Filtration of cohomology via symmetric semisimplicial spaces

Abstract

In the simplicial theory of hypercoverings we replace the indexing category $$\Delta $$ Δ by the symmetric simplicial category$$\Delta S$$ Δ S and study (a class of) $$\Delta _{\textrm{inj}}S$$ Δ 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 $$\textrm{Sym}\,M$$ Sym M for some space M. For $$\Delta S$$ Δ S -hypercoverings we construct a spectral sequence, somewhat like the Čech-to-derived category spectral sequence. The advantage of working with $$\Delta S$$ Δ S over $$\Delta $$ Δ is that various combinatorial complexities that come with working on $$\Delta $$ Δ are bypassed, giving simpler, unified proof of results like the computation of (in some cases, stable) singular cohomology (with $$\mathbb {Q}$$ Q coefficients) and étale cohomology (with $$\mathbb {Q}_{\ell }$$ Q ℓ coefficients) of the moduli space of degree n maps $$C\rightarrow \mathbb {P}^r$$ C → 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 $$\mathfrak {g}^r_d$$ g d r that is m-very ample for some m etc. In the special case when a $$\Delta _{\textrm{inj}}S$$ Δ inj S -object Xadmits a symmetric semisimplicial filtration byM, we relate these moduli spaces to a certain derived tensor.