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. (more…)