Let {k} be a field. The commutative cochain problem over {k} is to assign (contravariantly) functorially, to every simplicial set {K_\bullet}, a commutative (in the graded sense) {k}-algebra {A(K_\bullet)}, which is naturally weakly equivalent to the algebra {C^*(K_\bullet, k)} of singular cochains (with {k}-coefficients). We also require that {A(K_\bullet) \rightarrow A(L_\bullet)} is a surjection whenever {L_\bullet \subset K_\bullet}. Recall that {C^*(K_\bullet, k)} 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 {C^*(K_\bullet, k)}.

If {k} 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 {k = \mathbb{Q}}, given by the polynomial de Rham theory. In this post, I will explain this. (more…)