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# Oishee Banerjee: Filtration of cohomology via symmetric (semi-)simplicial spaces

## January 12, 2022 at 16:00 – 17:00 CET

Oberseminar Algebra und Geometrie

Talk by Oishee Banerjee (University of Bonn)

*Filtration of cohomology via symmetric (semi-)simplicial spaces*

Abstract: Inspired by Deligne’s use of the simplicial theory of hypercoverings in defining mixed Hodge structures we replace the indexing category ∆ by the symmetric simplicial category ∆S and study (a class of) ∆S-hypercoverings, which we call spaces admitting symmetric (semi)simplicial filtration. For ∆S-hypercoverings we construct a spectral sequence, somewhat like the Cˇech-to-derived category spectral sequence. The advantage of working on ∆S is that all of the combinatorial com- plexities that come with working on ∆ are bypassed, giving simpler, unified proof of known results like the computation of (in some cases, stable) singular cohomol- ogy (with rational coefficients) and étale cohomology (with Q_l coefficients) of the moduli space of degree n maps C to a projective space , C a smooth projective curve of genus g, of unordered configuration spaces, that of the moduli space of smooth sections of a fixed gdr that is m-very ample etc.