Let be a field. The commutative cochain problem over
is to assign (contravariantly) functorially, to every simplicial set
, a commutative (in the graded sense)
-algebra
, which is naturally weakly equivalent to the algebra
of singular cochains (with
-coefficients). We also require that
is a surjection whenever
. Recall that
is an associative algebra, but it is not commutative; the commutativity only appears after one takes cohomology. The commutative cochain problem attempts to find an improvement to
.
If has finite characteristic, this problem cannot be solved, owing to the existence of nontrivial cohomology operations. (The answers at this MathOverflow question are relevant here.) However, there is a solution for
, given by the polynomial de Rham theory. In this post, I will explain this.
First, we need to construct the relevant functor. By general nonsense, the following are equivalent for any cocomplete category :
- Cosimplicial objects in
.
- Colimit-preserving functors
.
This follows because a cosimplicial object defines a functor from standard simplices to , and any simplicial set is canonically a colimit of standard simplices.
Let be the category of (noncommutative) dgas over
. As a result, we can observe the equivalence of the following:
- Simplicial objects in
.
- Contravariant functors
that send colimits to limits.
The same holds when is replaced with the category
of commutative dgas over
. We can explicitly describe the map in the interesting direction as follows. If we have a simplicial dga
and want to associate to it a contravariant functor
, then we let
So, explicitly, this means that for each -simplex of
, we are given an element of
, and that these glue appropriately. This is a dga.
We shall now give an important example of a simplicial . Suppose that
has characteristic zero here.
The polynomial de Rham algebra is the following simplicial
.
- In level
,
is the free graded-commutative algebra over
on generators
in degree zero and
in degree one, quotiented by the relations
and
. (Freeness means that
is a tensor product of a polynomial ring on the
and an exterior algebra on the
.) The differential is given by
and
.In other words, to borrow intuition from differential geometry,
is the space of polynomial differential forms on the
-simplex
.
- Suppose given a map
of ordered sets. The map
is given by restriction to a face: algebraically, it sends
and similarly on the
.
Consequently, there is defined a contravariant functor given by
; we write this as
. So, given
, then
is the commutative dga whose elements can be described as follows: for every
-simplex of
,
, there has to be given a form on
; and these have to fit together compatibly based on maps of standard simplices.
We want to show that the polynomial de Rham functor is a solution to the commutative cochain problem.
Proposition 13 If
is an injection of simplicial sets, then
is surjective.
Proof: This states that any map of simplicial sets can be extended to a map
. In other words, the proposition can be phrased as stating that
(as a simplicial set) is a contractible Kan complex. As a simplicial group, it is a Kan complex. Checking that it is contractible, though, takes some work (it can be done using explicit homotopies).
This fact will be very useful to us, as it will allow us to obtain a model for the relative cohomology algebra. Namely, let be a simplicial dga which is also a contractible Kan complex, with associated functor
(not to be confused with
, though this is a candidate!). Given
, we define the relative cohomology ring
as the kernel of the surjection
. So there is a short exact sequence
Now, we are interested in a natural weak equivalence between the cochain algebra of a simplicial set and the polynomial de Rham algebra
. Since the cochain algebra is not commutative, we can’t produce a map between the two. However, we shall produce a chain of weak equivalences.
Note first that the map sends colimits to limits. In other words, it can be described by a simplicial
. This simplicial dga, which we denote
, is given by
. So
Let us note that the map is surjective for any inclusion
of simplicial sets; this means that
is also a contractible Kan complex. This is why we can define relative cohomology.
Let be the simplicial dga given by
; this is given degreewise as
. There is an associated functor from simplicial sets to
, which we write as
. Because there are natural inclusions of simplicial dgas
, we have natural transformations of functors
We shall need:
Lemma 14 Let
be simplicial dgas, leading to functors
. Suppose given a map of simplicial dgas
, leading to a natural transformation of functors
. Suppose both
are contractible Kan complexes (many sources say “
and
are extendable”) and that
is a quasi-isomorphism for each
. Then
is a quasi-isomorphism for each simplicial set
.
Proof: To see this, we then work by induction on the simplices, and then use a limiting argument.
Let us fix , and suppose that if
is any simplicial set whose nondegenerate simplices are in dimension
, then
is a quasi-isomorphism. Note that this implies that if
, then
is a quasi-isomorphism. Indeed, this follows because
is (by the inductive hypothesis), and
is (by the original hypothesis), so we can apply the five-lemma.
Now let be a simplicial set whose nondegenerate simplices are in dimensions
. We are going to show that
is a quasi-isomorphism. This will imply that the natural map is a quasi-isomorphism for any simplicial set concentrated in finitely many dimensions. There is a set
, and a push-out diagram
for
a simplicial set with nondegenerate simplices in dimensions
. We have then a diagram
Here we have used the extendability. By induction, the rightmost map is a weak equivalence (since has nondegenerate simplices in dimension
). We want to argue that the leftmost map is as well; this will follows because
and similarly for
.
Thus the lemma is true for any with nondegenerate simplices in only finitely many dimensions, by the above inductive argument. Let
now be any simplicial set. We have a filtration
such that is the filtered colimit of the
. Here each
is the
-skeleton functor, and consequently each
is concentrated in a finite number of dimensions. In particular,
is a quasi-isomorphism for each . Note that each of the maps
is surjective, and
is the inverse limit of these; similarly for
. Since these are homotopy inverse limits by the above surjectivity, we find that
is a quasi-isomorphism.
Let us return to the main goal.
Theorem 15 Both the above inclusions in (1) are weak equivalences. In particular,
is a solution to the commutative cochain algebra.
Proof: By the lemma, we have to verify that the three simplicial dgas are “extendable” (i.e. contractible Kan complexes), and the maps are quasi-isomorphisms in each dimension. We have already seen that and
are contractible Kan complexes. We have to show the same for
.
But because and
are contractible and
is a tensor product, we can tensor the contracting homotopies together to see that
is contractible. Finally, we have to check that
and
are quasi-isomorphic to
for each
, but this follows from the Künneth formula: we know (by direct computation) that
is acyclic except in dimension zero, where it has a
; we know by singular cohomology that
is acyclic except in dimension zero, where it has a
; and now we can apply the Künneth formula to
.
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