2013/04/19 by David A. Madore, David Madore, Fabrice Orgogozo · 1 voice · 2 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Homotopy and Cohomology in Algebraic Topology #Polynomial and algebraic computation #math.AG
paper · pdf · doi:10.2140/ant.2015.9.1647
published as Algebra Number Theory 9 (2015) 1647-1739 · In French. v2 has been considerably reworked and expanded. v3 incorporates slight corrections and simplifications and a few additions (notably: computability of the morphism from hyperčech cohomology, graded algebra structure, and a worked out example); submitted for publication
arxiv published 2013/04/19 · arxiv created 2014/07/04 · openalex publication_date 2015/09/22 · arxiv updated 2016/01/20 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Let X be an algebraic scheme over an algebraically closed field and ℓ a prime number invertible on X. According to classical results (due essentially to A. Grothendieck, M. Artin and P. Deligne), the étale cohomology groups Hi(X,ℤ/ℓℤ) are finite-dimensional. Using an ℓ-adic variant of M. Artin's good neighborhoods and elementary results on the cohomology of pro-ℓ groups, we express the cohomology of X as a well controlled colimit of that of toposes constructed on BG where the G are computable finite ℓ-groups. From this, we deduce that the Betti numbers modulo ℓ of X are algorithmically computable (in the sense of Church-Turing). The proof of this fact, along with certain related results, occupies the first part of this paper. This relies on the tools collected in the second part, which deals with computational algebraic geometry. Finally, in the third part, we present a "universal" formalism for computation on the elements of a field.