Derived binomial rings I: integral Betti cohomology of log schemes

TL;DR

This paper introduces a derived binomial monad on the derived category of integers, enabling new insights into singular cohomology of topological spaces and log schemes, extending classical formulas with integral coefficients.

Contribution

It defines a derived binomial monad acting on singular cohomology, computes free derived binomial rings, and applies these to log schemes and topological spaces, extending existing cohomological formulas.

Findings

Identified singular cohomology of Eilenberg–Mac Lane spaces with derived binomial rings.

Embedded the category of certain spaces into derived binomial rings via a fully faithful functor.

Provided a formula for the cohomology of log complex analytic spaces using derived binomial rings.

Abstract

We introduce and study a derived version $\mathbf L\mathrm{Bin}$ of the binomial monad on the unbounded derived category $\mathscr D(\mathbb Z)$ 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 $C_*^{\mathrm{sing}}(K(\mathbb Z,n),\mathbb Z)$ with $\mathbf L\mathrm{Bin}(\mathbb Z[-n])$ via a different argument than in works of To\"en and Horel. Using this we show that the singular cohomology functor $C_*^{\mathrm{sing}}(-,\mathbb Z)$ 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 $\mathbf L \mathcal Bin_X$ of the derived binomial monad on the $\infty$-category of $\mathscr D(\mathbb Z)$-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 $R\pi_*\underline{\mathbb Z}$ for the corresponding Kato-Nakayama space $\pi\colon X^{\mathrm{log}}\rightarrow X$ with the free coaugmented derived binomial ring on the 2-term exponential complex $\mathcal O_X\rightarrow \mathcal M^{\mathrm{gr}}$. This gives an extension of Steenbrink's formula and its generalization by the second author to $\mathbb Z$-coefficients.