Derived binomial rings and singular cohomology of K(Z,n) (Dmitry Kubrak)

Orateur : Dmitry Kubrak

Lieu : salle des Séminaires M3

Résumé :

A commutative ring A is called a binomial ring if it is torsion free over Z and is binomially closed: namely binomial coefficients of all elements of A (viewed as elements of A tensor Q) still lie in A. I will talk about the joint work https://arxiv.org/abs/2308.01110 with G.Shuklin and A.Zakharov where we studied a derived version of this notion. In the derived context a (derived) binomial ring structure is really an extra structure and not a property, and using it can really make a difference. A non-trivial example of derived binomial ring is given by Z-valued singular cohomology of a topological space. It turns out that the natural functor X --> C^*_sing(X,Z) to the category derived binomial rings is fully faithful when restricted to a certain natural subcategory of spaces (e.g. simply-connected spaces of finite type). This gives a reasonable "algebraic model" for such a space, which really loses almost no information about it. The key step which makes the above fully-faithfulness statement work is the computation of the free derived binomial ring LBin(Z[-n]): it turns out to match directly the singular cohomology of the Eilenberg-Maclane space K(Z,n).