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Transactions of the American Mathematical Society

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On the algebraic and analytic $q$-De Rham complexes attached to $q$-difference equations


Author: Julien Roques
Journal: Trans. Amer. Math. Soc. 375 (2022), 1461-1507
MSC (2020): Primary 39A06, 39A13, 39A45
DOI: https://doi.org/10.1090/tran/8540
Published electronically: October 28, 2021
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Abstract: This paper is concerned with the algebraic and analytic $q$-de Rham complexes attached to linear $q$-difference operators with Laurent polynomial coefficients over the field of complex numbers. There is a natural morphism from the former to the latter complex. Whether or not it is a quasi-isomorphism, i.e., whether or not the induced morphisms on the corresponding cohomology spaces are isomorphisms, is the basic question considered in the present paper. We study this question following three distinct approaches. The first one is based on duality, and leads to a direct connection between the problem considered in the present paper and the convergence of formal series solutions of $q$-difference equations. The second approach is sheaf theoretic, based on growth considerations. The third one relies on the local analytic theory of $q$-difference equations. The paper ends with an extension of our results to variants of the above $q$-de Rham complexes when certain $q$-spirals of poles are allowed. Our study includes the case $\vert q \vert = 1$.


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Additional Information

Julien Roques
Affiliation: Université de Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne, France
MR Author ID: 803167
ORCID: 0000-0002-2450-9085
Email: Julien.Roques@univ-lyon1.fr

Received by editor(s): May 25, 2021
Received by editor(s) in revised form: July 14, 2021
Published electronically: October 28, 2021
Additional Notes: This work was supported by the ANR De rerum natura project, grant ANR-19-CE40-0018 of the French Agence Nationale de la Recherche
Article copyright: © Copyright 2021 by the authors