Derived binomial rings I: Integral Betti cohomology of log schemes

Source abstract

Abstract We introduce and study a derived version of the binomial monad on the unbounded derived category of ‐modules. This monad acts naturally on singular cohomology of any topological space, and does so more efficiently than the more classical monad or the monad given by free ‐ring. We compute all free derived binomial rings on abelian groups concentrated in a single degree, in particular identifying with via a different argument than in [Horel, https://arxiv.org/pdf/2211.02349.pdf , 2022; Toën, Épijournal Géom. Algébrique 4 (2020), https://doi.org/10.46298/epiga.2020.volume4.6060 ]. Using this, we show that the singular cohomology functor induces a fully faithful embedding of the category of connected nilpotent spaces of finite type to the category of derived binomial rings, explicitly describing image of the subcategory of simply connected spaces. We then also define a version of the derived binomial monad on the ‐category of ‐valued sheaves on a sufficiently nice topological space . As an application we give a closed formula for the singular cohomology of an fs log complex analytic space : namely we identify the pushforward for the corresponding Kato–Nakayama space with the free coaugmented derived binomial ring on the 2‐term exponential complex . This gives an extension of Steenbrink's formula and its generalization in [Shuklin, https://arxiv.org/pdf/2209.03720.pdf , 2022] to ‐coefficients.