Connectivity of joins, cohomological quantifier elimination, and an algebraic Toda’s theorem

• Saugata Basu ^[1] ; Deepam Patel ^[1]
  1. [1] Purdue University

     Purdue University

     Township of Wabash, Estados Unidos

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 26, Nº. 5, 2020
• Idioma: inglés
• DOI: 10.1007/s00029-020-00596-0
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• Resumen

  • In this article, we use cohomological techniques to obtain an algebraic version of Toda’s theorem in complexity theory valid over algebraically closed fields of arbitrary characteristic. This result follows from a general ‘connectivity’ result in cohomology. More precisely, given a closed subvariety X⊂Pn over an algebraically closed field k, and denoting by J[p](X)=J(X,J(X,…,J(X,X)⋯) the p-fold iterated join of X with itself, we prove that the restriction homomorphism on (singular or ℓ-adic etale) cohomology Hi(PN)→Hi(J[p](X)), with N=(p+1)(n+1)−1, is an isomorphism for 0≤i