Filtration of cohomology via symmetric semisimplicial spaces

Abstract

In the simplicial theory of hypercoverings we replace the indexing category \(\Delta \) by the symmetric simplicial category \(\Delta S\) and study (a class of) \(\Delta _{\textrm{inj}}S\)-hypercoverings, which we call spaces admitting symmetric (semi)simplicial filtration—this special class happens to have a structure of a module over a graded commutative monoid of the form \(\textrm{Sym}\,M\) for some space M. For \(\Delta S\)-hypercoverings we construct a spectral sequence, somewhat like the Čech-to-derived category spectral sequence. The advantage of working with \(\Delta S\) over \(\Delta \) is that various combinatorial complexities that come with working on \(\Delta \) are bypassed, giving simpler, unified proof of results like the computation of (in some cases, stable) singular cohomology (with \(\mathbb {Q}\) coefficients) and étale cohomology (with \(\mathbb {Q}_{\ell }\) coefficients) of the moduli space of degree n maps \(C\rightarrow \mathbb {P}^r\) with C a smooth projective curve of genus g, of unordered configuration spaces, of the moduli space of smooth sections of a fixed \(\mathfrak {g}^r_d\) that is m-very ample for some m etc. In the special case when a \(\Delta _{\textrm{inj}}S\)-object X admits a symmetric semisimplicial filtration by M, we relate these moduli spaces to a certain derived tensor.