Derived binomial rings I: integral Betti cohomology of log schemes

Abstract

We introduce and study a derived version $\mathbf{L}\mathrm{Bin}$ of the binomial monad on the unbounded derived category $\mathcal{D}(\mathbb{Z})$ of $\mathbb{Z}$-modules. This monad acts naturally on singular cohomology of any topological space, and does so more efficiently than the more classical monad $\mathbf{L}{\mathrm{Sym}}_{\mathbb{Z}}$. We compute all free derived binomial rings on abelian groups concentrated in a single degree, in particular identifying $C_{\ast }^{\mathrm{sing}}(K(\mathbb{Z},n),\mathbb{Z})$ with $\mathbf{L}\mathrm{Bin}(\mathbb{Z}[-n])$ via a different argument than in works of Toën and Horel. Using this we show that the singular cohomology functor $C_{\ast }^{\mathrm{sing}}(-,\mathbb{Z})$ induces a fully faithful embedding of the category of connected nilpotent spaces of finite type to the category of derived binomial rings. We then also define a version $\mathbf{L}\mathcal{B}i{n}_X$ of the derived binomial monad on the $\infty$-category of $\mathcal{D}(\mathbb{Z})$-valued sheaves on a sufficiently nice topological space $X$. As an application we give a closed formula for the singular cohomology of an fs log complex analytic space $(X,\mathcal{M})$: namely we identify the pushforward $R{\pi }_{\ast }\underline{\mathbb{Z}}$ for the corresponding Kato-Nakayama space $\pi :X^{\log }\to X$ with the free coaugmented derived binomial ring on the 2-term exponential complex ${\mathcal{O}}_X\to {\mathcal{M}}^{\mathrm{gr}}$. This gives an extension of Steenbrink's formula and its generalization by the second author to $\mathbb{Z}$-coefficients.

68 pages, comments very welcome! this version accepted in the Journal of the London Mathematical Society