k^{∗} → N^{1}(Spec k)^{c}(1) is defined by

a ↦ C_{a} := Z(t(a − 1) − a) ⊆ □^{1}.

In other words, C_{a} is merely the point \frac{a}{a−1} ∈ □^{1} = 𝔸^{1}. To get a codimension 1 cycle in □^{1} × 𝔸^{1}, we pullback along the projection □^{1} × 𝔸^{1} → □^{1}, yielding the cycle C_{a} × 𝔸^{1} ⊆ □^{1} × 𝔸^{1}. Note, however, that C_{a} × 𝔸^{1} ∉ Z^{1}(k)^{c}(1) since this cycle is not quasifinite and dominant over □^{1}.

Remark 3.1. When a = 1, we have

C_{a} = Z(t(1 − 1) − 1) = Z(−1) = ∅ ⊆ □^{1},

so that [C_{1}] = 0 ∈ CH^{1}(Spec k).

Recall that the maps

Z^{1}(k)^{c}(1) ↪ N^{1}(𝔸^{1})^{c}(1) ⟵^{π^{∗}} N^{1}(k)^{c}(1)

are quasi-iso’s. We are looking for a cycle in Z^{1}(k)^{c}(1) which maps to the class of C_{a} × 𝔸^{1}.

Definition 3.2. Denoting the coordinate on □^{1} by x and the coordinate on 𝔸^{1} by t, the cycle

Γ_{a} := Z(tx(x − 1) − (x(a − 1) − a)) ⊆ □^{1} × 𝔸^{1}

is quasi-finite and dominant over □^{1}. This defines a map Γ_{(−)} : k^{∗} → Z^{1}(k)^{c}(1).

The cycle Γ_{a} is the graph of the rational function

φ(x) = \frac{x(a − 1) − a}{x(x − 1)} = (a − 1)\frac{x − \frac{a}{a−1}}{x(x − 1)}.

Moreover, we can see that it maps to the class of C_{a} × 𝔸^{1}, as an explicit homotopy C_{a} × 𝔸^{1} ≃ Γ_{a} is given by

Z(tx(x − 1)h − (x(1 − a) − a)) ⊆ □^{2} × 𝔸^{1}.

Here we are considering x and h as coordinates on □^{2}.

It remains to find an explicit cochain which bounds the cochain Γ_{a} · Γ_{1−a}. We do not know of an explicit such cycle.

In fact, one has more generally an isomorphism

CH^{1}(Y, 1) ≅ O^{∗}(Y).

Definition 3.3. We define a map O^{∗}(Y) ⟶^{Γ} Z^{1}(Y)^{c}(1) by

f ↦ Γ_{f} := Z(tx(x − 1) − [x(f(y) − 1) − f(y)]).

As above one can show, by use of an explicit homotopy, that this is compatible with the map O^{∗}(Y) → N^{1}(Y)^{c}(1). We will hereafter write [f] = Γ_{f}, as the notation is less cumbersome.

Though we do not give here an explicit construction of a cochain bounding [f] · [1 − f], let us fix a particular choice of T_{t} ∈ A^{1}(𝔾_{m} − {1})(2) such that d(T_{t}) = [t]·[1−t], where t is the parameter on 𝔾_{m} −{1}. Note since [1−t]·[t] = −[t]·[1−t], we can choose T_{1−t} := −T_{t}. Note furthermore that this choice of universal Totaro cycle T_{t} gives us Totaro cycles in general. For instance, an element a ∈ k − {0, 1}, corresponds to a map Spec k ⟶^{φ_{a}} 𝔾_{m} − {1}, and one can then define

T_{a} := φ^{∗}(T_{t}).

4. Towards the Beilinson Motive, n = 2

We assume the Beilinson-Soulé vanishing conjecture, so that

A^{∗} = A^{∗}(Spec k)(∗)

is a cohomologically connected (Adams-graded) cdga.

Let X = ℙ^{1} − {0, 1, ∞}, and let a ≠ b ∈ X(k). We will be interested in building a cell A-module representing the “Beilinson complex”

\begin{aligned}A(X × X) & ⟶^{β} A(\{a\} × X) ⊕ A(Δ_{X}) ⊕ A(X × \{b\}) \\ & ⟶^{β′} A((a, a)) ⊕ A((a, b)) ⊕ A((b, b)). \end{aligned}

Here the first map is given by the restriction along the inclusions, where restriction onto the diagonal is taken with a minus sign. The second map is given by restriction to a minus restriction to b.

We will write A{n} for A⟨n⟩[n], the complex obtained by shifting both the Adams grading and the homological grading by n. Recall that there is a quasi-isomorphism A ⊕ A{−1} ≃ A(𝔾_{m}). If π : 𝔾_{m} → Spec(k) is the structure morphism and if t is the standard parameter on 𝔾_{m}, this map is given by

(x, y) ↦ π^{∗}(x) + π^{∗}(y) · [t].

Together with the Mayer-Vietoris sequence associated to the covering

(𝔸^{1} − {0}) ∪ (𝔸^{1} − {1}) = 𝔸^{1}

(note that (𝔸^{1} − {0}) ∩ (𝔸^{1} − {1}) = X), this gives a quasi-isomorphism

A ⊕ A{−1} ⊕ A{−1} → A(X).

Explicitly, this is

(x, y, z) ↦ π^{∗}(x) + π^{∗}(y) · [t] + π^{∗}(z) · [1 − t].

Similarly, one gets a quasi-isomorphism

A ⊕ A{−1}^{4} ⊕ A{−2}^{4} → A(X × X)

given explicitly by

(x, y_{1}, …, y_{4}, z_{1}, …, z_{4}) ↦ π^{∗}x + π^{∗}y_{1} · [U] + π^{∗}y_{2} · [1 − U]
+ π^{∗}y_{3} · [V] + π^{∗}y_{4} · [1 − V]
+ π^{∗}z_{1} · [U] · [V] + π^{∗}z_{2} · [1 − U] · [V]
+ π^{∗}z_{3} · [U] · [1 − V] + π^{∗}z_{4} · [1 − U] · [1 − V],

where U is the parameter on the first factor and V is the parameter on the second factor. Keeping the above quasi-isomorphisms in mind, we will label the generators of the various summands by [U], [1 − U] · [V], etc.

Using these quasi-isomorphic free A-modules, we attempt to build a complex

A ⊕ A{−1}^{4} ⊕ A{−2}^{4} ⟶^{α} A^{3} ⊕ A{−1}^{6} ⟶^{α′} A^{3}.

There is no trouble defining α′; this is defined to be restriction to a minus restriction to b as above. Similarly, there is no trouble defining α on the A and A{−1} summands. The trouble comes when trying to define α on the A{−2} summands. Writing W for the parameter on the diagonal Δ_{X}, one wants to set

α([U] · [1 − V]) = [a] · [1 − V] − [W] · [1 − W] + [U] · [1 − b],

but of course we can’t since [W] is not an element of (A{−1})^{2}(2) = A^{1}(1). Instead, we have no choice but to define

α([U] · [1 − V]) = [a] · [1 − V] − [1 − b] · [U]

(we have switched the order of [U] and [1 − b] here, introducing a sign, because [U] denotes the generator of the left A-module A{−1}). We define α similarly on the other A{−2} summands.

Unfortunately, the forced definition of α does not yield a complex. As expected, we have α′ ∘ α = 0 on the A and A{−1} summands. On the A{−2} summands we have

α′ ∘ α([U] · [V]) = ([a]^{2}, 0, −[b]^{2})

and

α′ ∘ α([1 − U] · [1 − V]) = ([1 − a]^{2}, 0, −[1 − b]^{2}).

These both vanish by graded-commutativity (we are working over ℚ). On the other hand,

α′ ∘ α([1 − U] · [V]) = ([1 − a] · [a], 0, −[1 − b] · [b])

and

α′ ∘ α([U] · [1 − V]) = ([a] · [1 − a], 0, −[b] · [1 − b]).

These do not vanish, but they are coboundaries (since CH^{2}(k, 2) ≅ K_{2}^{M}(k)). Choices of bounding cycles allow one to define a contracting homotopy h : α′∘α ≃ 0.

Let us suppose for a moment that we have chosen appropriate bounding cycles and defined a contracting homotopy. This allows us to build a cell module out of our almost-complex. We must now compare this cell module to the original complex. At this point, we have the (not commutative!) solid arrow diagram

[figure: A(X × X); β; A(_{a}X) ⊕ A(Δ_{X}) ⊕ A(X_{b}); β′; A_{a,a} ⊕ A_{a,b} ⊕ A_{b,b}; A ⊕ A{−1}^{4} ⊕ A{−2}^{4}; α; f; H; h; A^{3} ⊕ A{−1}^{6}; α′; f′; A^{3}]

where, for instance, A_{a,a} denotes A(a, a) ≅ A. The right-hand square commutes, but the A{−2} summands cause the left-hand square to only commute up to homotopy. The composites β ∘ f and f′ ∘ α differ by [W] · [1 − W] on the summand A{−2}[U]·[1−V] and by [1−W]·[W] on the summand A{−2}[1−U]·[V]. Defining a commuting homotopy H : f′ ∘α ≃ β ∘f thus amounts to finding a bounding cycle for [W] · [1 − W] and for [1 − W] · [W].

Suppose that we have chosen such a homotopy H. To compare our cell module to the original complex, it suffices to find a homotopy Θ : h ≃ β′ ∘ H. But note that given any H, we can simply define h to be β′ ∘ H. Thus Θ is not necessary, and it remains only to construct H (and therefore h) explicitly.

We know that a choice of H exists, since the Steinberg relation holds, though we do not know of an explicit choice. Nevertheless, any such choice now produces for us a cell module, together with a quasi-isomorphism to Beilinson’s complex. Thus any two choices will produce homotopy equivalent cell modules.

We have seen in this section that to build the Beilinson motive in the case n = 2, it suffices that the relations [t] · [1 − t] = 0 hold in CH^{2}(𝔾_{m} − {1}, 2) (this is the universal case). We will generalize this to the n = 3 case in §6, and we give the general result in §7.

5. Massey products

In this section, we briefly discuss Massey products (see, e.g., [M]), as these will be needed in the following sections.

Throughout this section, C^{∗} will be a dga. Given a cochain c ∈ C^{n}, we let \overline{c} denote (−1)^{n−1}c. This will be convenient later.

Suppose given α_{1}, α_{2}, α_{3} ∈ H^{∗}(C) such that α_{1}α_{2} = 0 and α_{2}α_{3} = 0. Choosing representing cocycles a_{1}, a_{2}, a_{3} ∈ C^{∗}, so that [a_{i}] = α_{i}, this means that there are cochains a_{12}, a_{23} ∈ C^{∗} such that d(a_{12}) = a_{1}a_{2} and d(a_{23}) = a_{2}a_{3}. Actually, it is more convenient to choose a′_{12} and a′_{23} so that d(a′_{12}) = \overline{a_{1}}a_{2} and d(a′_{23}) = \overline{a_{2}}a_{3}. Then note that that cochain

\overline{a′_{12}}a_{3} + \overline{a_{1}}a′_{23}

is in fact a cocycle. We denote the cohomology class it represents by ⟨α_{1}, α_{2}, α_{3}⟩ and refer to it as the triple Massey product.

Unfortunately, the above class is not determined by α_{1}, α_{2}, and α_{3}. The cohomology class is sensitive to the choices of a′_{12} and a′_{23}; adding a cocycle to either of these cochains yields a new cohomology class which has equal right to be called the triple Massey product. Thus the triple Massey product ⟨α_{1}, α_{2}, α_{3}⟩ is not an element of H^{∗}(C) but rather a coset of α_{1}H^{∗}(C) + H^{∗}(C)α_{3} in H^{∗}(C). The subgroup α_{1}H^{∗}(C) + H^{∗}(C)α_{3} ⊆ H^{∗}(C) is called the indeterminacy of this Massey product.

We will need more generally the notion of an n-fold Massey product, which is defined inductively. Suppose given α_{1}, …, α_{n} ∈ H^{∗}(C) with choices of representing cocycles a_{i} ∈ C^{∗}, [a_{i}] = α_{i}. Suppose moreover that we have defined the notion of 3-fold, 4-fold, …, and n − 1-fold Massey products. Finally, suppose that all of the Massey products ⟨α_{i}, …, α_{j}⟩ are defined and contain zero for j − i < n − 1. That these Massey products contain zero means that we have bounding cochains a_{i,...,j} for ⟨α_{i}, …, α_{j}⟩. Then the cochain

∑_{i=1}^{n−1} \overline{a_{1,i}}a_{i+1,n}

is a cocyle, and we denote the resulting cohomology class by ⟨α_{1}, …, α_{n}⟩. Again, there is indeterminacy involved.

The Massey products that will arise later will be iterated Massey products of the elements [t] and [1 − t] in H(A(𝔾_{m} − {1})) ≅ CH^{∗}(𝔾_{m} − {1}, ∗). In fact, we will see that the necessary (and sufficient) condition for building a cellular approximation to the Beilinson complex will be that these iterated Massey products are defined and contain zero. As we have already used the notation T_{t} for a cochain bounding the Massey product ⟨[t], [1 − t]⟩, we will use the notation T_{x_{1},...,x_{n}} for a cochain bounding the Massey product ⟨x_{1}, …, x_{n}⟩, and we will refer to these as generalized Totaro cycles (do not confuse cycle with cocyle here!).

6. The n = 3 case

We now look at the next simplest case, when n = 3. We are interested in obtaining a cellular approximation to the complex

A(X^{3}) ⟶^{β_{1}} A({a} × X^{2}) ⊕ A(Δ_{X} × X) ⊕ A(X × Δ_{X}) ⊕ A(X^{2} × {b})

⟶^{β_{2}} A({a, a} × X) ⊕ A({a} × Δ_{X}) ⊕ A(Δ_{X^{3}})

⊕ A(Δ_{X} × {b}) ⊕ A(X × {b, b}) ⊕ A({a} × X × {b})

⟶^{β_{3}} A(a, a, a) ⊕ A(a, a, b) ⊕ A(a, b, b) ⊕ A(b, b, b)

A Mayer-Vietoris argument again gives us a cellular approximation

A ⊕ A{−1}^{6} ⊕ A{−2}^{12} ⊕ A{−3}^{8} ⟶^{∼} A(X^{3}).

A cellular approximation to the above complex will then consist of maps as in the diagram

(1)  [figure: A ⊕ A{−1}^{6} ⊕ A{−2}^{12} ⊕ A{−3}^{8}; f_{3}; ∼; α_{3}; A(X^{3}); β_{3}; A^{4} ⊕ A{−1}^{16} ⊕ A{−2}^{16}; f_{2}; ∼; α_{2}; A(X^{2})^{4}; β_{2}; A^{6} ⊕ A{−1}^{12}; f_{1}; ∼; α_{1}; A(X)^{6}; β_{1}; A^{4}; A^{4},]

together with appropriate homotopies. Repeated application of Lemma A.1, together with the fact that the bottom rectangle strictly commutes, gives that the required homotopies are

h_{3}^{2} : α_{2}α_{3} ≃ 0,
h_{2}^{2} : 0 ≃ α_{1}α_{2},
h_{3}^{3} : α_{1}h_{3}^{2} + h_{2}^{2}α_{3} ≃ 0,
H_{3}^{1} : f_{2}α_{3} ≃ β_{3}f_{3},
H_{2}^{1} : β_{2}f_{2} ≃ f_{1}α_{2},
H_{3}^{2} : f_{1}h_{3}^{2} + H_{2}^{1}α_{3} − β_{2}H_{3}^{1} ≃ 0,
H_{2}^{2} : β_{1}H_{2}^{1} − h_{2}^{2} ≃ 0,

and

H_{3}^{3} : h_{3}^{3} + H_{2}^{2}α_{3} − β_{1}H_{3}^{2} ≃ 0.

As we remark in §8, it is often convenient to write the differentials on cofibers in matrix form. Here the matrices for the two cell modules would be

A = ( \begin{matrix}d & 0 & 0 & 0 \\ \overline{α_{3}} & d & 0 & 0 \\ \overline{h_{3}^{2}} & \overline{α_{2}} & d & 0 \\ \overline{h_{3}^{3}} & \overline{h_{2}^{2}} & \overline{α_{1}} & d \end{matrix} )  and  B = ( \begin{matrix}d & 0 & 0 & 0 \\ \overline{β_{3}} & d & 0 & 0 \\ 0 & \overline{β_{2}} & d & 0 \\ 0 & 0 & \overline{β_{1}} & d \end{matrix} )

The desired quasi-isomorphism of cofibers would be a matrix

H = ( \begin{matrix}f_{3} & 0 & 0 & 0 \\ H_{3}^{1} & f_{2} & 0 & 0 \\ H_{3}^{2} & H_{2}^{1} & f_{1} & 0 \\ H_{3}^{3} & H_{2}^{2} & H_{1}^{1} & id \end{matrix} )

such that HA = BH.

Replacing the modules on the left column of (1) by A_{3}, …, A_{0} and those on the right column by B_{3}, …, B_{0}, the desired homotopies are represented in the diagram

[figure: A_{3}; f_{3}; α_{3}; h_{3}^{2}; h_{3}^{3}; H_{3}^{1}; H_{3}^{2}; H_{3}^{3}; B_{3}; β_{3}; A_{2}; f_{2}; α_{2}; h_{2}^{2}; H_{2}^{1}; H_{2}^{2}; B_{2}; β_{2}; A_{1}; f_{1}; α_{1}; B_{1}; β_{1}; A_{0}; B_{0}] .

As in the n = 2 case, the fact that the bottom rectangle commutes allows us to simplify the above somewhat. Given a choice of H_{2}^{1}, one can define h_{2}^{2} := β_{1}H_{2}^{1} and H_{2}^{2} = 0. Similarly, given a choice of H_{3}^{2}, one can define h_{3}^{3} := β_{1}H_{3}^{2} and H_{3}^{3} = 0. Thus it suffices to define explicitly homotopies

h_{3}^{2} : α_{2}α_{3} ≃ 0,  H_{3}^{1} : f_{2}α_{3} ≃ β_{3}f_{3},
H_{2}^{1} : β_{2}f_{2} ≃ f_{1}α_{2},  and  H_{3}^{2} : f_{1}h_{3}^{2} + H_{2}^{1}α_{3} − β_{2}H_{3}^{1} ≃ 0.

We now describe explicitly what is needed in order to define these homotopies. Let us begin with h_{3}^{2}. As in §4, the composite α_{2}α_{3} vanishes on the A and A{−1} summands, and so a homotopy is required only on the A{−2} and A{−3} summands. We will write U, V, and W for the parameters on the three copies of X. As in the n = 2 case, α_{2}α_{3} vanishes on the summands of the form [U] · [V] and [1 − V] · [1 − W]. However,

α_{2}α_{3}([U] · [1 − V]) = α_{2}(([a] · [1 − V])_{|aX^{2}} + ([U] · [1 − V])_{|XΔ_{X}} − ([U] · [1 − V])_{|X^{2}b})
= ([a] · [1 − a])_{|aaX} − ([a] · [1 − V])_{|aΔ_{X}} + ([a] · [1 − V])_{|aXb}
+ ([a] · [1 − V])_{|aΔ_{X}} − ([1 − b] · [U])_{|Xbb}
− ([a] · [1 − V])_{|aXb} + ([1 − b] · [U])_{|Xbb}
= ([a] · [1 − a])_{|aaX}

Thus we define

h_{3}^{2}([U] · [1 − V]) := (T_{a})_{|aaX}.

Also, we have

h_{3}^{2}([1 − U] · [V]) = T_{1−a} := −T_{a}

(note that, since a is fixed throughout this discussion, the above choice will not cause compatibility problems), and h_{3}^{2} is defined similarly on the other summands.

To see what is needed on the A{−3} summands, we first note that, for instance

α_{2}α_{3}([U] · [V] · [1 − W]) = −[b] · [1 − b] · [U].

Thus we may define

h_{3}^{2}([U] · [V] · [1 − W]) = T_{b} · [U]

(note that there is no sign since [U] · [V] · [1 − W] is of odd degree). Similarly, one finds that

α_{2}α_{3}([U] · [1 − V] · [W]) = [a] · [1 − a] · [W] + [b] · [1 − b] · [U].

Thus we may define

h_{3}^{2}([U] · [1 − V] · [W]) = −T_{a} · [W] − T_{b} · [U].

The nonzero values of h_{3}^{2} on the A{−2} and A{−3} summands are given by

h_{3}^{2}
[U] · [1 − V]  T_{a},
[1 − U] · [V]  −T_{a},
[V] · [1 − W]  −T_{b},
[1 − V] · [W]  T_{b},
[U] · [V][·[1 − W]  T_{b} · [U],
[U] · [1 − V] · [W]  −T_{a} · [W] − T_{b} · [U],
[U] · [1 − V] · [1 − W] −T_{a} · [1 − W],
[1 − U] · [V] · [W]  T_{a} · [W],
[1 − U] · [V] · [1 − W] T_{a} · [1 − W] + T_{b} · [1 − U],
[1 − U] · [1 − V] · [W] −T_{b} · [1 − U].

We do not go into as much detail for H_{3}^{1} and H_{2}^{1} but merely claim that suitable definitions are

H_{3}^{1}([U] · [1 − V] · [1 − W]) = −(T_{U} · [1 − W])_{|Δ_{X}×X},
H_{3}^{1}([U] · [1 − V] · [W]) = −(T_{U} · [W])_{|Δ_{X}×X} + ([U] · T_{V})_{|X×Δ_{X}} ,
H_{3}^{1}([U] · [V] · [1 − W]) = −([U] · T_{V})_{|X×Δ_{X}} ,

and

H_{2}^{1}([U] · [1 − V]) = −T_{U}.

Finally, let us see what is needed to define H_{3}^{2}. We compute

(f_{1}h_{3}^{2} + H_{2}^{1}α_{3} − β_{2}H_{3}^{1})([U] · [V] · [1 − W]) =
T_{b} · [U] − [a] · T_{U} + [a] · T_{U} − [U] · T_{U} + [U] · T_{b}
= −[U] · T_{U}

Thus H_{3}^{2}([U]·[V]·[1−W]) is a class T_{U,U,1−U} such that d(T_{U,U,1−U}) = [U]·T_{U}. Note that since [U]^{2} = 0, the class [U] · T_{U} is a representative for the Massey product ⟨[U], [U], [1 − U]⟩. Similarly,

(f_{1}h_{3}^{2} + H_{2}^{1}α_{3} − β_{2}H_{3}^{1})([U] · [1 − V] · [W]) = −T_{U} · [U] + [U] · T_{U} = 2 · [U] · T_{U}.

The class H_{3}^{2}([U] · [1 − V] · [W]) is then given by a class −2T_{U,U,1−U}. Note that, since we have T_{1−U} = −T_{U}, we have that −2[U]·T_{U} represents the Massey product ⟨[U], [1 − U], [U]⟩.

We have sketched a proof of the following

Proposition 6.1. In order to build the Beilinson motive in the case n = 3, it suffices that the Massey product ⟨[t], [t], [1 − t]⟩ in

H^{2}(A(𝔾_{m} − {1}))(3) ≅ CH^{3}(𝔾_{m} − {1}, 4)

is defined and contains zero.

7. The general case

From the discussion in the previous sections, it is not difficult to guess at the picture for general n. One starts with the diagram

[figure: B_{n}; β_{n}; B_{n−1}; β_{n−1}; …; B_{1}; β_{1}; B_{0}; A_{n}; f_{n}; α_{n}; A_{n−1}; f_{n−1}; α_{n−1}; …; A_{1}; f_{1}; α_{1}; A_{0}]

where the top row is the Beilinson complex, each A_{i} is a free A-module, and each f_{i} is a quasi-isomorphism. In addition, one needs homotopies h_{i}^{j} for each 3 ≤ i ≤ n and 2 ≤ j ≤ n − 1 and H_{k}^{l} for each 2 ≤ k ≤ n and 1 ≤ l ≤ n − 1. As in the previous cases, the right-hand square commutes on the nose, and no homotopies are required in order to deal with it.

In order to build the h_{i}^{2}’s and the H_{j}^{1}’s, one only needs the Totaro cycles. In order to build the h_{i}^{3}’s and the H_{j}^{2}’s, one needs in addition certain triple Massey products to be defined and to contain zero. In order to define the h_{i}^{4}’s and the H_{j}^{3}’s, one needs certain 4-fold Massey products to be defined and to contain zero. This pattern continues, so that in order to build the h_{1}^{n−2} and the H_{j}^{n−3}’s, one needs certain n − 2-fold Massey products to be defined and to contain zero, and in order to define H_{1}^{n−2} one needs certain n − 1-fold Massey products to be defined and to contain zero. Precise statements are given below.

In order to give precise statements, it will be convenient to introduce some notation. Given a function f on 𝔾_{m} − {1}, we will write Φ_{f} to mean either [f] or [1 − f]. Thus the product Φ_{U}^{1}Φ_{V}^{2} can mean any of the following four cohomology classes:[U][V], [1−U][V], [U][1−V], or [1−U][1−V]. Note that a set of generators of A_{n} is given by the elements of the form Φ_{U_{1}}^{1} ⋯ Φ_{U_{n}}^{n}.

Proposition 7.1. For 2 ≤ k ≤ n, the homotopy h_{n}^{k} : A_{n} → A_{n−k}[k − 1] satisfies

d_{n−k}(h_{n}^{k}Φ_{U_{1}}^{1} ⋯ Φ_{U_{n}}^{n}) =

(−1)^{λ}(⟨Φ_{a}^{1}, …, Φ_{a}^{k}⟩ · Φ_{U_{k+1}}^{k+1} ⋯ Φ_{U_{n}}^{n} − ⟨Φ_{b}^{n−k+1} … Φ_{b}^{n}⟩ · Φ_{U_{1}}^{1} ⋯ Φ_{U_{n−k}}^{n−k}),

where λ = n(k − 1) + \frac{k(k−1)}{2}. Thus h_{n}^{k} can be defined by

h_{n}^{k}(Φ_{U_{1}}^{1} ⋯ Φ_{U_{n}}^{n}) = (−1)^{λ}(T_{Φ_{a}^{1},...,Φ_{a}^{k}} · Φ_{U_{k}}^{k+1} ⋯ Φ_{U_{n}}^{n} − T_{Φ_{b}^{n−k+1},...,Φ_{b}^{n}} · Φ_{U_{1}}^{1} ⋯ Φ_{U_{n−k}}^{n−k}).

Proof. The proof is by double induction on k and n. The base case is k = n = 2, and we have

d_{0}h_{2}^{2}(Φ_{U_{1}}^{1}Φ_{U_{2}}^{2}) = −α_{1}α_{2}(Φ_{U_{1}}^{1}Φ_{U_{2}}^{2}) = −(Φ_{a}^{1}Φ_{a}^{2} − Φ_{b}^{1}Φ_{b}^{2})

as desired. In fact, we similarly find that

d_{n−2}h_{n}^{2}(Φ_{U_{1}}^{1} … Φ_{U_{n}}^{n}) = (−1)^{n−1}(Φ_{a}^{1}Φ_{a}^{2} · Φ_{U_{3}}^{3} ⋯ Φ_{U_{n}}^{n} − Φ_{b}^{n−1}Φ_{b}^{n} · Φ_{U_{1}}^{1} ⋯ Φ_{U_{n−2}}^{n−2}).

For the inductive step, assume the formula holds for all h_{j}^{i} with j < n or i < k. We have

d_{n−k}h_{n}^{k}Φ_{U_{1}}^{1} ⋯ Φ_{U_{n}}^{n} = −(−1)^{n}(h_{n−1}^{k−1}α_{n}Φ_{U_{1}}^{1} ⋯ Φ_{U_{n}}^{n} − h_{n−2}^{k−2}h_{n}^{2}Φ_{U_{1}}^{1} ⋯ Φ_{U_{n}}^{n}+
⋯ + (−1)^{k}α_{n−k+1}h_{n}^{k−1}Φ_{U_{1}}^{1} ⋯ Φ_{U_{n}}^{n}).

By induction, we know all of the terms in this sum. The general term is given by

h_{n−j}^{k−j}h_{n}^{j}(Φ_{U_{1}}^{1} ⋯ Φ_{U_{n}}^{n}) =(−1)^{nk+j+ \frac{k(k−1)}{2}} ·

(T_{Φ_{a}^{1},...,Φ_{a}^{j}}(T_{Φ_{a}^{j+1},...,Φ_{a}^{k}}Φ_{U_{k+1}}^{k+1} ⋯ Φ_{U_{n}}^{n} − T_{Φ_{b}^{n−k+j+1},...,Φ_{b}^{n}}Φ_{U_{j+1}}^{j+1} ⋯ Φ_{U_{n−k+j}}^{n−k+j})

− T_{Φ_{b}^{n−j+1},...,Φ_{b}^{n}}(T_{Φ_{a}^{1},...,Φ_{a}^{k−j}}Φ_{U_{k−j+1}}^{k−j+1} ⋯ Φ_{U_{k−j}}^{k−j} − T_{Φ_{b}^{n−k+1},...,Φ_{b}^{n−j}}Φ_{U_{1}}^{1} ⋯ Φ_{U_{n−k}}^{n−k}))

One checks that the terms involving a product of a Totaro cycle at a with a Totaro cycle b cancel the similar mixed terms appearing in the expression for h_{n−k+j}^{j}h_{n}^{k−j}Φ_{U_{1}}^{1}Φ_{U_{n}}^{n}, using that h_{n−j}^{k−j}h_{n}^{j} appears with sign (−1)^{n+j}. Summing the terms gives the result. ■

One can similarly obtain inductively a formula for the H_{n}^{k}’s. In order to make a precise statement, it is convenient to introduce the following definition.

Definition 7.2. We will say that the length of an n-fold Massey product is n. Similarly, if T_{x_{1},...,x_{n}} is a Totaro cycle bounding an n-fold Massey product, we will say that it is a Totaro cycle of length n. The reduced length of a Massey product or Totaro cycle will be defined to be the length minus 1.

Proposition 7.3. For 1 ≤ k ≤ n − 1, the homotopy H_{n}^{k} : A_{n} → B_{n−k}[k] can be defined by

H_{n}^{k}(Φ_{U_{1}}^{1} ⋯ Φ_{U_{n}}^{n}) = (−1)^{μ}(∑ Φ_{U_{1}}^{1} ⋯ T_{Φ_{U_{i}}^{i} ,...,Φ_{U_{i}}^{j}} Φ_{U_{j+1}}^{j+1} ⋯ T_{Φ_{U_{k}}^{k} ···Φ_{U_{k}}^{l}} ⋯ Φ_{U_{n}}^{n})

where μ = nk + \frac{k(k−1)}{2} and where the sum is over all such products such that the sum of the reduced lengths of the Totaro cycles is k.

The above proposition directly implies the following theorem

Theorem 7.4. In order to build the Beilinson motive for general n, it suffices that the iterated Massey products ⟨Φ_{t}^{1}, …, Φ_{t}^{k}⟩ in

H^{2}(A(𝔾_{m} − {1}))(k) ≅ CH^{k}(𝔾_{m} − {1}, 2k − 2)

are defined and contain zero for all k ≤ n.

8. Trimming the fat

Rather than just obtaining a cellular approximation to the Beilinson complex, one would like to obtain an approximation which is a minimal A-module. We do not go into the details, but the explicit cellular approximations we have obtained lend themselves easily to finding minimal cellular approximations. The resulting picture is

[figure: B_{n}; B_{n−1}; …; B_{1}; B_{0}; A_{n}; ∼; A_{n−1}; ∼; …; A_{1}; ∼; A_{0}; A{−n}^{2^{n}}; A{−(n − 1)}^{2^{n−1}}; …; A{−1}^{2}; A.]

The homotopies h_{i}^{k} descend to the quotients and allow one to build a minimal cell module out of A{−n}^{2^{n}} → ⋯ → A{−1} → A, and the epimorphisms A_{i} ↠ A{−i}^{2^{i}} assemble to give a quasi-isomorphism of cell modules.

Appendix: Review of cofibers

In this section we review the relevant homological algebra, including sign conventions. We are using the sign conventions of [KM], and we will reproduce here some of their discussion from section III.1. Throughout this section, A will denote a (cohomologically graded) commutative dga.

Recall the unit interval A-module I. It is free of rank one in degree −1, with generator denoted [I], and free of rank two in degree 0, with generators denoted [0] and [1]. The only nontrivial differential is d[I] = [0] − [1]. The unit interval I is introduced in order to define homotopies: given dg A-modules M and N and two maps f, g : M → N, a homotopy from f to g is a map h : M ⊗ I → N such that h(m ⊗ [0]) = f(m) and h(m ⊗ [1]) = g(m). Note that since we have chosen to write our unit interval on the right of M, we have the formula

d(m ⊗ [I]) = dm ⊗ [I] + (−1)^{deg m}m ⊗ ([0] − [1]).

This gives the formula

d(h(m ⊗ [I])) = h(dm ⊗ [I]) + (−1)^{deg m}(f(m) − g(m)).

We will follow standard usage and often write h(m) for h(m ⊗ [I].

We define the cone on M to be the pushout

\xymatrix{M ⊗ [1] \ar[r] \ar[d] & 0 \ar[d] \\ M ⊗ I \ar[r] & CM.}

The induced differential is d(m ⊗ [I]) = dm ⊗ [I] + (−1)^{deg m}m ⊗ [0]. Note that there is a canonical map M ≅ M ⊗ [0] ↪ CM. More generally, given any map f : M → N of dg A-modules, we define the cofiber of f by the pushout diagram

\xymatrix{M \ar[r]^{f} \ar[d] & N \ar[d] \\ CM \ar[r] & C(f).}

A typical element of C(f) can be written in the form n + m ⊗ [I], and the induced differential is

d(n + m ⊗ [I]) = dn + (−1)^{deg m}f(m) + dm ⊗ [I].

It is often useful to write the differential in the form of the matrix

( \begin{matrix}d_{M} & 0 \\ \overline{f} & d_{N} \end{matrix} ) ,

where \overline{f}(x) = (−1)^{|x|}f(x). Given the above definitions, it is now straightforward to verify

Lemma A.1. Suppose given dg A-modules and maps between them as shown in the diagram

\xymatrix{W \ar[r]^{α} \ar[d]_{f} \ar@{-->}[dr]^{h} & Y \ar[d]^{g} \\ X \ar[r]^{β} & Z,}

where h is a homotopy h : βf ≃ gα. Then the assignment

x + w ⊗ [I] ↦ β(x) + h(w) + α(w) ⊗ [I]

defines a map of dg A-modules C(f) → C(g).

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