Binomial rings and homotopy theory

The following statements hold.

1. Let 𝐶 be a chain complex of abelian groups. Assume that, for all prime 𝑝, the chain complex C ⊗ Z L F p is acyclic and that C ⊗ Z L Q is acyclic. Then 𝐶 is acyclic.

2. Let f : C → D be a map of chain complexes of abelian groups. Assume that, for all prime 𝑝, the map f ⊗ Z L F p is a quasi-isomorphism and the map f ⊗ Z L Q is a quasi-isomorphism, then 𝑓 is a quasi-isomorphism.

Proof

In this proof, we drop the 𝕃 superscript from the notation; all tensor products are derived. First, we observe that part (2) of the proposition follows immediately from part (1). Indeed, a map of chain complexes is a quasi-isomorphism if and only if its cofiber is acyclic and taking cofiber commutes with derived tensor product.

We now prove part (1). First, using the short exact sequences

and an obvious inductive argument, we can conclude that C ⊗ Z / p n is acyclic for all 𝑛. It follows that C ⊗ Z / N is acyclic for all integers 𝑁. Since the group Q / Z is a filtered colimit of finite cyclic groups, we deduce that C ⊗ Q / Z is also acyclic. Finally, the short exact sequence 0 → Z → Q → Q / Z → 0 lets us conclude that C = C ⊗ Z is also acyclic. ∎

The cobar construction sends quasi-isomorphisms between simply coconnected coalgebras (i.e. such that C i = 0 for i ≥ − 1 ) to quasi-isomorphisms.

Proof

The cobar construction is given by Ω ⁢ ( C ) = ( T ⁢ ( s − 1 ⁢ C ) , d C + d ′ ) , where d ′ is constructed using the coalgebra structure. There is an obvious decreasing filtration on Ω ⁢ ( C ) given by length of tensors

We claim that if 𝐶 is simply coconnected, the map

is a quasi-isomorphism. For this, it suffices to observe that

1. this limit is in fact a homotopy limit;

2. the obvious map

   H i ⁢ ( holim n Ω ⁢ ( C ) / F n ⁢ ( Ω ⁢ ( C ) ) ) → lim n H i ⁢ ( Ω ⁢ ( C ) / F n ⁢ ( Ω ⁢ ( C ) ) )

   is an isomorphism;

3. for each 𝑖, the composed map

   H i ⁢ ( Ω ⁢ ( C ) ) → lim n H i ⁢ ( Ω ⁢ ( C ) / F n ⁢ ( Ω ⁢ ( C ) ) )

   is an isomorphism.

The first claim follows from the fact that the transition maps are all epimorphisms in this tower. The second claim uses the Milnor short exact sequence and the fact that the lim 1 -term vanishes by the Mittag-Leffler criterion. For the last claim, we observe that F n ⁢ ( Ω ⁢ ( C ) ) is concentrated in homological degrees less than − n by the coconnectedness assumption. It follows that, for any 𝑖, the map H i ⁢ ( Ω ⁢ ( C ) ) → H i ⁢ ( Ω ⁢ ( C ) / F n ⁢ ( Ω ⁢ ( C ) ) ) is an isomorphism for 𝑛 large enough.

Now that we know that the map (A.1) is a quasi-isomorphism, the proposition is easy to prove. Let f : C → D be a quasi-isomorphism between simply coconnected coalgebras. It suffices to consider the following commutative square:

in which both vertical maps are induced by 𝑓. All we have to do is prove that the right vertical map is a quasi-isomorphism. Since we know that both limits are in fact homotopy limits, it suffices to prove that, for each 𝑛, the map Ω ⁢ ( C ) / F n ⁢ ( Ω ⁢ ( C ) ) → Ω ⁢ ( D ) / F n ⁢ ( Ω ⁢ ( D ) ) is a quasi-isomorphism. By an inductive argument, it is enough to show that, for all 𝑛, the map induced by 𝑓, F n − 1 ⁢ ( Ω ⁢ ( C ) ) / F n ⁢ ( Ω ⁢ ( C ) ) → F n − 1 ⁢ ( Ω ⁢ ( C ) ) / F n ⁢ ( Ω ⁢ ( C ) ) , is a quasi-isomorphism, which follows immediately from the fact that 𝑓 is a quasi-isomorphism. ∎

Let 𝕂 denote the ring 𝐙 or a field. Let f : A → A ′ be a map of augmented 0-coconnected dg-algebras over 𝕂. Assume that B ⁢ ( f ) : B ⁢ ( A ) → B ⁢ ( A ′ ) is a quasi-isomorphism. Then 𝑓 is a quasi-isomorphism.

Proof

Thanks to the first proposition, we may reduce the case of 𝐙 to the case of a field, so from now on, we assume that 𝕂 is a field.

Thanks to the second proposition, we know that Ω ⁢ ( B ⁢ ( f ) ) : Ω ⁢ B ⁢ ( A ) → Ω ⁢ B ⁢ ( A ′ ) is a quasi-isomorphism. Finally, the canonical maps Ω ⁢ B ⁢ ( A ) → A and Ω ⁢ B ⁢ ( A ′ ) → A ′ are quasi-isomorphisms by [15, Corollary 2.3.4], which concludes the proof. ∎